Existence of the Nontrivial Solution for a p-Kirchhoff Problem with Critical Growth and Logarithmic Nonlinearity

Abstract

In this paper, we mainly study the p-Kirchhoff type equations with logarithmic nonlinear terms and critical growth: $\left\{\begin{array}{c}-M\left({\displaystyle {\int }_{\mathsf{\Omega }}{\left|\nabla u\right|}^{p}dx}\right){\Delta }_{p}u={\left|u\right|}^{p^{\ast }-2}u+\lambda {\left|u\right|}^{p-2}u-{\left|u\right|}^{p-2}u\ln {\left|u\right|}^2\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}x\in \mathsf{\Omega },\\ u=0 x\in \mathsf{\partial }\mathsf{\Omega },\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\end{array}\right.$ where $\mathsf{\Omega }\subset {ℝ}^N$ is a bounded domain with a smooth boundary, $2<p<p^{\ast }<N$, and both $p$ and $N$ are positive integers. By using the Nehari manifold and the Mountain Pass Theorem without the Palais-Smale compactness condition, it was proved that the equation had at least one nontrivial solution under appropriate conditions. It addresses the challenges posed by the critical term, the Kirchhoff nonlocal term and the logarithmic nonlinear term. Additionally, it extends partial results of the Brézis–Nirenberg problem with logarithmic perturbation from p = 2 to more general p-Kirchhoff type problems.

1. Introduction

In this paper, we study the p-Kirchhoff problem

$$\left\{\begin{array}{l}-M\left({\displaystyle {\int }_{\mathsf{\Omega }}{\left|\nabla u\right|}^{p}dx}\right){\Delta }_{p}u={\left|u\right|}^{p^{\ast }-2}u+\lambda u^{p-2}u-{\left|u\right|}^{p-2}u\ln {\left|u\right|}^2\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em} x\in \mathsf{\Omega },\\ u=0 x\in \mathsf{\partial }\mathsf{\Omega },\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\end{array}\right.$$

(1)

with a critical exponent and nonlinear logarithmic terms, where $\mathsf{\Omega }\subset {ℝ}^N$ is the bounded smooth region, $2<p<p^{\ast }<N$, and both $p$ and $N$ are positive integers, ${\Delta }_{p}u=div({\left|\nabla u\right|}^{p-2}\nabla u)$ is the p-Laplace operator and $p<N,p^{\ast }={\displaystyle \frac{pN}{N-p}}$ is the Sobolev embedded critical exponent. $M\left(t\right)$:$\left[0,+\infty \right)\to ℝ$ is a continuously increasing function, and $M\left(t\right)$ satisfies the following assumptions:

M₀: There exists $m_0>0$ and $m_1>0$ such that $m_0\le M(t)\le m_1,\forall t\ge 0$;

M₁: There exists $\theta \in \left(p,p^{\ast }\right)$ and $\delta <{\displaystyle \frac{\theta }{p}}$ such that $m_1<\delta m_0<{\displaystyle \frac{\theta }{p}}{m}_0$.

As natural science advances and real-world demands evolve, a fascinating variant of the Kirchhoff equation has emerged, stemming from the D’Alembert wave equation for a freely vibrating elastic string. The significance of wave equations with logarithmic sources spans nuclear physics, geophysics, and optics. Delving into the existence and properties of solutions to these equations is not only highly relevant but also holds immense potential for meaningful discoveries in these fields.

Obviously, problem (1) is closely related to the following p-Laplace equation:

$$\left\{\begin{array}{c}-div\left({\left|\nabla u\right|}^{p-2}u\right)=\lambda {\left|u\right|}^{q-2}u\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x\in \mathsf{\Omega },\\ \hspace{1em}\hspace{1em}\hspace{1em} u\ge 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x\in \mathsf{\Omega },\\ \hspace{1em}\hspace{1em}\hspace{1em} u=0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x\in \mathsf{\partial }\mathsf{\Omega }.\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\end{array}\right.$$

(2)

where $1<p<N$, $1<p\le q$. In 1987, García Azorero and Peral Alonso in [1] first proved the existence and multiplicity of the solution of Equation (2) by using the Minimax Principle under the appropriate assumption of $p$, $q$, $\lambda$. Specifically, utilizing the solution of Equation (2) and the Concentration Compactness Principle, the following equations with critical terms were also studied in [1]:

$$\left\{\begin{array}{c}-div\left({\left|\nabla u\right|}^{p-2}u\right)=\lambda {\left|u\right|}^{q-2}u+\mu {\left|u\right|}^{p^{\ast }-2}u\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{1em}x\in \mathsf{\Omega },\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} u\ge 0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x\in \mathsf{\Omega },\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} u=0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x\in \mathsf{\partial }\mathsf{\Omega },\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\end{array}\right.$$

(3)

where $p\le q<p^{\ast }.$ and the following research results have been obtained: when $N\ge p^2$,$p=q$ and $\mu \equiv 1$, $0<\lambda <{\lambda }_1$, where ${\lambda }_1$ is the first eigenvalue of the problem, Equation (3) has a solution; when $N>p^2,p<q<p^{\ast }$, for all $\lambda >0$, Equation (3) has a solution. However, when $p<N<p^2$, the problem becomes challenging. In general domains (non-spherical), the issue of studying $p\ne 2$ remains open. In 1991, García Azorero and Peral Alonso addressed Equation (3) in [2] using Critical Point Theory. They demonstrated not only the existence of solutions but also their multiplicity. Comparing their work with others reveals differences not only in methods but also in the parameter ranges of the solutions. Further results on Equation (3) can be found in references [3,4].

In addition, a large number of scholars have studied the Kirchhoff equation. Fruitful research results have been obtained. For example, Figueiredo in [5] studied the equation:

$$\left\{\begin{array}{c}-M\left({\displaystyle {\int }_{\mathsf{\Omega }}{\left|\nabla u\right|}^{2}dx}\right)\Delta u={\left|u\right|}^{2^{\ast }-2}u+\lambda f(x,u)\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}x\in \mathsf{\Omega },\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}u=0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x\in \mathsf{\partial }\mathsf{\Omega },\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\end{array}\right.$$

(4)

where $\mathsf{\Omega }\subset {ℝ}^N$ is the bounded smooth region, $N\ge 3$, $\lambda$ is a positive parameter, $2^{\ast }={\displaystyle \frac{2N}{N-2}},$ $M:{ℝ}^{+}\to {ℝ}^{+}$ and $f:\mathsf{\Omega }\times ℝ\to ℝ$ are both continuous functions that satisfy the following assumption:

$\left({\mathrm{F}}_0\right)$ $f$ satisfies the Ambrosetti-Rabinowitz superlinear condition, that is, for some $2<\theta <2^{\ast }$,

$$0<\theta F(x,t)=\theta {\displaystyle {\int }_0^{t}f(x,s)ds\le t}f(x,t),\forall x\in \mathsf{\Omega },t>0,\mathrm{where}F(x,t)={\displaystyle {\int }_0^{t}f(x,s)}ds.$$

The existence of the positive solution of Equation (4) and the asymptotic behavior of the solution are obtained by using the variational method, appropriate truncation parameters and prior estimation method under the assumption condition. Furthermore, Chen et al. in [6] studied the following equation:

$$\left\{\begin{array}{c}-M\left({\displaystyle {\int }_{\mathsf{\Omega }}{\left|\nabla u\right|}^{2}dx}\right)\Delta u=\lambda f(x){\left|u\right|}^{q-2}u+\hspace{0.33em}g(x){\left|u\right|}^{p-2}u\hspace{0.33em}\hspace{0.33em}x\in \mathsf{\Omega },\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}u=0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x\in \mathsf{\partial }\mathsf{\Omega }\hspace{0.33em},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\end{array}\right.$$

(5)

where $1<q<2<p<2^{\ast }$ (if $N\ge 3$, then $2^{\ast }={\displaystyle \frac{2N}{N-2}}$; if $N=1,2$, then $2^{\ast }=\infty$), $M(s)=as+b$ and $a,b,\lambda >0$, $f,g\in C(\overline{\mathsf{\Omega }})$ are the weight functions. Using the Nehari manifold and fiber mapping, the authors investigated three cases: $4>p,p=4$ and $p<4$, each under different parameter ranges. They successfully established the existence and multiplicity of positive solutions for problem (5). Detailed results are available in [6].

With the emergence of p-Kirchhoff operators in numerous equations, many scholars have extensively researched a class of p-Kirchhoff equations, yielding significant findings. For instance, Li and Niu in [7] explored nonlinear p-Kirchhoff type equations with critical growth:

$$\left\{\begin{array}{c}-\left(a+b{\displaystyle {\int }_{{\mathbb{R}}^N}{\left|\nabla u\right|}^{p}dx}\right){\Delta }_{p}u={\left|u\right|}^{p^{\ast }-2}u+\hspace{0.33em}\mu f(x){\left|u\right|}^{p-2}u\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}x\in {\mathbb{R}}^N,\\ u \in D^{1,p}({\mathbb{R}}^N),\end{array}\right.$$

(6)

where $a\ge 0,b\ge 0,1<p<N,1<q<p,f\in L^{\frac{p·}{p^{·}-q}}({ℝ}^N)\backslash \left\{0\right\}$, and $f$ is a nonnegative function. Utilizing Ekeland’s variational principle, in conjunction with the Concentration Compactness Principle and the Mountain Pass Theorem, the author has demonstrated the existence of a solution to the equation under appropriate conditions in the whole space ${ℝ}^N$. Similarly, Chu and Sun in [8] studied the following Kirchhoff Dirichlet boundary value problem:

$$\left\{\begin{array}{c}-M\left({\displaystyle {\int }_{\mathsf{\Omega }}{\left|\nabla u\right|}^{p}dx}\right){\Delta }_{p}u=\hspace{0.33em}\lambda u^{s-1}+u^{p^{\ast }-1}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}in \mathsf{\Omega },\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}u =0 on \mathsf{\partial }\mathsf{\Omega },\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\end{array}\right.$$

(7)

where $\lambda >0,1<s<p<p^{\ast },p<N,M(t)=a+b{t}^m$ and $a,b>0,p(m+1)<p^{\ast }.$ Through the application of the Concentration Compactness Principle, Ekeland’s variational principle and the strong maximum principle, the author proved the existence of at least one positive solution to Equation (7). Furthermore, the author employed the Mountain Pass Theorem to establish the existence of a second positive solution to this equation. For more research on p-Kirchhoff equations, see [9,10,11,12] and references therein.

In recent years, the scholarly community has increasingly directed its research focus towards a specific class of equations characterized by nonlinear logarithmic terms. Notably, Wei in [13] studied semilinear elliptic equations with nonlinear logarithmic terms:

$$\left\{\begin{array}{c}-\Delta u=a(x)u\log \left|u\right|\hspace{0.33em}\hspace{0.33em} x\in \mathsf{\Omega },\\ \hspace{1em}\hspace{1em}\hspace{1em} u =0 x\in \mathsf{\partial } \mathsf{\Omega },\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\end{array}\right.$$

(8)

where $\mathsf{\Omega }\subset {ℝ}^N(N\ge 2)$ is the bounded smooth region, and $a(x)\in C(\mathsf{\Omega })$. The author found that the existence of the solution of Equation (8) is closely related to the sign of $a(x)$. He proved the existence and multiplicity of solutions for $a(x)>0$, $a(x)<0$ and $a(x)$ is a sign-changing function by the Symmetric Mountain Pass Theorem and the standard logarithmic Sobolev inequality (refer to [14]). In [15], Tian also studied the multiple solutions of Equation (8) and obtained that Equation (8) has at least two nontrivial solutions under certain conditions by using the Nehari manifold and logarithmic Sobolev inequality. In addition, Ji et al. in [16] mainly studied the periodic problems and asymptotic states of semilinear heat equations and pseudo-parabolic equations with logarithmic sources, and obtained the existence and instability of positive periodic solutions of the equations. For more studies with logarithmic terms $\left|u\right|\ln u^2$, interested scholars can refer to [17,18,19,20].

In particular, in 2022, Deng et al. in [21] studied the following equation:

$$\left\{\begin{array}{c}-\Delta u={\left|u\right|}^{2^{\ast }-2}u+\lambda u+\mu u\log u^2\hspace{0.33em}\hspace{0.33em}x\in \mathsf{\Omega },\\ \hspace{1em}\hspace{1em}\hspace{1em} u =0 x\in \mathsf{\partial }\mathsf{\Omega },\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\end{array}\right.$$

(9)

where $N\ge 3$, $\lambda ,\mu \in ℝ$. They proved the existence and nonexistence of positive solutions to Equation (9) as $\mu \ne 0$. That is, when $\mu >0$, $N\ge 4$, the equation has a positive Mountain pass solution, which is also a ground state solution; when $\mu <0$, $N=3, 4$, the equation has at least one positive solution; when $\mu <0$, $N\ge 3$, there is no positive solution to the equation. Subsequently, in 2023, Li et al. in [22] studied the critical biharmonic elliptic problem with logarithm, and also obtained the existence and nonexistence of nontrivial solutions of the equation by using the Mountain Pass Lemma and truncation function methods.

The above research is carried out in the case of p-Laplacian operators with $p=2$. When $p\ne 2$, some scholars also carried out research. For example, in [23], Li et al. studied the following p-Laplacian Kirchhoff type problem with nonlinear term:

$$\left\{\begin{array}{c}-(a+b{\displaystyle {\int }_{\mathsf{\Omega }}{\left|\nabla u\right|}^{p}dx}){\Delta }_{p}u={\left|u\right|}^{q-2}u\ln u^2 x\in \mathsf{\Omega },\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}u =0\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x\in \mathsf{\partial }\mathsf{\Omega },\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\end{array}\right.$$

(10)

and proved that the equation has a sign-changing solution by constrained variational method, topological theory and quantitative deformation principle. More papers with logarithmic terms ${\left|u\right|}^{q-2}u\ln u^2$ can be found in [24,25,26,27,28,29,30,31,32,33]. While many of the studies cited in [24,25,26,27,28,29,30,31,32,33] have focused on equations within the fractional Sobolev space and most of the solutions obtained are sign-changing solutions. And most of the equations studied have either only the p-Laplacian Kirchhoff term or only the critical term.

Inspired by the above papers, we study Equation (1) with logarithmic terms, p-Kirchhoff terms and critical terms. There are two main difficulties encountered in the research process. First of all, the presence of a critical Sobolev exponent causes the lack of compactness of the embedding $W_0^{1,p}\left(\mathsf{\Omega }\right)\subset L^{p^{\ast }}(\mathsf{\Omega })$, which makes it impossible to use the variational method directly, so we use the Mountain Pass Theorem without compactness condition to study the equation. Secondly, logarithmic terms do not satisfy classical (AR) conditions, making it difficult to verify the boundedness of ${(PS)}_c$ sequences, so we use some logarithmic inequalities and properties of logarithms to solve this problem.

The result of the existence of the solution is as follows.

Theorem 1.

If $N=p^2$, $\lambda \in A_0\cup B_0$, and

$${\displaystyle \frac{{(E_{p^2})}^2{e}^{(2^{\alpha }+1)(p-1)-(\lambda +\frac{2}{p})-\frac{2{(p-1)}^2}{p}{\displaystyle \sum _{k=1}^{\alpha }\frac{1}{k}}}{(1+2^{\frac{p}{p-1}})}^{\frac{2{(p-1)}^2}{p}}}{{\rho }_{\max }^2}}<1$$

is satisfied, where $\alpha =p(p-1)-1$, ${\rho }_{\max }:=\sup \left\{r>0:\exists x\in \mathsf{\Omega }, st\hspace{0.33em}B(x,r)\subset \mathsf{\Omega }\right\}$ and $E_{p^2}$ is defined later, then Equation (1) has a nontrivial solution.

Remark 1.

$A_0$ and $B_0$ in Theorem 1 are the following sets:

$$A_0=\left\{\lambda \in \left(0,m_0{\lambda }_1\left(\mathsf{\Omega }\right)\right)\left|{\displaystyle \frac{1}{N}}{\left({\displaystyle \frac{m_0{\lambda }_1\left(\mathsf{\Omega }\right)-\lambda }{{\lambda }_1\left(\mathsf{\Omega }\right)}}\right)}^{{\displaystyle \frac{N}{p}}}{S}^{{\displaystyle \frac{N}{p}}}-{\displaystyle \frac{2}{p^2}}\left|\mathsf{\Omega }\right|>0\right.\right\},$$

$$B_0=\left\{\lambda \in \mathbb{R},\left|{\displaystyle \frac{1}{N}}{m}_0^{{\displaystyle \frac{N}{p}}}{S}^{{\displaystyle \frac{N}{p}}}-{\displaystyle \frac{2}{p^2}}{e}^{\frac{\lambda p}{2}}\left|\mathsf{\Omega }\right|>0\right.\right\}.$$

2. Preliminaries

In this section, we would give some preliminaries, which are important in our proofs.

We first show some notations and definitions that will be used throughout the paper. In the following text, $L^p(\mathsf{\Omega })$ denotes the usual Lebesgue space, with norm ${\left|u\right|}_p={({{\displaystyle {\int }_{\mathsf{\Omega }}\left|u\right|}}^{p}dx)}^{\frac{1}{p}}$; $W_0^{1,p}(\mathsf{\Omega })$ denotes the usual Sobolev space, endowed with the norm $‖u‖:={\left({\displaystyle {\int }_{\mathsf{\Omega }}{\left|\nabla u\right|}^{p}dx}\right)}^{\frac{1}{p}}$; $\left|\mathsf{\Omega }\right|$ is the volume of $\mathsf{\Omega }$ in ${\mathbb{R}}^N$; ${\omega }_N$ denotes the area of the unit sphere surface in ${\mathbb{R}}^N$.

We use $S_r$ to denote the best embedding constant for continuous compact embedding from $W_0^{1,p}$ to $L^r(\mathsf{\Omega })$ and $1\le r<p^{\ast }$, i.e.,

$$S_r:=\underset{u\in W^{1,p}({ℝ}^N)\backslash \left\{0\right\}}{\inf }{\displaystyle \frac{{\displaystyle {\int }_{{ℝ}^N}{\left|\nabla u\right|}^{p}dx}}{{\left({\displaystyle {\int }_{{ℝ}^N}{\left|u\right|}^{r}dx}\right)}^{\frac{p}{r}}}}.$$

(11)

In particular, $S$ is the best continuous embedding constant from $W_0^{1,p}\left(\mathsf{\Omega }\right)$ to $L^{p^{\ast }}(\mathsf{\Omega })$, i.e.,

$$S:=\underset{u\in W^{1,p}({ℝ}^N)\backslash \left\{0\right\}}{\inf }{\displaystyle \frac{{\displaystyle {\int }_{{ℝ}^N}{\left|\nabla u\right|}^{p}dx}}{{\left({\displaystyle {\int }_{{ℝ}^N}{\left|u\right|}^{p^{\ast }}dx}\right)}^{\frac{p}{p^{\ast }}}}}.$$

(12)

${\lambda }_1\left(\mathsf{\Omega }\right)>0$ denotes the first eigenvalue of the eigenvalue problem:

$$\left\{\begin{array}{c}-{\Delta }_{p}u=\lambda {\left|u\right|}^{p-2}u\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}x\in \mathsf{\Omega },\\ u=0 x\in \mathsf{\partial }\mathsf{\Omega }\hspace{0.33em},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\end{array}\right.$$

and

$${\lambda }_1\left(\mathsf{\Omega }\right):=\underset{u\in W_0^{1,p}({ℝ}^N)\backslash \left\{0\right\}}{\inf }{\displaystyle \frac{{\displaystyle {\int }_{\mathsf{\Omega }}{\left|\nabla u\right|}^{p}dx}}{{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^{p}dx}}}.$$

(13)

Definition 1. 

If the following is true for all functions $\phi \in W_0^{1,p}\left(\mathsf{\Omega }\right)$

$$M\left({‖u‖}^p\right){\displaystyle {\int }_{\mathsf{\Omega }}{\left|\nabla u\right|}^{p-2}\nabla u}\nabla \phi dx-\lambda {\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^{p-2}}u\phi dx-{\displaystyle {\int }_{\mathsf{\Omega }}u}\phi {\left|u\right|}^{p^{\ast }-2}dx+{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^{p-2}u}\phi \ln {\left|u\right|}^{2}dx=0.$$

(14)

then the function $u\in W_0^{1,p}\left(\mathsf{\Omega }\right)$ is called the weak solution of problem (1).

We define a modified energy functional corresponding to problem (1) as follows:

$$J\left(u\right)={\displaystyle \frac{1}{p}}\widehat{M}\left({‖u‖}^p\right)-{\displaystyle \frac{\lambda }{p}}{{\displaystyle {\int }_{\mathsf{\Omega }}\left|u\right|}}^{p}dx-{\displaystyle \frac{1}{p^{\ast }}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^{p^{\ast }}}dx-{\displaystyle \frac{2}{p^2}}{{\displaystyle {\int }_{\mathsf{\Omega }}\left|u\right|}}^{p}dx+{\displaystyle \frac{1}{p}}{{\displaystyle {\int }_{\mathsf{\Omega }}\left|u\right|}}^p\ln {\left|u\right|}^{2}dx.$$

(15)

It can also be rewritten as

$$J\left(u\right)={\displaystyle \frac{1}{p}}\widehat{M}\left({‖u‖}^p\right)-{\displaystyle \frac{\lambda }{p}}{{\displaystyle {\int }_{\mathsf{\Omega }}\left|u\right|}}^{p}dx-{\displaystyle \frac{1}{p^{\ast }}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^{p^{\ast }}}dx+{\displaystyle \frac{1}{p}}{{\displaystyle {\int }_{\mathsf{\Omega }}\left|u\right|}}^p\left(\ln {\left|u\right|}^2-{\displaystyle \frac{2}{p}}\right)dx.$$

(16)

where $\widehat{M}\left(t\right)={\displaystyle {\int }_0^{t}M\left(s\right)}ds$. It is obvious that the energy functional is continuous in $W_0^{1,p}\left(\mathsf{\Omega }\right)$.

The purpose is to prove the existence of the weak solution of the equation, which is verified by the two commonly used tools of fiber mapping and the Nahari manifold. For arbitrary $u\in W_0^{1,p}\left(\mathsf{\Omega }\right)\backslash \left\{0\right\}$, we first consider the fiber mapping ${\varphi }_u\left(t\right):\left(0,+\infty \right)\to R$

$${\varphi }_u\left(t\right)=J\left(tu\right)={\displaystyle \frac{1}{p}}\widehat{M}\left(t^p{‖u‖}^p\right)-{\displaystyle \frac{t^{p^{\ast }}}{p^{\ast }}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^{p^{\ast }}}dx+{\displaystyle \frac{t^p}{p}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^p}\left[\ln (t^2{u}^2)-\lambda -{\displaystyle \frac{2}{p}}\right]dx.$$

(17)

Secondly, the Nahari manifold is introduced, i.e.,

$$\mathrm{E}=\left\{u\in H_0^1\left(\mathsf{\Omega }\right)\backslash \left\{0\right\}|\varphi (u)=〈J^{\prime }(u),u〉=0\right\}=\left\{u\in W_0^{1,p}\left(\mathsf{\Omega }\right)\backslash \left\{0\right\}|{{\varphi }^{\prime }}_u(1)=0\right\}.$$

(18)

where $〈 , 〉$ denotes the dual product of ${\tilde{W}}_0^{1,p}\left(\mathsf{\Omega }\right)$ and $W_0^{1,p}\left(\mathsf{\Omega }\right)$, and ${\tilde{W}}_0^{1,p}\left(\mathsf{\Omega }\right)$ is the dual space of $W_0^{1,p}\left(\mathsf{\Omega }\right)$. If $u\in W_0^{1,p}\left(\mathsf{\Omega }\right)$ is a nontrivial solution to problem (1), then $u\in \mathrm{E}$.

Definition 2 

(${(PS)}_c$ condition). Let $X$ be a Banach space, $J\in C^1(X,R)$ and $c\in ℝ$. The function $J$satisfies the ${(PS)}_c$ condition if any sequence $\left\{u_n\right\}\subset X$ such that

$$J(u_n)\to c,J^{\prime }(u_n)\to 0, in \tilde{X}, as n\to +\infty$$

has a convergent subsequence. Here, $\tilde{X}$ denotes the dual space of $X$, $c:=\underset{\gamma \in \mathsf{\Gamma }}{\inf }\underset{t\in [0,1]}{\max }J(\gamma (t))$ and $\mathsf{\Gamma }:=\left\{\gamma \in C(\left[0,1\right],X):\gamma (0)=0,\gamma (1)=e\right\}$.

Lemma 1

(Brézis-lieb’s lemma in [34]). When $p\in (1,+\infty )$ , assuming that the sequence $\left\{u_n\right\}$ is a bounded sequence in $L^p\left(\mathsf{\Omega }\right)$ and satisfies $u_n\to u a.e$ in $\mathsf{\Omega }$ , then

$$\underset{n\to \infty }{\lim }({\left|u_n\right|}_p^p-{\left|u_n-u\right|}_p^p)={\left|u\right|}_p^p.$$

Lemma 2

(Lemma 2.3 of [22]). (1) For all $t\in (0,1)$, there holds that

$$\left|t\ln t\right|\le {\displaystyle \frac{1}{e}} \forall t>0.$$

(19)

We deduce that

$$t\ln t\le {\displaystyle \frac{1}{e}} \forall t>0.$$

(20)

(2) For all $\sigma >0$, there holds that

$$\ln t\le {\displaystyle \frac{1}{e\sigma }} t^{\sigma } \forall t>0.$$

(21)

(3) For all  $\eta ,\beta >0$ , there exists a positive constant $C_{\eta ,\beta }$ such that

$$\left|\ln t\right|\le C_{\eta ,\beta }(t^{\eta }+t^{-\eta }) \forall t>0.$$

(22)

Lemma 3.

(1) $m_{0}t\le \widehat{M}(t)\le m_{1}t\le {\displaystyle \frac{\theta }{p}}{m}_{0}t, \forall t\ge 0;$

Proof. 

Obviously, it can be known by the sign-preserving property of integral

$$m_{0}t={\displaystyle {\int }_0^t{m}_{0}ds}\le \widehat{M}(t)={\displaystyle {\int }_0^{t}M(s)ds}\le {\displaystyle {\int }_0^t{m}_{1}ds}=m_{1}t\le {\displaystyle \frac{\theta }{p}}{m}_{0}t,\forall t\ge 0.$$

□

(2) $\widehat{M}(t)\ge {\displaystyle \frac{p}{\theta }}M(t)t,\forall t\ge 0;$

Proof. 

From the hypothesis condition ${\mathrm{M}}_1$, we can know $m_1<{\displaystyle \frac{\theta }{p}}{m}_0$, $\forall t\ge 0$, and

$$\left(m_0-{\displaystyle \frac{\theta }{p}}{m}_1\right)t\ge 0, \forall t\ge 0.$$

Since

$$\left(m_0-{\displaystyle \frac{\theta }{p}}{m}_1\right)t=m_{0}t-{\displaystyle \frac{\theta }{p}}{m}_{1}t\le \widehat{M}(t)-{\displaystyle \frac{\theta }{p}}M(t)t,\forall t\ge 0.$$

We have

$$0\le \widehat{M}(t)-{\displaystyle \frac{\theta }{p}}M(t)t, \forall t\ge 0,$$

that is

$$\widehat{M}(t)\ge {\displaystyle \frac{\theta }{p}}M(t)t, \forall t\ge 0.$$

□

Lemma 4.

(1) For all $n,m\in {\mathbb{Z}}^{+}$ (${\mathbb{Z}}^{+}$  represents the set of positive integers), there holds that  $C_n^m=C_n^{n-m}$,  $C_n^m=C_{n-1}^m+C_{n-1}^{m-1}$  where  $C_n^m={\displaystyle \frac{n!}{m!\left(n-m\right)!}}$ is a combinatorial number;

(2)  ${\displaystyle \frac{C_n^1}{1}}-{\displaystyle \frac{C_n^2}{2}}+{\displaystyle \frac{C_n^3}{3}}+\cdots +{\left(-1\right)}^{n-1}{\displaystyle \frac{C_n^n}{n}}=1+{\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{3}}+\cdots +{\displaystyle \frac{1}{n}}:={\displaystyle \sum _{k=1}^n\frac{1}{k}}(n\in {\mathbb{Z}}^{+})$;

Proof. 

Let $S_n={\displaystyle \frac{C_n^1}{1}}-{\displaystyle \frac{C_n^2}{2}}+{\displaystyle \frac{C_n^3}{3}}+\cdots +{\left(-1\right)}^{n-1}{\displaystyle \frac{C_n^n}{n}}.$

Since

$$\begin{array}{l}{\displaystyle \frac{1}{k}}{C}_n^k={\displaystyle \frac{1}{k}}{C}_{n-1}^k+{\displaystyle \frac{1}{k}}{C}_{n-1}^{k-1}={\displaystyle \frac{1}{k}}{C}_{n-1}^k+{\displaystyle \frac{(n-1)!}{k!(n-k)!}}\\ \hspace{1em}={\displaystyle \frac{1}{k}}{C}_{n-1}^k+{\displaystyle \frac{1}{n}}{\displaystyle \frac{n!}{k!(n-k)!}}\\ \hspace{1em}={\displaystyle \frac{1}{k}}{C}_{n-1}^k+{\displaystyle \frac{1}{n}}{C}_n^k(k=1,2,3,\cdots n).\end{array}$$

We deduce that

$$\begin{array}{l}{S}_n=\left[{\displaystyle \frac{C_{n-1}^1}{1}}-{\displaystyle \frac{C_{n-1}^2}{2}}+{\displaystyle \frac{C_{n-1}^3}{3}}+\cdots +{\left(-1\right)}^{n-2}{\displaystyle \frac{C_{n-1}^{n-1}}{n-1}}\right]+{\displaystyle \frac{1}{n}}\left[C_n^1-C_n^2+C_n^3+\cdots +{\left(-1\right)}^{n-1}{C}_n^n\right]\\ \hspace{1em}=S_{n-1}+{\displaystyle \frac{1}{n}}(C_{n-1}^0+C_{n-1}^n)=S_{n-1}+{\displaystyle \frac{1}{n}}(C_{n-1}^0+0)=S_{n-1}+{\displaystyle \frac{1}{n}}.\end{array}$$

So

$$S_n-S_{n-1}={\displaystyle \frac{1}{n}}(n\ge 2).$$

And because of $S_1=C_1^1=1$, we have

$$\begin{array}{l}{S}_n=\left(S_n-S_{n-1}\right)+\left(S_{n-1}-S_{n-2}\right)+\left(S_{n-2}-S_{n-3}\right)+\cdots +\left(S_4-S_3\right)+\left(S_3-S_2\right)+\left(S_2-S_1\right)+S_1\\ \hspace{1em}={\displaystyle \frac{1}{n}}+{\displaystyle \frac{1}{n-1}}+{\displaystyle \frac{1}{n-2}}+\cdots +{\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{3}}+{\displaystyle \frac{1}{2}}+1.\end{array}$$

□

(3) $C_n^1+C_n^3+C_n^5+\cdots =C_n^0+C_n^1+C_n^2+\cdots =2^{n-1}$.

Proof. 

From the Binomial Theorem, we have

$${(a+b)}^n={\displaystyle \sum _{k=0}^n{C}_n^k{a}^{n-k}}{b}^k.$$

Let $a=1, b=1,$ we have

$${\displaystyle \sum _{k=0}^n{C}_n^k}=2^n.$$

Since $C_n^k=C_n^{n-k}.$, then

$$C_n^1+C_n^3+C_n^5+\cdots =C_n^0+C_n^1+C_n^2+\cdots =2^{n-1}.$$

□

3. The Existence of Nontrivial Solutions to Equation

Lemma 5.

If $N>p$ and $\lambda \in A_0\cup B_0$ , then functional $J\left(u\right)$ satisfies the mountain pass geometry: 

(1)

there exist  $\mu ,\tau >0$ such that  $J\left(u\right)\ge \mu >0$  for all  $‖v‖=\tau$;

(2)

there exist$\omega \in W_0^{1,p}\left(\mathsf{\Omega }\right)$ such that $‖\omega ‖\ge \tau$ and $J\left(\omega \right)<0$.

Proof. 

According to the division of the set, there are two kinds of proof.

Case 1: $\lambda \in A_0$

First, verify that condition (1) is satisfied. Using the logarithmic algorithm, we have

$$\begin{array}{l}-{\displaystyle \frac{\lambda }{p}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^p}dx+{\displaystyle \frac{1}{p}}{{\displaystyle {\int }_{\mathsf{\Omega }}\left|u\right|}}^p(\ln {\left|u\right|}^2-{\displaystyle \frac{2}{p}})dx\\ =-{\displaystyle \frac{\lambda }{p}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^p}dx+{\displaystyle \frac{1}{p}}{{\displaystyle {\int }_{\mathsf{\Omega }}\left|u\right|}}^p\left[\ln \left({\left|u\right|}^2{e}^{-\frac{2}{p}}\right)\right]dx\\ \ge -{\displaystyle \frac{\lambda }{p{\lambda }_1}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|\nabla u\right|}^p}dx+{\displaystyle \frac{1}{p}}{{\displaystyle {\int }_{\mathsf{\Omega }}\left|u\right|}}^p\left[\ln \left({\left|u\right|}^2{e}^{-\frac{2}{p}}\right)\right]dx\\ :=-{\displaystyle \frac{\lambda }{p{\lambda }_1}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|\nabla u\right|}^p}dx+I_1.\end{array}$$

By Lemma 2 of (19), we obtain

$$\begin{array}{l}{\displaystyle \frac{p}{2}}{I}_1={\displaystyle \frac{1}{p}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^p\ln }\left(e^{-1}{\left|u\right|}^p\right)dx\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\\ \hspace{1em}={\displaystyle \frac{1}{p}}{\displaystyle {\int }_{\left\{\right.e^{-1}{\left|u\right|}^p\left.\ge 1\right\}}{\left|u\right|}^p}\ln \left(e^{-1}\right.\left.{\left|u\right|}^p\right)dx+{\displaystyle \frac{1}{p}}{\displaystyle {\int }_{\left\{\right.e^{-1}{\left|u\right|}^p\left.\le 1\right\}}{\left|u\right|}^p}\ln \left(e^{-1}\right.\left.{\left|u\right|}^p\right)dx\\ \hspace{1em}\ge {\displaystyle \frac{1}{p}}{\displaystyle {\int }_{\left\{\right.e^{-1}{\left|u\right|}^p\left.\le 1\right\}}{\left|u\right|}^p}\ln \left(e^{-1}\right.\left.{\left|u\right|}^p\right)dx\\ \hspace{1em}\ge {\displaystyle \frac{1}{p}}e{\displaystyle {\int }_{\left\{\right.e^{-1}{\left|u\right|}^p\left.\le 1\right\}}-e^{-1}}dx\\ \hspace{1em}\ge {\displaystyle \frac{1}{p}}e{\displaystyle {\int }_{\mathsf{\Omega }}-e^{-1}}dx\\ \hspace{1em}=-{\displaystyle \frac{1}{p}}\left|\mathsf{\Omega }\right|,\end{array}$$

which implies that $I_1\ge -{\displaystyle \frac{2}{p^2}}\left|\mathsf{\Omega }\right|$.

Combining the definition of functional $J\left(u\right)$ and the definition of the best embedding constant $S$, we have

$$J\left(u\right)\ge {\displaystyle \frac{1}{p}}{\displaystyle \frac{m_0{\lambda }_1\left(\mathsf{\Omega }\right)-\lambda }{{\lambda }_1\left(\mathsf{\Omega }\right)}}{‖u‖}^p-{\displaystyle \frac{1}{p^{\ast }}}{S}^{-\frac{p^{\ast }}{p}}{‖u‖}^{p^{\ast }}-{\displaystyle \frac{2}{p^2}}\left|\mathsf{\Omega }\right|.$$

Let

$$g_1(t)={\displaystyle \frac{1}{p}}\left({\displaystyle \frac{m_0{\lambda }_1\left(\mathsf{\Omega }\right)-\lambda }{{\lambda }_1\left(\mathsf{\Omega }\right)}}\right)t^p-{\displaystyle \frac{1}{p^{\ast }}}{S}^{-\frac{p^{\ast }}{p}}{t}^{p^{\ast }}-{\displaystyle \frac{2}{p^2}}\left|\mathsf{\Omega }\right| t>0.$$

Through direct calculation, we can see that $g_1\left(t\right)$ takes the maximum value at $t_0={\left({\displaystyle \frac{m_0{\lambda }_1\left(\mathsf{\Omega }\right)-\lambda }{{\lambda }_1\left(\mathsf{\Omega }\right)}}\right)}^{\frac{N-p}{p^2}}{S}^{\frac{N}{p^2}}$ and there is

$$g_1(t_0)= {\displaystyle \frac{1}{N}}{\left({\displaystyle \frac{m_0{\lambda }_1\left(\mathsf{\Omega }\right)-\lambda }{{\lambda }_1\left(\mathsf{\Omega }\right)}}\right)}^{\frac{N}{p}}{S}^{\frac{N}{p}}-{\displaystyle \frac{2}{p^2}}\left|\mathsf{\Omega }\right|.$$

Combined with $\lambda \in A_0$, we have

$$\begin{array}{l}J\left(u\right)\ge {\displaystyle \frac{1}{p}}{\displaystyle \frac{m_0{\lambda }_1\left(\mathsf{\Omega }\right)-\lambda }{{\lambda }_1\left(\mathsf{\Omega }\right)}}{\tau }^p-{\displaystyle \frac{1}{p^{\ast }}}{S}^{-\frac{p^{\ast }}{p}}{\tau }^{p^{\ast }}-{\displaystyle \frac{2}{p^2}}\left|\mathsf{\Omega }\right|\\ \hspace{1em}={\displaystyle \frac{1}{N}}{S}^{\frac{N}{p}}{\left({\displaystyle \frac{m_0{\lambda }_1\left(\mathsf{\Omega }\right)-\lambda }{{\lambda }_1\left(\mathsf{\Omega }\right)}}\right)}^{\frac{N}{p}}-{\displaystyle \frac{2}{p^2}}\left|\mathsf{\Omega }\right|\\ =\mu >0.\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \end{array}$$

Therefore, if we take $\mu ={\displaystyle \frac{1}{N}}{\left({\displaystyle \frac{m_0{\lambda }_1\left(\mathsf{\Omega }\right)-\lambda }{{\lambda }_1\left(\mathsf{\Omega }\right)}}\right)}^{\frac{N}{p}}{S}^{\frac{N}{p}}-{\displaystyle \frac{2}{p^2}}\left|\mathsf{\Omega }\right|>0$ and $\tau ={\left({\displaystyle \frac{m_0{\lambda }_1\left(\mathsf{\Omega }\right)-\lambda }{{\lambda }_1\left(\mathsf{\Omega }\right)}}\right)}^{\frac{N-p}{p^2}}{S}^{\frac{N}{p^2}}>0$, then for all $‖v‖=\tau$ such that condition (1) holds.

Second, prove that condition (2) is true.

According to the definition of ${\varphi }_u\left(t\right)$, for any given $u\in W_0^{1,p}\left(\mathsf{\Omega }\right)\backslash \left\{0\right\}$, we can obviously get $\underset{t\to 0}{\lim }{\varphi }_u\left(t\right)=0$. Using Lemma 3 of (1), it can be deduced that

$$\begin{array}{c}{\varphi }_u\left(t\right)={\displaystyle \frac{1}{p}}\widehat{M}\left(t^p{‖u‖}^p\right)-{\displaystyle \frac{t^{p^{\ast }}}{p^{\ast }}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^{p^{\ast }}}dx+{\displaystyle \frac{t^p}{p}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^p}\left[\ln (t^2{u}^2)-\lambda -{\displaystyle \frac{2}{p}}\right]dx\\ \le {\displaystyle \frac{\theta }{p^2}}{m}_0{t}^p{‖u‖}^p-{\displaystyle \frac{t^{p^{\ast }}}{p^{\ast }}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^{p^{\ast }}}dx-{\displaystyle \frac{\lambda }{p}}{t}^p{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^p}dx-{\displaystyle \frac{2}{p^2}}{t}^p{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^p}dx\\ +{\displaystyle \frac{1}{p}}{t}^p{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^p\ln {\left|u\right|}^{2}dx}+{\displaystyle \frac{1}{p}}{t}^p\ln {\left|t\right|}^2{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^p}dx.\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\end{array}$$

Since $\underset{t\to +\infty }{\lim }{\displaystyle \frac{t^{p^{\ast }}}{t^p\ln t^2}}=+\infty$, we derive that $\underset{t\to +\infty }{\lim }{\varphi }_u\left(t\right)=-\infty$. This indicates that condition (2) holds: choose $t_1\in {\mathbb{R}}^{+}$ sufficiently large so that $J\left(t_{1}u\right)<0 and ‖t_{1}u‖>\tau$.

Case 2: $\lambda \in B_0$

In a similar way, by direct calculation there are

$$\begin{array}{c}{\displaystyle \frac{1}{p}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^p\left(\ln {\left|u\right|}^2-\frac{2}{p}-\lambda \right)}dx={\displaystyle \frac{1}{p}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^p\ln \left(e^{-\lambda -\frac{2}{p}},{\left|u\right|}^2\right)}dx\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{2}{p^2}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^p\ln \left(e^{\frac{-p\lambda }{2}-1},{\left|u\right|}^p\right)}dx\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{2}{p^2}}{\displaystyle {\int }_{\left\{\right.e^{\frac{-p\lambda }{2}-1}{\left|u\right|}^p\left.\ge 1\right\}}{\left|u\right|}^p\ln \left(e^{-\frac{p\lambda }{2}-1},{\left|u\right|}^p\right)}dx+{\displaystyle \frac{2}{p^2}}{\displaystyle {\int }_{\left\{\right.e^{\frac{-p\lambda }{2}-1}{\left|u\right|}^p\left.\le 1\right\}}{\left|u\right|}^p\ln \left(e^{-\frac{p\lambda }{2}-1},{\left|u\right|}^p\right)}dx\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ge {\displaystyle \frac{2}{p^2}}{e}^{\frac{p\lambda }{2}+1}{\displaystyle {\int }_{\left\{\right.e^{\frac{-p\lambda }{2}-1}{\left|u\right|}^p\left.\le 1\right\}}{e}^{-\frac{p\lambda }{2}-1}{\left|u\right|}^p\ln \left(e^{-\frac{p\lambda }{2}-1},{\left|u\right|}^p\right)}dx\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ge {\displaystyle \frac{2}{p^2}}{e}^{\frac{p\lambda }{2}+1}{\displaystyle {\int }_{\left\{\right.e^{\frac{-p\lambda }{2}-1}{\left|u\right|}^p\left.\le 1\right\}}-e^{-1}}dx\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ge {\displaystyle \frac{-2}{p^2}}{e}^{\frac{p\lambda }{2}}\left|\mathsf{\Omega }\right|.\end{array}$$

Similarly, combining the definition of functional $J\left(u\right)$ with the definition of the best embedding constant $S$, we have

$$J\left(u\right)\ge {\displaystyle \frac{1}{p}}{m}_0{‖u‖}^p-{\displaystyle \frac{1}{p^{\ast }}}{S}^{-\frac{p^{\ast }}{p}}{‖u‖}^{p^{\ast }}-{\displaystyle \frac{2}{p^2}}{e}^{\frac{p\lambda }{2}}\left|\mathsf{\Omega }\right|.$$

Let

$$g_2(t)={\displaystyle \frac{1}{p}}{m}_0{t}^p-{\displaystyle \frac{1}{p^{\ast }}}{S}^{-\frac{p^{\ast }}{p}}{t}^{p^{\ast }}-{\displaystyle \frac{2}{p^2}}{e}^{\frac{p\lambda }{2}}\left|\mathsf{\Omega }\right| t>0.$$

Through simple calculation, we can see that $g_2\left(t\right)$ obtains the maximum value at ${\widehat{t}}_0={\left(m_0\right)}^{\frac{N-p}{p^2}}{S}^{\frac{N}{p^2}}$, and has

$$g_2({\widehat{t}}_0)={\displaystyle \frac{1}{N}}{\left(m_0\right)}^{\frac{N}{p}}{S}^{\frac{N}{p}}-{\displaystyle \frac{2}{p^2}} e^{\frac{p\lambda }{2}}\left|\mathsf{\Omega }\right| .$$

Combined with $\lambda \in B_0$, we know

$$\begin{array}{c}J\left(v\right)\ge {\displaystyle \frac{1}{p}}{m}_0{\tau }^p-{\displaystyle \frac{1}{p^{\ast }}}{S}^{-\frac{p^{\ast }}{p}}{\tau }^{p^{\ast }}-{\displaystyle \frac{2}{p^2}}{e}^{\frac{p\lambda }{2}}\left|\mathsf{\Omega }\right|\\ ={\displaystyle \frac{1}{N}}{(m_0)}^{\frac{N}{p}}{S}^{\frac{N}{p}}-{\displaystyle \frac{2}{p^2}}{e}^{\frac{p\lambda }{2}}\left|\mathsf{\Omega }\right|=\mu >0.\end{array}$$

Therefore, if we take $\mu ={\displaystyle \frac{1}{N}}{\left(m_0\right)}^{\frac{N}{p}}{S}^{\frac{N}{p}}-{\displaystyle \frac{2}{p^2}} e^{\frac{p\lambda }{2}}\left|\mathsf{\Omega }\right|>0$ and $\tau ={\left(m_0\right)}^{\frac{N-p}{p^2}}{S}^{\frac{N}{p^2}}>0$, then condition (1) is true for all $‖v‖=\tau$. The verification of condition (2) is similar to case 1 and will not be repeated.

□

Lemma 6.

Assume that $N>p, \lambda \in \mathbb{R}$ , if sequence $\left\{u_n\right\}$ is a sequence ${(PS)}_c$  of functional $J(u)$, then sequence $\left\{u_n\right\}$ must be bounded in $W_0^{1,p}\left(\mathsf{\Omega }\right)$ for all $c\in R$.

Proof. 

According to the ${(PS)}_c$ sequence definition, that is

$$J(u_n)\to c,J^{\prime }(u_n)\to 0, in {\tilde{W}}_0^{1,p}(\mathsf{\Omega }) , n\to +\infty .$$

and the definition of functional $J\left(u\right)$, we obtain

$$J\left(u_n\right)={\displaystyle \frac{1}{p}}\widehat{M}\left({‖u_n‖}^p\right)-{\displaystyle \frac{\lambda }{p}}{{\displaystyle {\int }_{\mathsf{\Omega }}\left|u_n\right|}}^{p}dx-{\displaystyle \frac{1}{p^{\ast }}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^{p^{\ast }}}dx-{\displaystyle \frac{2}{p^2}}{{\displaystyle {\int }_{\mathsf{\Omega }}\left|u_n\right|}}^{p}dx+{\displaystyle \frac{1}{p}}{{\displaystyle {\int }_{\mathsf{\Omega }}\left|u_n\right|}}^p\ln {\left|u_n\right|}^{2}dx=c+o_n(1),$$

(23)

and

$$〈J^{\prime }\left(u_n\right),u_n〉=M\left({‖u_n‖}^p\right){‖u_n‖}^p-\lambda {\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^p}dx-{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^{p^{\ast }}}dx+{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^p}\ln {\left|u_n\right|}^{2}dx=o_n(1)‖u_n‖.$$

(24)

By Lemma 3, (23) and (24), we have

$$\begin{array}{l}c+o_n(1)+{\displaystyle \frac{o_n(1)}{\theta }}‖u_n‖=J\left(u_n\right)-{\displaystyle \frac{1}{\theta }}〈J^{\prime }\left(u_n\right),u_n〉\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{1}{p}}\widehat{M}({‖u_n‖}^p)-{\displaystyle \frac{1}{\theta }}M({‖u_n‖}^p){‖u_n‖}^p+\lambda \left({\displaystyle \frac{1}{\theta }}-{\displaystyle \frac{1}{p}}\right){\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^{p}dx}+\left({\displaystyle \frac{1}{\theta }}-{\displaystyle \frac{1}{p^{\ast }}}\right){\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^{p^{\ast }}dx}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left({\displaystyle \frac{1}{\theta }}-{\displaystyle \frac{1}{p}}\right){\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^p\ln {\left|u_n\right|}^{2}dx}-{\displaystyle \frac{2}{p^2}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^{p}dx}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ge \left({\displaystyle \frac{1}{p}}{m}_0-{\displaystyle \frac{1}{\theta }}{m}_1\right){‖u_n‖}^p+\lambda \left({\displaystyle \frac{1}{\theta }}-{\displaystyle \frac{1}{p}}\right){\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^{p}dx}+\left({\displaystyle \frac{1}{\theta }}-{\displaystyle \frac{1}{p^{\ast }}}\right){\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^{p^{\ast }}dx}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left({\displaystyle \frac{1}{\theta }}-{\displaystyle \frac{1}{p}}\right){\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^p\ln {\left|u_n\right|}^{2}dx}-{\displaystyle \frac{2}{p^2}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^{p}dx}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\ge \left({\displaystyle \frac{1}{p}}{m}_0-{\displaystyle \frac{1}{\theta }}{m}_1\right){‖u_n‖}^p-\left({\displaystyle \frac{1}{\theta }}-{\displaystyle \frac{1}{p}}\right){\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^p\ln {\left|u_n\right|}^{2}dx}+\lambda \left({\displaystyle \frac{1}{\theta }}-{\displaystyle \frac{1}{p}}\right){\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^{p}dx}-{\displaystyle \frac{2}{p^2}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^{p}dx}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}:=\left({\displaystyle \frac{1}{p}}{m}_0-{\displaystyle \frac{1}{\theta }}{m}_1\right){‖u_n‖}^p+I_2.\end{array}$$

Using Lemma 2 of (19), when $n$ is sufficiently large, we have

$$\begin{array}{l}{I}_2=-{\displaystyle \frac{(p-\theta )}{p\theta }}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^p}\ln (e^{\frac{2\theta }{p^2-\theta p}-\lambda }{\left|u_n\right|}^2)dx\\ \hspace{1em}=-{\displaystyle \frac{2(p-\theta )}{p^2\theta }}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^p}\ln (e^{\frac{\theta }{p-\theta }-\frac{p\lambda }{2}}{\left|u_n\right|}^p)dx\\ \hspace{1em}=-{\displaystyle \frac{2(p-\theta )}{p^2\theta }}{e}^{\frac{p\lambda }{2}-\frac{\theta }{p-\theta }}{\displaystyle {\int }_{\mathsf{\Omega }}{e}^{\frac{\theta }{p-\theta }-\frac{p\lambda }{2}}{\left|u_n\right|}^p}\ln (e^{\frac{\theta }{p-\theta }-\frac{p\lambda }{2}}{\left|u_n\right|}^p)dx\\ \hspace{1em}=-{\displaystyle \frac{2(p-\theta )}{p^2\theta }}{e}^{\frac{p\lambda }{2}-\frac{\theta }{p-\theta }}{\displaystyle {\int }_{\left\{\right.e^{\frac{\theta }{p-\theta }-\frac{p\lambda }{2}}{\left|u_n\right|}^p\le 1\left.\right\}}{e}^{\frac{\theta }{p-\theta }-\frac{p\lambda }{2}}{\left|u_n\right|}^p\ln (e^{\frac{\theta }{p-\theta }-\frac{p\lambda }{2}}{\left|u_n\right|}^p)}dx\\ \hspace{1em}-{\displaystyle \frac{2(p-\theta )}{p^2\theta }}{e}^{\frac{p\lambda }{2}-\frac{\theta }{p-\theta }}{\displaystyle {\int }_{\left\{\right.e^{\frac{\theta }{p-\theta }-\frac{p\lambda }{2}}{\left|u_n\right|}^p\ge 1\left.\right\}}{e}^{\frac{\theta }{p-\theta }-\frac{p\lambda }{2}}{\left|u_n\right|}^p\ln (e^{\frac{\theta }{p-\theta }-\frac{p\lambda }{2}}{\left|u_n\right|}^p)}dx\\ \hspace{1em}\ge -{\displaystyle \frac{2(p-\theta )}{p^2\theta }}{e}^{\frac{p\lambda }{2}-\frac{\theta }{p-\theta }}{\displaystyle {\int }_{\left\{\right.e^{\frac{\theta }{p-\theta }-\frac{p\lambda }{2}}{\left|u_n\right|}^p\le 1\left.\right\}}{e}^{\frac{\theta }{p-\theta }-\frac{p\lambda }{2}}{\left|u_n\right|}^p\ln (e^{\frac{\theta }{p-\theta }-\frac{p\lambda }{2}}{\left|u_n\right|}^p)}dx\\ \hspace{1em}\ge -{\displaystyle \frac{2(p-\theta )}{p^2\theta }}{e}^{\frac{p\lambda }{2}-\frac{\theta }{p-\theta }}{\displaystyle {\int }_{\left\{\right.e^{\frac{\theta }{p-\theta }-\frac{p\lambda }{2}}{\left|u_n\right|}^p\le 1\left.\right\}}-e^{-1}}dx\\ \hspace{1em}\ge -{\displaystyle \frac{2(p-\theta )}{p^2\theta }}{e}^{\frac{p\lambda }{2}-\frac{\theta }{p-\theta }}{\displaystyle {\int }_{\mathsf{\Omega }}-e^{-1}}dx\\ \hspace{1em}={\displaystyle \frac{2(p-\theta )}{p^2\theta }}{e}^{\frac{p\lambda }{2}-\frac{\theta }{p-\theta }-1}\left|\mathsf{\Omega }\right|.\end{array}$$

Combined with the above formulas, we can see that

$$c+o_n(1)+\frac{o_n(1)}{\theta }‖u_n‖\ge (\frac{1}{p}{m}_0-\frac{1}{\theta }{m}_1){‖u_n‖}^p+\frac{2\left(p-\theta \right)}{p^2\theta }{e}^{\frac{p\lambda }{2}-\frac{\theta }{p-\theta }-1}\left|\mathsf{\Omega }\right|.$$

So there exists a constant $C>0$ such that $‖u_n‖<C$. This indicates that the sequence $\left\{u_n\right\}$ is bounded in $W_0^{1,p}\left(\mathsf{\Omega }\right)$. □

Lemma 7.

Let $\left\{u_n\right\}$ be the sequence bounded in  $W_0^{1,p}\left(\mathsf{\Omega }\right)$, and satisfies   $u_n\to u a.e. in \mathsf{\Omega }$  as   $n\to \infty$, then 

$$\underset{n\to \infty }{\lim }{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^p\ln {\left|u_n\right|}^{2}dx={\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^p\ln {\left|u\right|}^{2}dx.}}$$

Proof. 

From Lemma 2 of (19) and (21), we know for any $\sigma >0,$ we get

$$\begin{array}{c}{\displaystyle {\int }_{\mathsf{\Omega }}\left|{\left|u_n\right|}^p\ln {\left|u_n\right|}^2\right|dx=2{\displaystyle {\int }_{\left\{\right.\left|u_n\right|\le 1\left.\right\}}\left|{\left|u_n\right|}^p\ln \left|u_n\right|\right|dx}}+2{\displaystyle {\int }_{\left\{\right.\left|u_n\right|\ge 1\left.\right\}}\left|{\left|u_n\right|}^p\ln \left|u_n\right|\right|dx}\hfill \\ \le \frac{2}{pe}\left|\mathsf{\Omega }\right|+\frac{2}{e\sigma }{\left|u_n\right|}_{p+\sigma }^{p+\sigma }.\hfill \end{array}$$

In particular, let us take ${\sigma }_1={\displaystyle \frac{p^{\ast }-p}{p}}$, then $r=p+{\sigma }_1<p^{\ast }$. By definition of $S_r$, we have

$$\frac{2}{pe}\left|\mathsf{\Omega }\right|+\frac{2}{e\sigma }{\left|u_n\right|}_{p+\sigma }^{p+\sigma }\le \frac{2}{pe}\left|\mathsf{\Omega }\right|+\frac{2}{e{\sigma }_1}{S_r}^{{\displaystyle \frac{-(p+{\sigma }_1)}{p}}}{‖u_n‖}^{p+{\sigma }_1},$$

This shows that $\left\{{\left|u\right|}^{p}In{\left|u\right|}^2\right\}$ is bounded integrable in $L^1\left(\mathsf{\Omega }\right)$.

According to Brézis-lieb’s Lemma 1, we have

$$\underset{n\to \infty }{\lim }{\displaystyle {\int }_{\mathsf{\Omega }}\left({\left|u_n\right|}^p\ln {\left|u_n\right|}^2-{\left|u_n-u\right|}^p\ln {\left|u_n-u\right|}^2\right)}dx={\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^p\ln {\left|u\right|}^{2}dx}.$$

Using Lemma 2 of (3) with $0<\eta ={\displaystyle \frac{p^{\ast }-p}{p}}<{\displaystyle \frac{p^{\ast }-p}{2}}$, we have

$${\left|x\right|}^p\ln {\left|x\right|}^2\le C\left({\left|x\right|}^{p+2\eta }+{\left|x\right|}^{p-2\eta }\right) where C>0.$$

And since $W_0^{1,p}\left(\mathsf{\Omega }\right)$ is embedded compactly in $L^r\left(\mathsf{\Omega }\right) \left(1\le r<p^{\ast }\right)$ and $r=p+2\eta <p^{\ast }$, as $n\to \infty$, there is

$$\left|{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n-u\right|}^p\ln {\left|u_n-u\right|}^2}dx\right|\le C{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n-u\right|}^{p+2\eta }dx}+C{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n-u\right|}^{p-2\eta }dx}\to 0,$$

so

$$\underset{n\to \infty }{\lim }{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^p\ln {\left|u_n\right|}^{2}dx={\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^p\ln {\left|u\right|}^{2}dx}}.$$

 □

Lemma 8. 

If $N>p,\lambda \in \mathbb{R}$, $\mathrm{c}\in \left(-\infty ,0\right)\cup \left(0,\frac{p^{\ast }-\theta }{\theta p^{*}}{\left(m_{0}S\right)}^{\frac{N}{p}}\right)$, and $\left\{u_n\right\}$ is a ${(PS)}_{\mathrm{c}}$ sequence of $J(u)$, then there exists a $u\in W_0^{1,p}\left(\mathsf{\Omega }\right)\backslash \left\{0\right\}$ such that $u_n\to u weakly in W_0^{1,p}(\mathsf{\Omega })$, and $u$ is a nontrivial solution to Equation (1).

Proof. 

Supposing sequence $\left\{u_n\right\}$ is sequence ${(PS)}_{\mathrm{c}}$ of functional $J(u)$. From Lemma 6, we know that sequence $\left\{u_n\right\}$ is bounded in $W_0^{1,p}\left(\mathsf{\Omega }\right)$, and we take a subsequence of $\left\{u_n\right\}$, which is still denoted as $\left\{u_n\right\}$. So there is $u\in W_0^{1,p}\left(\mathsf{\Omega }\right)$ such that

$$\begin{gathered} u_n\to u weakly in W_0^{1,p}(\mathsf{\Omega }), \\ {u}_n\to u strongly in L^r\left(\mathsf{\Omega }\right), 1\le r<p^{\ast }, \\ {u}_n\to u a.e. in \mathsf{\Omega }. \end{gathered}$$

Since for any $\phi \in C_0^{\infty }(\mathsf{\Omega })$, there is $〈J^{\prime }(u_n),u_n〉\to 0$ as $n\to \infty$. Therefore, $u$ is a weak solution to the following equation:

$$-M\left({\displaystyle {\int }_{\mathsf{\Omega }}{\left|\nabla u\right|}^{p}dx}\right){\Delta }_{p}u=\lambda {\left|u\right|}^{p-2}u+{\left|u\right|}^{p^{\ast }-2}u-{\left|u\right|}^{p-2}\ln {\left|u\right|}^2.$$

(25)

From the above formula, it can be deduced

$$M\left({\displaystyle {\int }_{\mathsf{\Omega }}{\left|\nabla u\right|}^{p}dx}\right){\displaystyle {\int }_{\mathsf{\Omega }}{\left|\nabla u\right|}^{p}dx}=\lambda {\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^{p}dx}+{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^{p^{\ast }}dx}-{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^p\ln {\left|u\right|}^{2}dx}.$$

(26)

Suppose $u=0$ and $v_n=u_n-u$, by (24) and (26), then we get

$$\begin{array}{l}\left[M({‖u_n‖}^p)\right.{‖u_n‖}^p-\left.M({‖u‖}^p){‖u‖}^p\right]-{\displaystyle {\int }_{\mathsf{\Omega }}({\left|u_n\right|}^{p^{\ast }}-{\left|u\right|}^{p^{\ast }})dx}-\lambda {\displaystyle {\int }_{\mathsf{\Omega }}\left[{\left|u_n\right|}^p-{\left|u\right|}^p\right]dx}\\ +{\displaystyle {\int }_{\mathsf{\Omega }}\left[{\left|u_n\right|}^p\ln {\left|u_n\right|}^2-{\left|u\right|}^p\ln {\left|u\right|}^2\right]dx}=o_n(1).\end{array}$$

According to the Brézis-lieb’s Lemma 1, we can see

$$M({‖u_n‖}^p){‖u_n‖}^p=M({‖u‖}^p){‖u‖}^p+M({‖v_n‖}^p){‖v_n‖}^p+o_n(1),$$

(27)

and

$${\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^{p^{\ast }}dx}={\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^{p^{\ast }}dx}+{\displaystyle {\int }_{\mathsf{\Omega }}{\left|v_n\right|}^{p^{\ast }}dx}+o_n(1).$$

(28)

Due to $W_0^{1,p}\left(\mathsf{\Omega }\right)$ embedded compactly in $L^r\left(\mathsf{\Omega }\right) \left(1\le r<p^{\ast }\right)$, we have

$${\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^{p}dx}={\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^{p}dx}+o_n(1).$$

(29)

According to Lemma 7, we know

$${\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_n\right|}^p\ln {\left|u_n\right|}^{2}dx={\displaystyle {\int }_{\mathsf{\Omega }}{\left|u\right|}^p\ln {\left|u\right|}^{2}dx}}+o_n(1).$$

(30)

Similarly, according to the Brézis-lieb’s Lemma 1, we can see

$$\widehat{M}({‖u_n‖}^p)=\widehat{M}({‖u‖}^p)+\widehat{M}({‖v_n‖}^p)+o_n(1).$$

(31)

Combined with formulas (27)–(31), we can get

$$M\left({‖v_n‖}^p\right){‖v_n‖}^p-{\displaystyle {\int }_{\mathsf{\Omega }}{\left|v_n\right|}^{p^{*}}dx}=o_n(1).$$

(32)

In a similar way, using (23) and functional $J(u)$, we also get

$${\displaystyle \frac{1}{p}}\widehat{M}({‖v_n‖}^p)-{\displaystyle \frac{1}{p^{\ast }}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|v_n\right|}^{p^{\ast }}}dx=c+o_n(1).$$

(33)

We can assume that

$$M\left({‖v_n‖}^p\right){‖v_n‖}^p\to k as n\to \infty ,$$

(34)

then

$${\displaystyle {\int }_{\mathsf{\Omega }}{\left|v_n\right|}^{p^{*}}dx}\to k as n\to \infty .$$

(35)

Obviously $k\ge 0$. Suppose $k=0$, then according to the Brézis-lieb’s Lemma 1, when $n$ is sufficiently large, we have

$$M({‖v_n‖}^p){‖v_n‖}^p=M({‖u_n‖}^p){‖u_n‖}^p\to 0,$$

and

$$\widehat{M}({‖v_n‖}^p){‖v_n‖}^p=\widehat{M}({‖u_n‖}^p){‖u_n‖}^p\to 0.$$

Therefore, the above formula indicates $J(u_n)\to 0$, which is in contradiction with $c\ne 0$.

From the definition of the best embedding constant $S$, ${\left|\nabla v_n\right|}_p^p\ge S{\left|v_n\right|}_{p^{\ast }}^p, v_n\in W_0^{1,p}\left(\mathsf{\Omega }\right)$.

By (32), we get

$$k+o_n\left(1\right)=M\left({‖v_n‖}^p\right){‖v_n‖}^p\ge m_{0}S{\left({\displaystyle {\int }_{\mathsf{\Omega }}{\left|v_n\right|}^{p^{*}}dx}\right)}^{\frac{p}{p^{*}}}=m_{0}S{k}^{\frac{N-p}{N}}+o_n(1).$$

If $k>0$, then $k\ge {\left(m_{0}S\right)}^{\frac{N}{p}}$. By (33) and Lemma 3 of (2), we can see that there is a $\theta \in \left(p,p^{\ast }\right)$, which satisfies

$$\begin{array}{l}{\displaystyle \frac{p^{\ast }-\theta }{\theta p\ast }}{(m_{0}S)}^{\frac{N}{p}}\le ({\displaystyle \frac{1}{\theta }}-{\displaystyle \frac{1}{p^{\ast }}})k\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{1}{\theta }}M({‖v_n‖}^p){‖v_n‖}^p-{\displaystyle \frac{1}{p^{\ast }}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|v_n\right|}^{p^{\ast }}}dx\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le {\displaystyle \frac{1}{p}}\widehat{M}({‖v_n‖}^p)-{\displaystyle \frac{1}{p^{\ast }}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|v_n\right|}^{p^{\ast }}}dx\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=c+o_n(1)<({\displaystyle \frac{1}{\theta }}-{\displaystyle \frac{1}{p^{\ast }}}){(m_{0}S)}^{\frac{N}{p}}={\displaystyle \frac{p^{\ast }-\theta }{\theta p\ast }}{(m_{0}S)}^{\frac{N}{p}}.\end{array}$$

From the above comes the contradiction, so the hypothesis is not valid. From all of this, we know that $u=0$ is impossible. This contradicts the assumption that $u=0$. Therefore, we proved that $u\in W_0^{1,p}\left(\mathsf{\Omega }\right)\backslash \left\{0\right\}$ and $u$ are non-trivial solutions of Equation (1). □

4. Proofs of the Main Results

The following needs to verify $c<{\displaystyle \frac{p^{\ast }-\theta }{\theta p^{\ast }}}{\left(m_{0}S\right)}^{\frac{N}{p}}$, from [22] of Lemma 3.6, we know that there exist $u_{\epsilon }\in W_0^{1,p}\left(\mathsf{\Omega }\right)$ such that $c<\underset{t\ge 0}{\sup }J\left(t{u}_{\epsilon }\right)$. So, we can see that we only need to find the appropriate $u_{\epsilon }\in W_0^{1,p}\left(\mathsf{\Omega }\right)$ to make $\underset{t\ge 0}{\sup }J(t{u}_{\epsilon })\le {\displaystyle \frac{p^{\ast }-\theta }{\theta p^{\ast }}}{\left(m_{0}S\right)}^{\frac{N}{p}}$.

Step 1. Find the appropriate $u_{\epsilon }\in W_0^{1,p}\left({\mathbb{R}}^N\right)$.

It can be seen from the literature [8] of p-Laplace equation:

$$\left\{\begin{array}{c}-{\Delta }_{p}u={\left|u\right|}^{p^{\ast }-2}u\hspace{0.33em},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}x\in {ℝ}^N,\\ u\in W_0^{1,p}({ℝ}^N).\end{array}\right.$$

(36)

If $u\in W_0^{1,p}\left({ℝ}^N\right)$ is a solution to Equation (36), then there exists $\epsilon >0$ such that

$$u\left(x\right)=U_{\epsilon ,x_0}\left(x\right)={\left[{\displaystyle \frac{{\epsilon }^{\frac{1}{p-1}}{N}^{\frac{1}{p}}{(\frac{N-p}{p-1})}^{\frac{p-1}{p}}}{{\epsilon }^{\frac{p}{p-1}}+{\left|x-x_0\right|}^{\frac{p}{p-1}}}}\right]}^{\frac{N-p}{p}},$$

when $x_0=0$, set $U_{\epsilon ,0}\left(x\right)=U_{\epsilon }\left(x\right)$, we have

$$U_{\epsilon }\left(x\right)=E_N{\left[{\displaystyle \frac{{\epsilon }^{\frac{1}{p-1}}}{{\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}}}}\right]}^{\frac{N-p}{p}},\mathrm{where}{E}_N=N^{\frac{N-p}{p^2}}{({\displaystyle \frac{N-p}{p-1}})}^{\frac{\left(p-1\right)\left(N-p\right)}{p^2}}.$$

(37)

and there are ${‖U_{\epsilon }‖}^p={\left|U_{\epsilon }\right|}_{p^{\ast }}^{p^{\ast }}=S^{\frac{N}{p}}$, where $S=\underset{u\in W_0^{1,p}\left(\mathsf{\Omega }\right)\backslash \left\{0\right\}}{\inf }{\displaystyle \frac{{‖u‖}^p}{{\left|u\right|}_{p^{\ast }}^p}}={\displaystyle \frac{{‖U_{\epsilon }‖}^p}{{\left|U_{\epsilon }\right|}_{p^{\ast }}^p}}$.

Step 2. Estimate

Lemma 9 

(see [8]). Suppose $N>p$ and $\varphi \in C_0^{\infty }\left(\mathsf{\Omega }\right)$ be a truncation function such that $\varphi \left(x\right)=\varphi \left(\left|x\right|\right)$, $0\le \varphi \left(x\right)\le 1$, for any $x\in \mathsf{\Omega },$ and

$$\varphi \left(x\right)=\left\{\begin{array}{c}1\hspace{0.33em}\hspace{0.33em},\hspace{0.33em}\hspace{0.33em}\left|x\right|<\rho ,\\ 0\hspace{0.33em}\hspace{0.33em},\hspace{0.33em}\hspace{0.33em}\left|x\right|<2\rho ,\end{array}\right.$$

where $\rho >0$ is a normal constant and $B_{2\rho }\left(0\right)\subset \mathsf{\Omega }$. Set $u_{\epsilon }\left(x\right)=\varphi \left(x\right)U_{\epsilon }\left(x\right)$, assuming, for any $N>p$, then there is

$${‖u_{\epsilon }‖}^p={\displaystyle {\int }_{\mathsf{\Omega }}{\left|\nabla u_{\epsilon }\right|}^p}dx=S^{\frac{N}{p}}+O\left({\epsilon }^{\frac{N-P}{p-1}}\right),$$

(38)

$${\left|u_{\epsilon }\right|}_{p^{\ast }}^{p^{\ast }}={\displaystyle {\int }_{\mathsf{\Omega }}{\left|\nabla u_{\epsilon }\right|}^p}dx=S^{\frac{N}{p}}+O\left({\epsilon }^{\frac{N-P}{p-1}}\right).$$

(39)

Lemma 10. 

If $N=p^2,$ when $\epsilon \to 0$, then

$${\left|u_{\epsilon }\right|}_p^p={\left(E_{p^2}\right)}^p{\omega }_{p^2}{\epsilon }^p\ln ({\displaystyle \frac{1}{\epsilon }})+O\left({\epsilon }^p\right).$$

Proof. 

Using $u_{\epsilon }\left(x\right)=\varphi \left(x\right)U_{\epsilon }\left(x\right)$ and the properties of the cut-off function $\varphi \left(x\right)$, we have

$$\begin{array}{l}{\left|u_{\epsilon }\right|}_p^p={\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_{\epsilon }\right|}^p}dx={\displaystyle {\int }_{\mathsf{\Omega }}{\left|\varphi (x)U_{\epsilon }(x)\right|}^p}dx\\ \hspace{1em}\hspace{1em}={\displaystyle {\int }_{\mathsf{\Omega }\backslash B_{2\rho }(0)}{\left|\varphi (x)U_{\epsilon }(x)\right|}^p}dx+{\displaystyle {\int }_{B_{2\rho }(0)}{\left|\varphi (x)U_{\epsilon }(x)\right|}^p}dx\\ \hspace{1em}\hspace{1em}={\displaystyle {\int }_{B_{2\rho }(0)}{\left|\varphi (x)U_{\epsilon }(x)\right|}^p}dx\\ \hspace{1em}\hspace{1em}={\displaystyle {\int }_{B_{\rho }(0)}{\left|U_{\epsilon }(x)\right|}^p}dx+{\displaystyle {\int }_{B_{2\rho }(0)\backslash B_{\rho }(0)}{\left|\varphi (x)U_{\epsilon }(x)\right|}^p}dx\\ \hspace{1em}\hspace{1em}:=I_3+I_4.\end{array}$$

According to the definition of $U_{\epsilon }\left(x\right)$, $N=p^2$ and N-dimensional spherical coordinate transformation, we can see

$$\begin{array}{l}{I}_3={(E_N)}^p{\displaystyle {\int }_{B_{\rho }(0)}\frac{{\epsilon }^{\frac{N-P}{(p-1)}}}{{\left({\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}}\right)}^{N-p}}}dx\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={(E_{p^2})}^p{\displaystyle {\int }_{B_{\rho }(0)}\frac{{\epsilon }^p}{{\left({\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}}\right)}^{p^2-p}}}dx\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={(E_{p^2})}^p{\epsilon }^p{\omega }_{p^2}{\displaystyle {\int }_0^{\rho }\frac{r^{p^2-1}}{{\left({\epsilon }^{\frac{p}{p-1}}+r^{\frac{p}{p-1}}\right)}^{p^2-p}}}dr\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{p-1}{p}}{(E_{p^2})}^p{\epsilon }^p{\omega }_{p^2}{\displaystyle {\int }_0^{\rho }\frac{r^{p^2-\frac{p}{p-1}}}{{\left({\epsilon }^{\frac{p}{p-1}}+r^{\frac{p}{p-1}}\right)}^{p^2-p}}}d({\epsilon }^{\frac{p}{p-1}}+r^{\frac{p}{p-1}})\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}:={\displaystyle \frac{p-1}{p}}{(E_{p^2})}^p{\epsilon }^p{\omega }_{p^2}\cdot H.\end{array}$$

Let

$$\begin{array}{c}H={\displaystyle {\int }_0^{\rho }\frac{r^{p^2-\frac{p}{p-1}}}{{\left({\epsilon }^{\frac{p}{p-1}}+r^{\frac{p}{p-1}}\right)}^{p^2-p}}}d({\epsilon }^{\frac{p}{p-1}}+r^{\frac{p}{p-1}})\\ ={\displaystyle {\int }_{{\epsilon }^{\frac{p}{p-1}}}^{{\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}}}\frac{{(Z-{\epsilon }^{\frac{p}{p-1}})}^{p(p-1)-1}}{{\left(Z\right)}^{p^2-p}}}dZ.\end{array}$$

Set $\alpha =p(p-1)-1$, it is obvious that $\alpha$ is a positive integer, $\alpha$ is odd and $\alpha -1$ is even. On the basis of the Binomial Theorem, we have

$$\begin{array}{l}{(Z-{\epsilon }^{\frac{p}{p-1}})}^{p(p-1)-1}={(Z-{\epsilon }^{\frac{p}{p-1}})}^{\alpha }\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \sum _{k=0}^{\alpha }{\left(-1\right)}^k{C}_{\alpha }^k}{Z}^{\alpha -k}{({\epsilon }^{\frac{p}{p-1}})}^k\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=C_{\alpha }^0{Z}^{\alpha }-C_{\alpha }^1{Z}^{\alpha -1}({\epsilon }^{\frac{p}{p-1}})+C_{\alpha }^2{Z}^{\alpha -2}{({\epsilon }^{\frac{p}{p-1}})}^2-C_{\alpha }^3{Z}^{\alpha -3}{({\epsilon }^{\frac{p}{p-1}})}^3+\cdots +C_{\alpha }^{\alpha -1}Z{({\epsilon }^{\frac{p}{p-1}})}^{\alpha -1}-C_{\alpha }^{\alpha }{({\epsilon }^{\frac{p}{p-1}})}^{\alpha }.\end{array}$$

which implies that

$$\begin{array}{l}H={\displaystyle {\int }_{{\epsilon }^{\frac{p}{p-1}}}^{{\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}}}\frac{C_{\alpha }^0{Z}^{\alpha }-C_{\alpha }^1{Z}^{\alpha -1}({\epsilon }^{\frac{p}{p-1}})+C_{\alpha }^2{Z}^{\alpha -2}{({\epsilon }^{\frac{p}{p-1}})}^2-C_{\alpha }^3{Z}^{\alpha -3}{({\epsilon }^{\frac{p}{p-1}})}^3+\cdots +C_{\alpha }^{\alpha -1}Z{({\epsilon }^{\frac{p}{p-1}})}^{\alpha -1}-C_{\alpha }^{\alpha }{({\epsilon }^{\frac{p}{p-1}})}^{\alpha }}{Z^{\alpha +1}}}dZ\\ \hspace{1em}\hspace{1em}={\displaystyle {\int }_{{\epsilon }^{\frac{p}{p-1}}}^{{\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}}}\left[\frac{C_{\alpha }^0}{Z}-\frac{C_{\alpha }^1({\epsilon }^{\frac{p}{p-1}})}{Z^2}+\frac{C_{\alpha }^2{({\epsilon }^{\frac{p}{p-1}})}^2}{Z^3}-\frac{C_{\alpha }^3{({\epsilon }^{\frac{p}{p-1}})}^3}{Z^4}+\cdots +\frac{C_{\alpha }^{\alpha -1}{({\epsilon }^{\frac{p}{p-1}})}^{\alpha -1}}{Z^{\alpha }}-\frac{C_{\alpha }^{\alpha }{({\epsilon }^{\frac{p}{p-1}})}^{\alpha }}{Z^{\alpha +1}}\right]}dZ\\ \hspace{1em}\hspace{1em}=(\ln Z+{\displaystyle \frac{C_{\alpha }^1({\epsilon }^{\frac{p}{p-1}})}{Z}}-{\displaystyle \frac{C_{\alpha }^2{({\epsilon }^{\frac{p}{p-1}})}^2}{2Z^2}}+{\displaystyle \frac{C_{\alpha }^3{({\epsilon }^{\frac{p}{p-1}})}^3}{3Z^3}}+\cdots -{\displaystyle \frac{C_{\alpha }^{\alpha -1}{({\epsilon }^{\frac{p}{p-1}})}^{\alpha -1}}{(\alpha -1)Z^{\alpha -1}}}+{\displaystyle \frac{C_{\alpha }^{\alpha }{({\epsilon }^{\frac{p}{p-1}})}^{\alpha }}{\alpha Z^{\alpha }}})\left|\begin{array}{c}{\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}}\\ {\epsilon }^{\frac{p}{p-1}}\end{array}\right.\\ \hspace{1em}\hspace{1em}={\displaystyle \frac{p}{p-1}}\ln ({\displaystyle \frac{1}{\epsilon }})+\left[\begin{array}{l}\ln ({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})+{\displaystyle \frac{C_{\alpha }^1({\epsilon }^{\frac{p}{p-1}})}{{\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}}}}-{\displaystyle \frac{C_{\alpha }^2{({\epsilon }^{\frac{p}{p-1}})}^2}{2{({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})}^2}}+{\displaystyle \frac{C_{\alpha }^3{({\epsilon }^{\frac{p}{p-1}})}^3}{3{({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})}^3}}+\cdots \\ -{\displaystyle \frac{C_{\alpha }^{\alpha -1}{({\epsilon }^{\frac{p}{p-1}})}^{\alpha -1}}{(\alpha -1){({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})}^{\alpha -1}}}+{\displaystyle \frac{C_{\alpha }^{\alpha }{({\epsilon }^{\frac{p}{p-1}})}^{\alpha }}{\alpha {({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})}^{\alpha }}}\end{array}\right]\\ \hspace{1em}\hspace{1em}-\left[{\displaystyle \frac{C_{\alpha }^1({\epsilon }^{\frac{p}{p-1}})}{{\epsilon }^{\frac{p}{p-1}}}}-{\displaystyle \frac{C_{\alpha }^2{({\epsilon }^{\frac{p}{p-1}})}^2}{2{({\epsilon }^{\frac{p}{p-1}})}^2}}+{\displaystyle \frac{C_{\alpha }^3{({\epsilon }^{\frac{p}{p-1}})}^3}{3{({\epsilon }^{\frac{p}{p-1}})}^3}}+\cdots -{\displaystyle \frac{C_{\alpha }^{\alpha -1}{({\epsilon }^{\frac{p}{p-1}})}^{\alpha }}{(\alpha -1){({\epsilon }^{\frac{p}{p-1}})}^{\alpha }}}+{\displaystyle \frac{C_{\alpha }^{\alpha }{({\epsilon }^{\frac{p}{p-1}})}^{\alpha }}{\alpha {({\epsilon }^{\frac{p}{p-1}})}^{\alpha }}}\right].\end{array}$$

According to Lemma 4 of (2), we have

$$\begin{array}{l}H={\displaystyle \frac{p}{p-1}}\ln ({\displaystyle \frac{1}{\epsilon }})+\left[\begin{array}{l}In({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})+{\displaystyle \frac{C_{\alpha }^1({\epsilon }^{\frac{p}{p-1}})}{{\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}}}}-{\displaystyle \frac{C_{\alpha }^2{({\epsilon }^{\frac{p}{p-1}})}^2}{2{({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})}^2}}+{\displaystyle \frac{C_{\alpha }^3{({\epsilon }^{\frac{p}{p-1}})}^3}{3{({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})}^3}}+\cdots \\ -{\displaystyle \frac{C_{\alpha }^{\alpha -1}{({\epsilon }^{\frac{p}{p-1}})}^{\alpha -1}}{(\alpha -1){({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})}^{\alpha -1}}}+{\displaystyle \frac{C_{\alpha }^{\alpha }{({\epsilon }^{\frac{p}{p-1}})}^{\alpha }}{\alpha {({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})}^{\alpha }}}\end{array}\right]\\ \hspace{1em}\hspace{1em}-{\displaystyle \sum _{k=1}^{\alpha }\frac{1}{k}}.\end{array}$$

Hence, as $\epsilon \to 0$, we get

$$\begin{array}{l}{I}_3={\displaystyle \frac{p-1}{p}}{\epsilon }^p{\omega }_{p^2}{(E_{p^2})}^p·\left[\begin{array}{l}\ln ({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})+{\displaystyle \frac{C_{\alpha }^1({\epsilon }^{\frac{p}{p-1}})}{{\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}}}}-{\displaystyle \frac{C_{\alpha }^2{({\epsilon }^{\frac{p}{p-1}})}^2}{2{({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})}^2}}+{\displaystyle \frac{C_{\alpha }^3{({\epsilon }^{\frac{p}{p-1}})}^3}{3{({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})}^3}}+\cdots \\ -{\displaystyle \frac{C_{\alpha }^{\alpha -1}{({\epsilon }^{\frac{p}{p-1}})}^{\alpha -1}}{(\alpha -1){({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})}^{\alpha -1}}}+{\displaystyle \frac{C_{\alpha }^{\alpha }{({\epsilon }^{\frac{p}{p-1}})}^{\alpha }}{\alpha {({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})}^{\alpha }}}\end{array}\right]\\ \hspace{1em}\hspace{1em}+{(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p\ln ({\displaystyle \frac{1}{\epsilon }})-{\displaystyle \frac{p-1}{p}}{\epsilon }^p{\omega }_{p^2}{(E_{p^2})}^p{\displaystyle \sum _{k=1}^{\alpha }\frac{1}{k}}\\ \hspace{1em}\hspace{1em}={(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p\ln ({\displaystyle \frac{1}{\epsilon }})+O({\epsilon }^p).\end{array}$$

By direct calculation, we know

$$\begin{array}{l}{I}_4={\displaystyle {\int }_{B_{2\rho }(0)\backslash B_{\rho }(0)}{\left|\varphi (x)\right|}^p{\left|U_{\epsilon }(x)\right|}^p}dx\\ \hspace{1em}={\displaystyle {\int }_{B_{2\rho }(0)\backslash B_{\rho }(0)}{\left|\varphi (x)\right|}^p\frac{{\epsilon }^{\frac{N-p}{p-1}}}{{({\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}})}^{N-p}}}dx\\ \hspace{1em}\le {\displaystyle {\int }_{B_{2\rho }(0)\backslash B_{\rho }(0)}\frac{{\epsilon }^{\frac{N-p}{p-1}}}{{\left|x\right|}^{\frac{p(N-p)}{p-1}}}}dx\\ \hspace{1em}={\epsilon }^p{\displaystyle {\int }_{B_{2\rho }(0)\backslash B_{\rho }(0)}\frac{1}{{\left|x\right|}^{p^2}}}dx\\ \hspace{1em}=O({\epsilon }^p).\end{array}$$

So

$${\left|u_{\epsilon }\right|}_p^p=(E_{p^2}{)}^p{\omega }_{p^2}{\epsilon }^p\ln ({\displaystyle \frac{1}{\epsilon }})+O\left({\epsilon }^p\right).$$

□

Lemma 11. 

If $N=p^2,$ as $\epsilon \to 0$, then

$$\begin{array}{l}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_{\epsilon }\right|}^p\ln {\left|u_{\epsilon }\right|}^2}\le 2{\left(E_{p^2}\right)}^p{\omega }_{p^2}{\epsilon }^p\ln {\displaystyle \frac{1}{\epsilon }}\ln \left[{\displaystyle \frac{E_{p^2}{({\epsilon }^{\frac{p}{p-1}}+{(2\rho )}^{\frac{p}{p-1}})}^{\frac{{(p-1)}^2}{p}}{e}^{(2^{\alpha -1})(p-1)-\frac{{(p-1)}^2}{p}{\displaystyle \sum _{k=1}^{\alpha }\frac{1}{k}}}}{{({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})}^{p-1}}}\right]\\ +(p-2){\left(E_{p^2}\right)}^p{\omega }_{p^2}{\epsilon }^p{\left(\ln {\displaystyle \frac{1}{\epsilon }}\right)}^2+O\left({\epsilon }^p\right).\end{array}$$

Proof. 

Similar to Lemma 10, we have

$$\begin{array}{l}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_{\epsilon }\right|}^p\ln {\left|u_{\epsilon }\right|}^2}dx={\displaystyle \frac{2}{p}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_{\epsilon }\right|}^p\ln {\left|u_{\epsilon }\right|}^p}dx\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{2}{p}}{\displaystyle {\int }_{\mathsf{\Omega }\backslash B_{2\rho }(0)}{\left|\varphi (x)U_{\epsilon }(x)\right|}^p\ln {\left|\varphi (x)U_{\epsilon }(x)\right|}^p}dx+{\displaystyle \frac{2}{p}}{\displaystyle {\int }_{B_{2\rho }(0)}{\left|\varphi (x)U_{\epsilon }(x)\right|}^p\ln {\left|\varphi (x)U_{\epsilon }(x)\right|}^p}dx\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{2}{p}}{\displaystyle {\int }_{B_{2\rho }(0)}{\left|\varphi (x)U_{\epsilon }(x)\right|}^p\ln {\left|\varphi (x)U_{\epsilon }(x)\right|}^p}dx\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{2}{p}}{\displaystyle {\int }_{B_{2\rho }(0)}(E_{p^2}{)}^p{\left|\varphi (x)\right|}^p{\left(\frac{{\epsilon }^{\frac{1}{p-1}}}{{\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}}}\right)}^{p^2-p}\ln \left[{\left|\varphi (x)\right|}^p(E_{p^2}{)}^p{\left(\frac{{\epsilon }^{\frac{1}{p-1}}}{{\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}}}\right)}^{p^2-p}\right]}dx\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{2}{p}}(E_{p^2}{)}^p{\displaystyle {\int }_{B_{2\rho }(0)}{\left|\varphi (x)\right|}^p{\left(\frac{{\epsilon }^{\frac{1}{p-1}}}{{\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}}}\right)}^{p^2-p}\ln {\left(\frac{{\epsilon }^{\frac{1}{p-1}}}{{\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}}}\right)}^{p^2-p}}dx\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{2}{p}}(E_{p^2}{)}^p{\displaystyle {\int }_{B_{2\rho }(0)}{\left|\varphi (x)\right|}^p{\left(\frac{{\epsilon }^{\frac{1}{p-1}}}{{\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}}}\right)}^{p^2-p}\ln (E_{p^2}{)}^p}dx\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{2}{p}}(E_{p^2}{)}^p{\displaystyle {\int }_{B_{2\rho }(0)\backslash B_{\rho }(0)}{\left|\varphi (x)\right|}^p{\left(\frac{{\epsilon }^{\frac{1}{p-1}}}{{\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}}}\right)}^{p^2-p}\ln {\left|\varphi (x)\right|}^p}dx\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}:=I_5+I_6+I_7.\end{array}$$

Using logarithmic inequalities in Lemma 2 of (1), we deduce that

$$\begin{array}{l}\left|I_7\right|=\left|{\displaystyle \frac{2}{p}}(E_{p^2}{)}^p{\displaystyle {\int }_{B_{2\rho }(0)\backslash B_{\rho }(0)}{\left|\varphi (x)\right|}^p{\left(\frac{{\epsilon }^{\frac{1}{p-1}}}{{\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}}}\right)}^{p^2-p}\ln {\left|\varphi (x)\right|}^p}dx\right|\hfill \\ \hspace{1em}\le {\displaystyle \frac{2}{pe}}(E_{p^2}{)}^p{\epsilon }^p{\displaystyle {\int }_{B_{2\rho }(0)\backslash B_{\rho }(0)}\frac{1}{{({\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}})}^{p^2-p}}}dx\hfill \\ \hspace{1em}\le {\displaystyle \frac{2}{pe}}(E_{p^2}{)}^p{\epsilon }^p{\displaystyle {\int }_{B_{2\rho }(0)\backslash B_{\rho }(0)}\frac{1}{{\left|x\right|}^p}}dx\hfill \\ \hspace{1em}=O({\epsilon }^p),\hfill \end{array}$$

and we have

$$\begin{array}{l}{I}_6={\displaystyle \frac{2}{p}}\ln (E_{p^2}{)}^p{\displaystyle {\int }_{B_{2\rho }(0)}{(\varphi (x))}^p(E_{p^2}{)}^p{\left(\frac{{\epsilon }^{\frac{1}{p-1}}}{{\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}}}\right)}^{p^2-p}}dx\\ \hspace{1em}={\displaystyle \frac{2}{p}}\ln (E_{p^2}{)}^p·(I_3+I_4)\\ \hspace{1em}={\displaystyle \frac{2}{p}}\left(\ln (E_{p^2}{)}^p\right){\omega }_{p^2}(E_{p^2}{)}^p{\epsilon }^p\ln ({\displaystyle \frac{1}{\epsilon }})+O({\epsilon }^p).\end{array}$$

In addition, by a direct computation, we have

$$\begin{array}{l}{I}_5={\displaystyle \frac{2}{p}}(E_{p^2}{)}^p{\displaystyle {\int }_{B_{2\rho }(0)}{\left|\varphi (x)\right|}^p{\left(\frac{{\epsilon }^{\frac{1}{p-1}}}{{\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}}}\right)}^{p^2-p}\ln {\left(\frac{{\epsilon }^{\frac{1}{p-1}}}{{\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}}}\right)}^{p^2-p}}dx\\ \hspace{1em}={\displaystyle \frac{2}{p}}(E_{p^2}{)}^p{\displaystyle {\int }_{B_{2\rho }(0)}{\left|\varphi (x)\right|}^p\frac{{\epsilon }^p}{{\left({\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}}\right)}^{p^2-p}}\ln }{\displaystyle \frac{1}{{\epsilon }^{p^2-p}{\left(1+{(\frac{\left|x\right|}{\epsilon })}^{\frac{p}{p-1}}\right)}^{p^2-p}}}dx\\ \hspace{1em}={\displaystyle \frac{2}{p}}(E_{p^2}{)}^p(p^2-p)\ln ({\displaystyle \frac{1}{\epsilon }}){\displaystyle {\int }_{B_{2\rho }(0)}{\left|\varphi (x)\right|}^p\frac{{\epsilon }^p}{{\left({\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}}\right)}^{p^2-p}}}dx\\ \hspace{1em}-{\displaystyle \frac{2}{p}}(E_{p^2}{)}^p(p^2-p){\displaystyle {\int }_{B_{2\rho }(0)}{\left|\varphi (x)\right|}^p\frac{{\epsilon }^p}{{\left({\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}}\right)}^{p^2-p}}\ln }\left(1+{(\frac{\left|x\right|}{\epsilon })}^{\frac{p}{p-1}}\right)dx\\ \hspace{1em}:=I_{51}+I_{52}.\end{array}$$

So, we have

$$\begin{array}{l}{I}_{51}={\displaystyle \frac{2}{p}}(E_{p^2}{)}^p(p^2-p)\ln ({\displaystyle \frac{1}{\epsilon }}){\displaystyle {\int }_{B_{2\rho }(0)}{\left|\varphi (x)\right|}^p\frac{{\epsilon }^p}{{\left({\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}}\right)}^{p^2-p}}}dx\\ \hspace{1em}\hspace{1em}\le {\displaystyle \frac{2}{p}}(E_{p^2}{)}^p(p^2-p)\ln ({\displaystyle \frac{1}{\epsilon }}){\displaystyle {\int }_{B_{2\rho }(0)}\frac{{\epsilon }^p}{{\left({\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}}\right)}^{p^2-p}}}dx\\ \hspace{1em}\hspace{1em}={\displaystyle \frac{2}{p}}(E_{p^2}{)}^p(p^2-p)\ln ({\displaystyle \frac{1}{\epsilon }}){\displaystyle \frac{p-1}{p}}{\epsilon }^p{\omega }_{p^2}{\displaystyle {\int }_0^{2\rho }\frac{r^{p^2-\frac{p}{p-1}}}{{\left({\epsilon }^{\frac{p}{p-1}}+r^{\frac{p}{p-1}}\right)}^{p^2-p}}}d({\epsilon }^{\frac{p}{p-1}}+r^{\frac{p}{p-1}})\\ \hspace{1em}\hspace{1em}={\displaystyle \frac{2}{p}}{(E_{p^2})}^p(p^2-p)\ln ({\displaystyle \frac{1}{\epsilon }}){\displaystyle \frac{p-1}{p}}{\epsilon }^p{\omega }_{p^2}·{\displaystyle \frac{p}{p-1}}\ln ({\displaystyle \frac{1}{\epsilon }})+{\displaystyle \frac{2}{p}}{(E_{p^2})}^p(p^2-p)\ln ({\displaystyle \frac{1}{\epsilon }}){\displaystyle \frac{p-1}{p}}{\epsilon }^p{\omega }_{p^2}·\\ \hspace{1em}\hspace{1em}\left[\begin{array}{l}\ln ({\epsilon }^{\frac{p}{p-1}}+{(2\rho )}^{\frac{p}{p-1}})+{\displaystyle \frac{C_{\alpha }^1({\epsilon }^{\frac{p}{p-1}})}{{\epsilon }^{\frac{p}{p-1}}+{(2\rho )}^{\frac{p}{p-1}}}}-{\displaystyle \frac{C_{\alpha }^2{({\epsilon }^{\frac{p}{p-1}})}^2}{2{({\epsilon }^{\frac{p}{p-1}}+{(2\rho )}^{\frac{p}{p-1}})}^2}}+{\displaystyle \frac{C_{\alpha }^3{({\epsilon }^{\frac{p}{p-1}})}^3}{3{({\epsilon }^{\frac{p}{p-1}}+{(2\rho )}^{\frac{p}{p-1}})}^3}}+\cdots \\ -{\displaystyle \frac{C_{\alpha }^{\alpha -1}{({\epsilon }^{\frac{p}{p-1}})}^{\alpha -1}}{(\alpha -1){({\epsilon }^{\frac{p}{p-1}}+{(2\rho )}^{\frac{p}{p-1}})}^{\alpha -1}}}+{\displaystyle \frac{C_{\alpha }^{\alpha }{({\epsilon }^{\frac{p}{p-1}})}^{\alpha }}{\alpha {({\epsilon }^{\frac{p}{p-1}}+{(2\rho )}^{\frac{p}{p-1}})}^{\alpha }}}\end{array}\right]\\ \hspace{1em}\hspace{1em}-{\displaystyle \frac{2}{p}}{(E_{p^2})}^p(p^2-p)\ln ({\displaystyle \frac{1}{\epsilon }}){\displaystyle \frac{p-1}{p}}{\epsilon }^p{\omega }_{p^2}·{\displaystyle \sum _{k=1}^{\alpha }\frac{1}{k}}\\ \hspace{1em}\hspace{1em}=2(p-1){(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p{(\ln {\displaystyle \frac{1}{\epsilon }})}^2+2{\displaystyle \frac{(p-1{)}^2}{p}}{(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p\ln ({\displaystyle \frac{1}{\epsilon }})\ln ({\displaystyle \frac{{\epsilon }^{\frac{p}{p-1}}+{(2\rho )}^{\frac{p}{p-1}}}{e^{{\displaystyle \sum _{k=1}^{\alpha }\frac{1}{k}}}}})\\ \hspace{1em}\hspace{1em}+O\left({\epsilon }^{p+\frac{p}{p-1}}\ln ({\displaystyle \frac{1}{\epsilon }})\right).\end{array}$$

and

$$\begin{array}{l}{I}_{52}=-{\displaystyle \frac{2}{p}}(E_{p^2}{)}^p(p^2-p){\displaystyle {\int }_{B_{2\rho }(0)}{\left|\varphi (x)\right|}^p\frac{{\epsilon }^p}{{\left({\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}}\right)}^{p^2-p}}\ln }\left(1+{(\frac{\left|x\right|}{\epsilon })}^{\frac{p}{p-1}}\right)dx\\ \hspace{1em}\hspace{1em}\le -{\displaystyle \frac{2}{p}}(E_{p^2}{)}^p(p^2-p){\displaystyle {\int }_{B_{\rho }(0)}{\left|\varphi (x)\right|}^p\frac{{\epsilon }^p}{{\left({\epsilon }^{\frac{p}{p-1}}+{\left|x\right|}^{\frac{p}{p-1}}\right)}^{p^2-p}}\ln }\left(1+{(\frac{\left|x\right|}{\epsilon })}^{\frac{p}{p-1}}\right)dx\\ \hspace{1em}\hspace{1em}=-{\displaystyle \frac{2}{p}}(E_{p^2}{)}^p(p^2-p){\displaystyle \frac{p-1}{p}}{\omega }_{p^2}{\epsilon }^p{\displaystyle {\int }_0^{\frac{\rho }{\epsilon }}\frac{r^{p^2-\frac{p}{p-1}}}{{\left(1+r^{\frac{p}{p-1}}\right)}^{p^2-p}}\ln (1+r^{\frac{p}{p-1}})d(1+r^{\frac{p}{p-1}})}\\ \hspace{1em}\hspace{1em}=-{\displaystyle \frac{2{(p-1)}^2}{p}}(E_{p^2}{)}^p{\omega }_{p^2}{\epsilon }^p{\displaystyle {\int }_0^{\frac{\rho }{\epsilon }}\begin{array}{l}\left[\frac{1}{1+r^{\frac{p}{p-1}}}-\frac{C_{\alpha }^1}{{(1+r^{\frac{p}{p-1}})}^2}+\frac{C_{\alpha }^2}{{(1+r^{\frac{p}{p-1}})}^3}-\frac{C_{\alpha }^3}{{(1+r^{\frac{p}{p-1}})}^4}+\cdots +\frac{C_{\alpha }^{\alpha -1}}{{(1+r^{\frac{p}{p-1}})}^{\alpha }}-\frac{C_{\alpha }^{\alpha }}{{(1+r^{\frac{p}{p-1}})}^{\alpha +1}}\right]\\ ·\ln (1+r^{\frac{p}{p-1}})d(1+r^{\frac{p}{p-1}})\end{array}}\\ \hspace{1em}\hspace{1em}\le -{\displaystyle \frac{2{(p-1)}^2}{p}}(E_{p^2}{)}^p{\omega }_{p^2}{\epsilon }^p{\displaystyle {\int }_0^{\frac{\rho }{\epsilon }}\left[\frac{1}{1+r^{\frac{p}{p-1}}}-\frac{C_{\alpha }^1}{{(1+r^{\frac{p}{p-1}})}^2}-\frac{C_{\alpha }^3}{{(1+r^{\frac{p}{p-1}})}^4}+\cdots -\frac{C_{\alpha }^{\alpha }}{{(1+r^{\frac{p}{p-1}})}^{\alpha +1}}\right]\ln (1+r^{\frac{p}{p-1}})d(1+r^{\frac{p}{p-1}})}\\ \hspace{1em}\hspace{1em}\le -{\displaystyle \frac{2{(p-1)}^2}{p}}(E_{p^2}{)}^p{\omega }_{p^2}{\epsilon }^p{\displaystyle {\int }_0^{\frac{\rho }{\epsilon }}\left[\frac{1}{1+r^{\frac{p}{p-1}}}\ln (1+r^{\frac{p}{p-1}})-\frac{\left(C_{\alpha }^1+C_{\alpha }^3+C_{\alpha }^5\cdots +C_{\alpha }^{\alpha -2}+C_{\alpha }^{\alpha }\right)}{1+r^{\frac{p}{p-1}}}\right]d(1+r^{\frac{p}{p-1}})}\\ \hspace{1em}\hspace{1em}=-{\displaystyle \frac{2{(p-1)}^2}{p}}(E_{p^2}{)}^p{\omega }_{p^2}{\epsilon }^p\left[{\displaystyle \frac{1}{2}}{(\ln (1+r^{\frac{p}{p-1}}))}^2-{\displaystyle \frac{2^{\alpha }}{2}}\ln (1+r^{\frac{p}{p-1}})\right]\left|\begin{array}{c}\frac{\rho }{\epsilon }\\ 0\end{array}\right.\\ \hspace{1em}\hspace{1em}=-{\displaystyle \frac{2{(p-1)}^2}{p}}(E_{p^2}{)}^p{\omega }_{p^2}{\epsilon }^p\left[{\displaystyle \frac{1}{2}}{(\ln ({\displaystyle \frac{{\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}}}{{\epsilon }^{\frac{p}{p-1}}}}))}^2-{\displaystyle \frac{2^{\alpha }}{2}}\ln ({\displaystyle \frac{{\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}}}{{\epsilon }^{\frac{p}{p-1}}}})\right]\\ \hspace{1em}\hspace{1em}=-{\displaystyle \frac{2{(p-1)}^2}{p}}(E_{p^2}{)}^p{\omega }_{p^2}{\epsilon }^p\left\{{\displaystyle \frac{1}{2}}{\left[\ln ({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})+{\displaystyle \frac{p}{p-1}}\ln {\displaystyle \frac{1}{\epsilon }}\right]}^2-{\displaystyle \frac{2^{\alpha }}{2}}\left[\ln ({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})+{\displaystyle \frac{p}{p-1}}\ln {\displaystyle \frac{1}{\epsilon }}\right]\right\}\\ \hspace{1em}\hspace{1em}=-p(E_{p^2}{)}^p{\omega }_{p^2}{\epsilon }^p{(\ln {\displaystyle \frac{1}{\epsilon }})}^2-2(p-1)(E_{p^2}{)}^p{\omega }_{p^2}{\epsilon }^p\ln {\displaystyle \frac{1}{\epsilon }}\ln ({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})\\ \hspace{1em}\hspace{1em}-{\displaystyle \frac{{(p-1)}^2}{p}}(E_{p^2}{)}^p{\omega }_{p^2}{\epsilon }^p{\left[\ln ({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})\right]}^2+{\displaystyle \frac{2^{\alpha }{(p-1)}^2}{p}}{\omega }_{p^2}(E_{p^2}{)}^p{\epsilon }^p\ln ({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})\\ \hspace{1em}\hspace{1em}+2^{\alpha }(p-1){\omega }_{p^2}(E_{p^2}{)}^p{\epsilon }^p\ln {\displaystyle \frac{1}{\epsilon }}\\ \hspace{1em}\hspace{1em}=-p(E_{p^2}{)}^p{\omega }_{p^2}{\epsilon }^p{(\ln {\displaystyle \frac{1}{\epsilon }})}^2-2(p-1)(E_{p^2}{)}^p{\omega }_{p^2}{\epsilon }^p\ln {\displaystyle \frac{1}{\epsilon }}\ln {\displaystyle \frac{{\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}}}{e^{2^{\alpha -1}}}}+O({\epsilon }^p).\end{array}$$

Hence

$$\begin{array}{l}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_{\epsilon }\right|}^p\ln {\left|u_{\epsilon }\right|}^2}dx\le I_{51}+I_{52}+I_6+I_7\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=2(p-1){(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p{(\ln {\displaystyle \frac{1}{\epsilon }})}^2+2{\displaystyle \frac{{(p-1)}^2}{p}}{(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p\ln {\displaystyle \frac{1}{\epsilon }}\ln ({\displaystyle \frac{{\epsilon }^{\frac{p}{p-1}}+{(2\rho )}^{\frac{p}{p-1}}}{e^{{\displaystyle \sum _{k=1}^{\alpha }\frac{1}{k}}}}})+O({\epsilon }^{p+\frac{p}{p-1}}\ln {\displaystyle \frac{1}{\epsilon }})\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-p{(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p{(\ln {\displaystyle \frac{1}{\epsilon }})}^2-2(p-1){(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p\ln {\displaystyle \frac{1}{\epsilon }}\ln ({\displaystyle \frac{{\epsilon }^{\frac{p}{p-1}}+{(\rho )}^{\frac{p}{p-1}}}{e^{2^{\alpha -1}}}})+O({\epsilon }^p)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{2}{p}}{(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p\ln {\displaystyle \frac{1}{\epsilon }}\ln {(E_{p^2})}^p+O({\epsilon }^p)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=2{(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p\ln {\displaystyle \frac{1}{\epsilon }}·\ln \left[{\displaystyle \frac{E_{p^2}{({\epsilon }^{\frac{p}{p-1}}+{(2\rho )}^{\frac{p}{p-1}})}^{\frac{{(p-1)}^2}{p}}{e}^{(2^{\alpha -1})(p-1)-\frac{{(p-1)}^2}{p}{\displaystyle \sum _{k=1}^{\alpha }\frac{1}{k}}}}{{({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})}^{p-1}}}\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+(p-2){(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p{(\ln {\displaystyle \frac{1}{\epsilon }})}^2+O({\epsilon }^p).\end{array}$$

□

Lemma 12.

If $N=p^2$, $\lambda \in A_0\cup B_0$, and 

$${\displaystyle \frac{{(E_{p^2})}^2{e}^{(2^{\alpha }+1)(p-1)-(\lambda +\frac{2}{p})-\frac{2{(p-1)}^2}{p}{\displaystyle \sum _{k=1}^{\alpha }\frac{1}{k}}}{(1+2^{\frac{p}{p-1}})}^{\frac{2{(p-1)}^2}{p}}}{{\rho }_{\max }^2}}<1$$

are satisfied, where $\alpha =p(p-1)-1$ and ${\rho }_{\max }:=\sup \left\{r>0:\exists x\in \mathsf{\Omega },st\hspace{0.33em}B(x,r)\subset \mathsf{\Omega }\right\}$, then $\underset{t\ge 0}{\sup }J\left(t{u}_{\epsilon }\right)\le {\displaystyle \frac{p^{\ast }-\theta }{p^{\ast }\theta }}{\left(m_{0}S\right)}^{\frac{N}{p}}$ holds.

Proof. 

Since $\underset{t\to 0}{\lim }{\varphi }_{u_{\epsilon }}(t)=0$, $\underset{t\to +\infty }{\lim }{\varphi }_{u_{\epsilon }}(t)=-\infty$, where $u_{\epsilon }$ is the value given in Lemma 9, this indicates the existence of a $t_{\epsilon }\in (0,+\infty )$, such that

$${\varphi }_{u_{\epsilon }}(t_{\epsilon })=\underset{t\ge 0}{\sup }{\varphi }_{u_{\epsilon }}(t).$$

(40)

According to the definition of fiber mapping, we have

$${\varphi }_{u_{\epsilon }}(t_{\epsilon })=J(t_{\epsilon }{u}_{\epsilon })=\underset{t\ge 0}{\sup }{\varphi }_{u_{\epsilon }}(t)=\underset{t\ge 0}{\sup }J(t{u}_{\epsilon }).$$

(41)

And when $\epsilon$ is sufficiently small, the following shows that $t_{\epsilon }$ is bounded.

Taking the derivative of ${\varphi }_{u_{\epsilon }}(t_{\epsilon })$ with respect to $t_{\epsilon }$, we can get

$$M(t_{\epsilon }^p{‖u_{\epsilon }‖}^p){‖u_{\epsilon }‖}^p-\lambda {\left|u_{\epsilon }\right|}_p^p+{\displaystyle {\int }_{\mathsf{\Omega }}{u}_{\epsilon }^p}\ln u_{\epsilon }^{2}dx-t_{\epsilon }^{p^{\ast }-p}{\left|u_{\epsilon }\right|}_{p^{\ast }}^{p^{\ast }}+\ln t_{\epsilon }^2{\left|u_{\epsilon }\right|}_p^p=0.$$

(42)

With the help of assumptions ${\mathrm{M}}_1$, estimates and Lemma 2 of (2) take $\sigma =\frac{p^{\ast }-p}{2}>0$, we have

$$\ln t_{\epsilon }^2\le {\displaystyle \frac{2}{e(p^{\ast }-p)}}{t}_{\epsilon }^{p^{\ast }-p} :=C t_{\epsilon }^{p^{\ast }-p} .$$

(43)

and according to $t^p\ln t\ge -{\displaystyle \frac{1}{ep}} , \forall t>0,p\ge 1$, we get

$$-\ln t_{\epsilon }^2\le {\displaystyle \frac{2}{ep}}{t}_{\epsilon }^{-p} :=C t_{\epsilon }^{-p} .$$

(44)

By simple derivation, there are

$$\begin{array}{l}{m}_1{S}^{\frac{N}{p}}\ge M(t_{\epsilon }^p{‖u_{\epsilon }‖}^p){‖u_{\epsilon }‖}^p-\lambda {\left|u_{\epsilon }\right|}_p^p+{\displaystyle {\int }_{\mathsf{\Omega }}{u}_{\epsilon }^p}\ln u_{\epsilon }^{2}dx\\ \hspace{1em}\hspace{1em}=t_{\epsilon }^{p^{\ast }-p}{\left|u_{\epsilon }\right|}_{p^{\ast }}^{p^{\ast }}-\ln t_{\epsilon }^2{\left|u_{\epsilon }\right|}_p^p\\ \hspace{1em}\hspace{1em}\ge t_{\epsilon }^{p^{\ast }-p}(\frac{1}{2}{S}^{\frac{N}{p}})-C{t}_{\epsilon }^{p^{\ast }-p},\end{array}$$

The above formula shows that $t_{\epsilon }$ has an upper bound.

Similarly, we have

$$\begin{array}{l}{m}_0{S}^{\frac{N}{p}}\le M(t_{\epsilon }^p{‖u_{\epsilon }‖}^p){‖u_{\epsilon }‖}^p-\lambda {\left|u_{\epsilon }\right|}_p^p+{\displaystyle {\int }_{\mathsf{\Omega }}{u}_{\epsilon }^p}\ln u_{\epsilon }^{2}dx\\ \hspace{1em}\hspace{1em}=t_{\epsilon }^{p^{\ast }-p}{\left|u_{\epsilon }\right|}_{p^{\ast }}^{p^{\ast }}-\ln t_{\epsilon }^2{\left|u_{\epsilon }\right|}_p^p\\ \hspace{1em}\hspace{1em}\le t_{\epsilon }^{p^{\ast }-p}(2S^{\frac{N}{p}})+C{t}_{\epsilon }^{-p},\end{array}$$

The above formula shows that $t_{\epsilon }$ has a lower bound. Therefore, $t_{\epsilon }$ is a bounded.

Suppose we take $\rho ={\rho }_{\max }$, set $\epsilon <\rho$, and when $\epsilon$ is sufficiently small, there is a $\delta <{\displaystyle \frac{\theta }{p}}$ such that

$$\begin{array}{l}\underset{t\ge 0}{\sup }J(t{u}_{\epsilon })={\displaystyle \frac{1}{p}}\widehat{M}\left(t_{\epsilon }^p{‖u_{\epsilon }‖}^p\right)-{\displaystyle \frac{t_{\epsilon }^{p^{\ast }}}{p^{\ast }}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_{\epsilon }\right|}^{p^{\ast }}}dx+{\displaystyle \frac{t_{\epsilon }^p}{p}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_{\epsilon }\right|}^p}\left[\ln (t_{\epsilon }^2{u}_{\epsilon }^2)-\lambda -{\displaystyle \frac{2}{p}}\right]dx\\ \hspace{1em}\hspace{1em}\le {\displaystyle \frac{\delta }{p}}{m}_0{t}_{\epsilon }^p{‖u_{\epsilon }‖}^p-{\displaystyle \frac{t_{\epsilon }^{p^{\ast }}}{p^{\ast }}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_{\epsilon }\right|}^{p^{\ast }}}dx+{\displaystyle \frac{t_{\epsilon }^p}{p}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_{\epsilon }\right|}^p}\left[\ln (u_{\epsilon }^2)-\lambda -{\displaystyle \frac{2}{p}}\right]dx+{\displaystyle \frac{1}{p}}{t}_{\epsilon }^p\ln (t_{\epsilon }^2){\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_{\epsilon }\right|}^{p}dx}.\end{array}$$

Let

$$h(t_{\epsilon })={\displaystyle \frac{\delta }{p}}{m}_0{t}_{\epsilon }^p{‖u_{\epsilon }‖}^p-{\displaystyle \frac{1}{p^{\ast }}}{t}_{\epsilon }^{p^{\ast }}{\left|u_{\epsilon }\right|}_{p^{\ast }}^{p^{\ast }}.$$

and by taking the derivative, we let

$$h^{\prime }(t_{\epsilon })=\delta m_0{t}_{\epsilon }^{p-1}{‖u_{\epsilon }‖}^p-t_{\epsilon }^{p^{\ast }-1}{\left|u_{\epsilon }\right|}_{p^{\ast }}^{p^{\ast }}=0.$$

After calculation, we can see $h(t_{\epsilon })$ gets the maximum value at $t_{\epsilon }^{\ast }$ and

$$t_{\epsilon }^{\ast }=t_{\epsilon }={\left({\displaystyle \frac{\delta m_0{‖u_{\epsilon }‖}^p}{{\left|u_{\epsilon }\right|}_{p^{\ast }}^{p^{\ast }}}}\right)}^{{\displaystyle \frac{N-p}{p^2}}}.$$

Combine the definition of $S$ and take $\delta ={\left[(\frac{1}{\theta }-\frac{1}{p^{\ast }})p^2\right]}^{\frac{p}{N}}$, we know

$$h(t_{\epsilon }^{\ast })=(\frac{1}{p}-\frac{1}{p^{\ast }}){(\delta m_0)}^{\frac{N}{p}}{S}^{\frac{N}{p}}={\displaystyle \frac{1}{N}}{(\delta m_0)}^{\frac{N}{p}}{S}^{\frac{N}{p}}={\displaystyle \frac{1}{N}}\left[(\frac{1}{\theta }-\frac{1}{p^{\ast }})N\right]{(m_0)}^{\frac{N}{p}}{S}^{\frac{N}{p}}={\displaystyle \frac{p^{\ast }-\theta }{\theta p^{\ast }}}{(m_{0}S)}^{\frac{N}{p}}.$$

So, we get

$$\begin{array}{l}\underset{t\ge 0}{\sup }J(t{u}_{\epsilon })\le h\left(t_{\epsilon }^{\ast }\right)+{\displaystyle \frac{t_{\epsilon }^p}{p}}{\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_{\epsilon }\right|}^p}\left[\ln (u_{\epsilon }^2)-\lambda -{\displaystyle \frac{2}{p}}\right]dx+{\displaystyle \frac{1}{p}}{t}_{\epsilon }^p\ln (t_{\epsilon }^2){\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_{\epsilon }\right|}^{p}dx}\\ \hspace{1em}={\displaystyle \frac{p^{\ast }-\theta }{\theta p^{\ast }}}{(m_{0}S)}^{\frac{N}{p}}+{\displaystyle \frac{t_{\epsilon }^p}{p}}{\displaystyle {\int }_{\mathsf{\Omega }}\left[{\left|u_{\epsilon }\right|}^p\ln (u_{\epsilon }^2)-(\lambda +\frac{2}{p}){\left|u_{\epsilon }\right|}^p\right]}dx+{\displaystyle \frac{1}{p}}{t}_{\epsilon }^p\ln (t_{\epsilon }^2){\displaystyle {\int }_{\mathsf{\Omega }}{\left|u_{\epsilon }\right|}^{p}dx}\\ \hspace{1em}\le {\displaystyle \frac{p^{\ast }-\theta }{\theta p^{\ast }}}{(m_{0}S)}^{\frac{N}{p}}+{\displaystyle \frac{t_{\epsilon }^p}{p}}(p-2){(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p{(\ln \frac{1}{\epsilon })}^2+{\displaystyle \frac{t_{\epsilon }^p}{p}}{(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p(\ln \frac{1}{\epsilon })\\ \hspace{1em}·\ln {\displaystyle \frac{(E_{p^2}{)}^2{e}^{2^{\alpha }\left(p-1\right)-\frac{2{\left(p-1\right)}^2}{p}{\displaystyle \sum _{k=1}^{\alpha }\frac{1}{k}}}{({\epsilon }^{\frac{p}{p-1}}+{(2\rho )}^{\frac{p}{p-1}})}^{\frac{2{(p-1)}^2}{p}}}{{({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})}^{2(p-1)}}}-{\displaystyle \frac{t_{\epsilon }^p}{p}}(\lambda +\frac{2}{p}){(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p(\ln \frac{1}{\epsilon })\\ \hspace{1em}+O({\epsilon }^p)+{\displaystyle \frac{1}{p}}{t}_{\epsilon }^p\ln (t_{\epsilon }^2){(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p(\ln \frac{1}{\epsilon })\\ \hspace{1em}\le {\displaystyle \frac{p^{\ast }-\theta }{\theta p^{\ast }}}{(m_{0}S)}^{\frac{N}{p}}+{\displaystyle \frac{C}{p}}(p-2){(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p{(\ln \frac{1}{\epsilon })}^2+{\displaystyle \frac{C}{p}}{(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p(\ln \frac{1}{\epsilon })\\ \hspace{1em}·\ln {\displaystyle \frac{(E_{p^2}{)}^2{e}^{2^{\alpha }\left(p-1\right)-(\lambda +\frac{2}{p})-\frac{2{\left(p-1\right)}^2}{p}{\displaystyle \sum _{k=1}^{\alpha }\frac{1}{k}}}{({\epsilon }^{\frac{p}{p-1}}+{(2\rho )}^{\frac{p}{p-1}})}^{\frac{2{(p-1)}^2}{p}}}{{({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})}^{2(p-1)}}}+O({\epsilon }^p)+{\displaystyle \frac{C}{p}}{(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p(\ln \frac{1}{\epsilon })\\ \hspace{1em}={\displaystyle \frac{p^{\ast }-\theta }{\theta p^{\ast }}}{(m_{0}S)}^{\frac{N}{p}}+{\displaystyle \frac{C}{p}}(p-2){(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p{(\ln \frac{1}{\epsilon })}^2+{\displaystyle \frac{C}{p}}{(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p(\ln \frac{1}{\epsilon })\\ \hspace{1em}·\ln {\displaystyle \frac{(E_{p^2}{)}^2{e}^{2^{\alpha }\left(p-1\right)-(\lambda +\frac{2}{p})+1-\frac{2{\left(p-1\right)}^2}{p}{\displaystyle \sum _{k=1}^{\alpha }\frac{1}{k}}}{({\epsilon }^{\frac{p}{p-1}}+{(2\rho )}^{\frac{p}{p-1}})}^{\frac{2{(p-1)}^2}{p}}}{{({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})}^{2(p-1)}}}+O({\epsilon }^p).\end{array}$$

The last inequality of the above equation is based on the following facts. That is, by virtue of the boundedness of $t_{\epsilon }$, we may as well set $t_{\epsilon }^p\le C_1$, $\ln (t_{\epsilon }^2)\le C_2$ and $C=\max \left\{C_1,C_1{C}_2\right\}$, where $C_1,C_2,C$ is a positive constant.

So, we have

$$\begin{array}{c}\frac{1}{p}{t}_{\epsilon }^p\ln (t_{\epsilon }^2){{\displaystyle {\int }_{\mathsf{\Omega }}\left|u_{\epsilon }\right|}}^{p}dx=\frac{1}{p}{t}_{\epsilon }^p\ln (t_{\epsilon }^2)\left[{(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p\ln \frac{1}{\epsilon }+O({\epsilon }^p)\right]\\ \le \frac{C_1{C}_2}{p}{(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p\ln \frac{1}{\epsilon }+O({\epsilon }^p)\\ \le \frac{C}{p}{(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p\ln \frac{1}{\epsilon }+O({\epsilon }^p).\end{array}$$

In addition, because of $\epsilon <\rho$, there is

$$\begin{array}{c}{\displaystyle \frac{{({\epsilon }^{\frac{p}{p-1}}+{(2\rho )}^{\frac{p}{p-1}})}^{\frac{2{(p-1)}^2}{p}}}{({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}}{)}^{2(p-1)}}}\le {\displaystyle \frac{{({\epsilon }^{\frac{p}{p-1}}+{(2\rho )}^{\frac{p}{p-1}})}^{\frac{2{(p-1)}^2}{p}}}{{\rho }^p}}\le {\displaystyle \frac{{({\rho }^{\frac{p}{p-1}}+{(2\rho )}^{\frac{p}{p-1}})}^{\frac{2{(p-1)}^2}{p}}}{{\rho }^p}}\\ ={\displaystyle \frac{{(1+2^{\frac{p}{p-1}})}^{\frac{2{(p-1)}^2}{p}}{\rho }^{2(p-1)}}{{\rho }^{2p}}}={\displaystyle \frac{{(1+2^{\frac{p}{p-1}})}^{\frac{2{(p-1)}^2}{p}}}{{\rho }^2}}.\end{array}$$

(45)

and according to the $O$ and $o$ algorithm, we have then

(1)

If $f(x)=O\left(\varphi \right),\varphi =o(\psi ),$ then $f(x)=o(\psi )$;

(2)

If $f(x)=O\left(\varphi \right),\varphi =O(\psi ),$ then $f(x)=O(\psi )$;

(3)

$O\left(f\right)+o(f)=O\left(f\right)$.

We know

$$\underset{\epsilon \to 0}{\lim }{\displaystyle \frac{{\epsilon }^p\ln \frac{1}{\epsilon }}{{\epsilon }^p}}=+\infty \iff {\epsilon }^p=o({\epsilon }^p\ln \frac{1}{\epsilon }),$$

that is

$$f(x)=O\left({\epsilon }^p\right),{\epsilon }^p=o({\epsilon }^p\ln \frac{1}{\epsilon })\Rightarrow f(x)=O\left({\epsilon }^p\right)=o({\epsilon }^p\ln \frac{1}{\epsilon }).$$

In the same way, we have

$$\underset{\epsilon \to 0}{\lim }{\displaystyle \frac{{\epsilon }^p(\ln \frac{1}{\epsilon }{)}^2}{{\epsilon }^p}}=+\infty \iff {\epsilon }^p=o({\epsilon }^p(\ln \frac{1}{\epsilon }{)}^2),$$

that is

$$f(x)=O\left({\epsilon }^p\right),{\epsilon }^p=o\left({\epsilon }^p{\left(\ln \frac{1}{\epsilon }\right)}^2\right)\Rightarrow f(x)=O\left({\epsilon }^p\right)=o\left({\epsilon }^p{\left(\ln \frac{1}{\epsilon }\right)}^2\right).$$

Therefore, when $\epsilon >0$ is sufficiently small, combining (45) and ${\epsilon }^p\ln \frac{1}{\epsilon }=o\left({\epsilon }^p{\left(\ln \frac{1}{\epsilon }\right)}^2\right)$, we obtain

$$\begin{array}{l}\underset{t\ge 0}{\sup }J(t{u}_{\epsilon })\le {\displaystyle \frac{p^{\ast }-\theta }{\theta p^{\ast }}}{(m_{0}S)}^{\frac{N}{p}}+{\displaystyle \frac{C}{p}}(p-2){(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p{(\ln \frac{1}{\epsilon })}^2+{\displaystyle \frac{C}{p}}{(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p(\ln \frac{1}{\epsilon })\\ \hspace{1em}·\ln {\displaystyle \frac{(E_{p^2}{)}^2{e}^{2^{\alpha }\left(p-1\right)-(\lambda +\frac{2}{p})+1-\frac{2{\left(p-1\right)}^2}{p}{\displaystyle \sum _{k=1}^{\alpha }\frac{1}{k}}}{({\epsilon }^{\frac{p}{p-1}}+{(2\rho )}^{\frac{p}{p-1}})}^{\frac{2{(p-1)}^2}{p}}}{{({\epsilon }^{\frac{p}{p-1}}+{\rho }^{\frac{p}{p-1}})}^{2(p-1)}}}+O({\epsilon }^p)\\ \hspace{1em}\le {\displaystyle \frac{p^{\ast }-\theta }{\theta p^{\ast }}}{(m_{0}S)}^{\frac{N}{p}}+{\displaystyle \frac{C}{p}}(p-2){(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p{(\ln \frac{1}{\epsilon })}^2+{\displaystyle \frac{C}{p}}{(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p(\ln \frac{1}{\epsilon })\\ \hspace{1em}·\ln {\displaystyle \frac{(E_{p^2}{)}^2{e}^{2^{\alpha }\left(p-1\right)-(\lambda +\frac{2}{p})+1-\frac{2{\left(p-1\right)}^2}{p}{\displaystyle \sum _{k=1}^{\alpha }\frac{1}{k}}}{(1+2^{\frac{p}{p-1}})}^{\frac{2{(p-1)}^2}{p}}}{{\rho }^2}}+O({\epsilon }^p)\\ \hspace{1em}\le {\displaystyle \frac{p^{\ast }-\theta }{\theta p^{\ast }}}{(m_{0}S)}^{\frac{N}{p}}+{\displaystyle \frac{C}{p}}(p-2){(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p{(\ln \frac{1}{\epsilon })}^2+{\displaystyle \frac{C}{p}}{(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p(\ln \frac{1}{\epsilon }{)}^2\\ \hspace{1em}·\ln {\displaystyle \frac{(E_{p^2}{)}^2{e}^{2^{\alpha }\left(p-1\right)-(\lambda +\frac{2}{p})+1-\frac{2{\left(p-1\right)}^2}{p}{\displaystyle \sum _{k=1}^{\alpha }\frac{1}{k}}}{(1+2^{\frac{p}{p-1}})}^{\frac{2{(p-1)}^2}{p}}}{{\rho }^2}}+O({\epsilon }^p)\\ \hspace{1em}={\displaystyle \frac{p^{\ast }-\theta }{\theta p^{\ast }}}{(m_{0}S)}^{\frac{N}{p}}+{\displaystyle \frac{C}{p}}{(E_{p^2})}^p{\omega }_{p^2}{\epsilon }^p(\ln \frac{1}{\epsilon }{)}^2\\ \hspace{1em}·\ln {\displaystyle \frac{(E_{p^2}{)}^2{e}^{(2^{\alpha }+1)\left(p-1\right)-(\lambda +\frac{2}{p})-\frac{2{\left(p-1\right)}^2}{p}{\displaystyle \sum _{k=1}^{\alpha }\frac{1}{k}}}{(1+2^{\frac{p}{p-1}})}^{\frac{2{(p-1)}^2}{p}}}{{\rho }^2}}+O({\epsilon }^p).\end{array}$$

From the given assumptions, it is not difficult to find that when $\epsilon >0$ is sufficiently small, there exists a function $u^{\ast }=u_{\epsilon }\in W_0^{1,p}(\mathsf{\Omega })\backslash \left\{0\right\}$ such that

$$\underset{t\ge 0}{\sup }J(t{u}_{\epsilon })\le {\displaystyle \frac{p^{\ast }-\theta }{\theta p^{\ast }}}{(m_{0}S)}^{\frac{N}{p}}.$$

Therefore, we have completed the proof. □

5. Conclusions

In this section, we will use the knowledge of variational method, combined with the preparatory work of the third and fourth parts, summarize the proof process of Theorem 1, and fully show that the proof process of Theorem 1 is complete.

According to Lemma 9, Lemma 10, Lemma 11 and Lemma 12, we show that under the appropriate assumptions of Theorem 1 condition ${(PS)}_c$ holds. And combined with Lemma 8, we obtain that Equation (1) has at least one nontrivial solution.