Existence and Multiplicity of Nontrivial Solutions for Semilinear Elliptic Equations Involving Hardy–Sobolev Critical Exponents

Abstract

A class of semi-linear elliptic equations with the critical Hardy–Sobolev exponent has been considered. This model is widely used in hydrodynamics and glaciology, gas combustion in thermodynamics, quantum field theory, and statistical mechanics, as well as in gravity balance problems in galaxies. The ${\left(PS\right)}_c$ sequence of energy functional was investigated, and then the mountain pass lemma was used to prove the existence of at least one nontrivial solution. Also a multiplicity result was obtained. Some known results were generalized.

1. Introduction

Consider the problem:

$$\left\{\begin{array}{cc}-\Delta u-\mu \frac{u}{{\left|x\right|}^2}=\frac{{\left|u\right|}^{2^{*}\left(s\right)-2}}{{\left|x\right|}^s}u+f(x,u),\hfill & x\in \mathrm{\Omega }\setminus \left\{0\right\},\\ u=0,\hfill & x\in \partial \mathrm{\Omega }.\end{array}\right.$$

(1)

Here, $\mathrm{\Omega }$ is an open bounded domain with smooth boundary $\partial \mathrm{\Omega }$ in ${\mathbb{R}}^N\left(N\ge 3\right)$; $0\in \mathrm{\Omega }.0\le \mu <\overline{\mu }:={\left(\frac{N-2}{2}\right)}^2,2^{*}\left(s\right)=2\left(N-s\right)/\left(N-2\right)(0<s<2)$ is the Hardy–Sobolev critical exponent; $2^{*}=2^{*}\left(0\right)$ is the Sobolev critical exponent. $f\in C\left(\mathrm{\Omega }\times \mathbb{R},\mathbb{R}\right)$, $F\left(x,t\right)={\int }_0^{t}f\left(x,s\right)ds$. We point out that (1) is related to the application of hydrodynamics and glaciology [1]. It is also used in some physical or mathematical problems, such as the theory of gas combustion in thermodynamics [2], quantum field theory, and statistical mechanics [3,4,5], as well as gravity balance problems in galaxies [2,6]. For more investigations into solutions for nonlinear equations with Hardy potential, one can see [7,8,9], etc. Furthermore, for the applications of PDEs in physical or mathematical problems, we prefer [10,11] and the references cited therein.

The modern variational method [12,13,14,15] plays a significant role in the study of PDEs (see [16,17,18,19,20]). In 1973, the mountain pass lemma was proposed by A. Ambrosetti and P. Rabinwitz in [16]; this is a milestone in the history of the development of critical point theory. However, in the process of studying the properties of certain equations, there are a lot of phenomena that lose compactness conditions, such as semilinear elliptic equations that involve a Sobolev critical exponent or Hardy–Sobolev critical exponent on a bounded domain. In 1983, H. Brezis and L. Nirenberg first chose the special mountain pass and selected energy estimates to prove the existence of a critical point if the energy functional satisfies the local $\left(PS\right)$ condition (see [17]); they investigated the following problem:

$$\left\{\begin{array}{cc}-\Delta u={\left|u\right|}^{2^{*}-2}u+\lambda u,\hfill & x\in \mathrm{\Omega },\hfill \\ u=0,\hfill & x\in \partial \mathrm{\Omega },\hfill \end{array}\right.$$

(2)

and determined that there exists a ${\lambda }_{*}\in \left(0,{\lambda }_1\right)$ such that, for any $\lambda \in \left({\lambda }_{*},{\lambda }_1\right),$ problem (2) has a positive solution.This is a special case of Equation (1) ($s=0,\mu =0$ and $f\left(x,u\right)=\lambda u$). Since then, many excellent results based on the above methods (see [13,21,22,23]) have appeared.

In previous decades, the semilinear elliptic equation with the Hardy term and Sobolev critical exponent (i.e., when $s=0$ and $\mu \ne 0$) was investigated by many mathematicians; one can refer to [24,25,26,27], etc. For example, the following elliptic problem:

$$\left\{\begin{array}{cc}-\Delta u-\mu \frac{u}{{\left|x\right|}^2}={\left|u\right|}^{2^{*}-2}u+\lambda u,\hfill & x\in \mathrm{\Omega },\hfill \\ u=0,\hfill & x\in \partial \mathrm{\Omega },\hfill \end{array}\right.$$

(3)

is considered in [24,25,26].

For simplicity, in the following, we denote the conditions (H1) and (H2), as follows:

(H1) $0<\lambda <{\lambda }_1\left(\mu \right)$ and $0\le \mu \le \overline{\mu }-1;$

(H2) $\overline{\mu }-1<\mu <\overline{\mu }$ and ${\lambda }_{*}\left(\mu \right)<\lambda <{\lambda }_1\left(\mu \right).$

where $\beta =\sqrt{\overline{\mu }}+\sqrt{\overline{\mu }-\mu },$

$${\lambda }_{*}\left(\mu \right)={\displaystyle \underset{\phi \in H_0^1\left(\mathrm{\Omega }\right)\setminus \left\{0\right\}}{min}}\frac{{\int }_{\mathrm{\Omega }}{\left|\nabla \phi \right|}^2/{\left|x\right|}^{2\beta }dx}{{\int }_{\mathrm{\Omega }}{\phi }^2/{\left|x\right|}^{2\beta }dx},$$

and

$${\lambda }_1\left(\mu \right)={\displaystyle \underset{u\in H_0^1\left(\mathrm{\Omega }\right)\setminus \left\{0\right\}}{inf}}\frac{{\int }_{\mathrm{\Omega }}\left({\left|\nabla u\right|}^2-\mu \frac{u^2}{{\left|x\right|}^2}\right)dx}{{\int }_{\mathrm{\Omega }}{\left|u\right|}^{2}dx}.$$

In [26], using the variational method, E. Jannelli proved that if (H1) or (H2) holds, then (3) has at least one positive solution in $H_0^1\left(\mathrm{\Omega }\right)$. Later, in [28], the authors investigated problem (1) with $f\left(x,u\right)=\lambda {\left|u\right|}^{q-2}u$ or $f\left(x,u\right)=\lambda u$ and obtained the following conclusion:

Theorem A

([28]). Assume $0\le s<2$ and $f\left(x,u\right)=\lambda u.$ If (H1) or (H2) holds, then problem (1) has a positive solution u in $H_0^1\left(\mathrm{\Omega }\right).$

Also, there are some results dealing with the case $\mu \ne 0,$ $s\ne 0$ and the general form $f\left(x,u\right)$ (see [29,30]). In [29], M.C. Wang and Q. Zhang showed that problem (1) has at least one nonnegative solution. In [30], L. Ding and C.L. Tang also investigated problem (1) and obtained the existence result. Inspired by [28,29,30], we study the existence of nontrivial solutions for problem (1). Our main conclusions are as follows:

Theorem 1.

Suppose that $f\left(x,t\right)$ satisfies

$\left(f_1\right)$ $f\in C\left(\overline{\mathrm{\Omega }}\times {\mathbb{R}}^{+},{\mathbb{R}}^{+}\right)$ and ${\displaystyle \underset{t\to 0^{+}}{lim}}\frac{f\left(x,t\right)}{t}=\lambda ,$ ${\displaystyle \underset{t\to +\infty }{lim}}\frac{f\left(x,t\right)}{t^{2^{*}\left(s\right)-1}}=\eta$ uniformly for $x\in \overline{\mathrm{\Omega }}$, where $\lambda ,\eta >0$.

$\left(f_2\right)$ There exists $2<\rho \le 2^{*}\left(s\right)$, such that $\frac{1}{\rho }f\left(x,t\right)t-F\left(x,t\right)\ge -\left(\frac{1}{2}-\frac{1}{\rho }\right)\lambda t^2$ for any $x\in \overline{\mathrm{\Omega }}$, $t\in {\mathbb{R}}^{+}$.

If (H1) or (H2) holds, then (1) has a positive solution u in $H_0^1\left(\mathrm{\Omega }\right)$.

Theorem 2.

Suppose that $f\left(x,t\right)$ satisfies $\left(f_2\right)$ and

$\left(f_3\right)$ $f\in C\left(\overline{\mathrm{\Omega }}\times \mathbb{R},\mathbb{R}\right),$ ${\displaystyle \underset{\left|t\right|\to 0^{+}}{lim}}\frac{f\left(x,t\right)}{t}=\lambda ,$ ${\displaystyle \underset{\left|t\right|\to +\infty }{lim}}\frac{f\left(x,t\right)}{{\left|t\right|}^{2^{*}\left(s\right)-2}t}=\eta$ uniformly for $x\in \overline{\mathrm{\Omega }},$ where $\lambda ,\eta >0.$

If (H1) or (H2) holds, then (1) has at least two distinct nontrivial solutions in $H_0^1\left(\mathrm{\Omega }\right)$.

Remark 1.

(i) 

Let $f\left(x,u\right)=\lambda u;$ we can obtain $\frac{1}{\rho }f\left(x,u\right)u-F\left(x,u\right)=-\left(\frac{1}{2}-\frac{1}{\rho }\right)\lambda u^2.$ Thus, $-\left(\frac{1}{2}-\frac{1}{\rho }\right)\lambda$ is the best constant.

(ii) 

Compared with [29,30], the restrictions on the nonlinear term $f\left(x,u\right)$ are weaker.

(iii) 

If $f\left(x,u\right)=\lambda u+\eta u^{2^{*}\left(s\right)-1}$, then it is easy to verify that $f\left(x,u\right)$ satisfies $\left(f_1\right)$–$\left(f_3\right).$

2. Proof of Theorems

Obviously, in Theorem 1, the values of $f\left(x,t\right)$ are irrelevant for $t<0$; therefore, we define the following:

$$f\left(x,t\right)=0\mathrm{for}x\in \overline{\mathrm{\Omega }}\mathrm{and}t\le 0.$$

Using the Hardy inequality and Hardy–Sobolev inequality (see [31]), we define the equivalent norm and inner product in $H_0^1\left(\mathrm{\Omega }\right):$

$$∥u∥:={\left[{\int }_{\mathrm{\Omega }}\left({\left|\nabla u\right|}^2-\mu \frac{u^2}{{\left|x\right|}^2}\right)dx\right]}^{\frac{1}{2}},\left(u,v\right):={\int }_{\mathrm{\Omega }}\left(\nabla u\nabla v-\mu \frac{uv}{{\left|x\right|}^2}\right)dx,\forall u,v\in H_0^1\left(\mathrm{\Omega }\right).$$

Let

$$u^{+}:=max\left\{0,u\right\},F^{+}\left(x,t\right):={\int }_0^t{f}^{+}\left(x,s\right)ds,f^{+}\left(x,t\right):=\left\{\begin{array}{c}f\left(x,t\right),t\ge 0,\\ 0,t<0.\end{array}\right.$$

The energy functional $J:H_0^1\left(\mathrm{\Omega }\right)\to \mathbb{R}$ to (1) is given by

$$J\left(u\right)=\frac{1}{2}{∥u∥}^2-\frac{1}{2^{*}\left(s\right)}{\int }_{\mathrm{\Omega }}\frac{{\left(u^{+}\right)}^{2^{*}\left(s\right)}}{{\left|x\right|}^s}dx-{\int }_{\mathrm{\Omega }}{F}^{+}\left(x,u\right)dx,u\in H_0^1\left(\mathrm{\Omega }\right).$$

We can easily obtain that $J\left(u\right)$ is well defined with $J\in C^1\left(H_0^1\left(\mathrm{\Omega }\right),\mathbb{R}\right)$ and

$$〈J^{\prime }\left(u\right),v〉=\left(u,v\right)-{\int }_{\mathrm{\Omega }}\frac{{\left(u^{+}\right)}^{2^{*}\left(s\right)-1}}{{\left|x\right|}^s}vdx-{\int }_{\mathrm{\Omega }}{f}^{+}\left(x,u\right)vdx,u,v\in H_0^1\left(\mathrm{\Omega }\right).$$

When $0\le \mu <\overline{\mu }$ and $0<s<2$, the best constant can be defined as follows (see [32]):

$$A_{\mu ,s}:={\displaystyle \underset{u\in H_0^1\left(\mathrm{\Omega }\right)\setminus \left\{0\right\}}{inf}}\frac{{\int }_{\mathrm{\Omega }}\left({\left|\nabla u\right|}^2-\mu \frac{u^2}{{\left|x\right|}^2}\right)dx}{{\left({\int }_{\mathrm{\Omega }}\frac{{\left|u\right|}^{2^{*}\left(s\right)}}{{\left|x\right|}^s}dx\right)}^{\frac{2}{2^{*}\left(s\right)}}}.$$

(4)

Lemma 1.

Suppose $\left(f_1\right)$ holds. For any $0<{\epsilon }_1<min\{\lambda ,\eta \}$ and ${\alpha }_1,{\alpha }_2\in \left(1,2^{*}\left(s\right)-1\right)$, there exists $\xi >0$ such that

$$f\left(x,t\right)\ge \left(\lambda -{\epsilon }_1\right)t+\left(\eta -{\epsilon }_1\right)t^{2^{*}\left(s\right)-1}-\xi t^{{\alpha }_1},\mathit{for}t\ge 0\mathit{and}x\in \overline{\mathrm{\Omega }},$$

(5)

and

$$f\left(x,t\right)\le \left(\lambda +{\epsilon }_1\right)t+\left(\eta +{\epsilon }_1\right)t^{2^{*}\left(s\right)-1}+\xi t^{{\alpha }_2},\mathit{for}t\ge 0\mathit{and}x\in \overline{\mathrm{\Omega }}.$$

(6)

Proof. 

It follows from $\left(f_1\right)$ that $\forall {\epsilon }_1>0$, $\exists \delta >0$ and $M_1>0$:

$$\left|\frac{f\left(x,t\right)}{t}-\lambda \right|\le {\epsilon }_1,\mathrm{for}\left(t,x\right)\in \left[0,\delta \right]\times \overline{\mathrm{\Omega }},$$

(7)

and

$$\left|\frac{f\left(x,t\right)}{t^{2^{*}\left(s\right)-1}}-\eta \right|\le {\epsilon }_1,\mathrm{for}\left(t,x\right)\in \left[M_1,+\infty \right)\times \overline{\mathrm{\Omega }},$$

(8)

from (7), we can obtain:

$$f\left(x,t\right)\ge \left(\lambda -{\epsilon }_1\right)t,\mathrm{for}\left(t,x\right)\in \left[0,\delta \right]\times \overline{\mathrm{\Omega }},$$

for ${\alpha }_1\in \left(1,2^{*}\left(s\right)-1\right)$, if we take $\xi \ge max\left\{0,\left(\eta -{\epsilon }_1\right){\delta }^{2^{*}\left(s\right)-1-{\alpha }_1}\right\}$, then for any $t\in \left[0,\delta \right]$, we have $\left(\eta -{\epsilon }_1\right)t^{2^{*}\left(s\right)-1}-\xi t^{{\alpha }_1}\le 0$; thus,

$$f\left(x,t\right)\ge \left(\lambda -{\epsilon }_1\right)t+\left(\eta -{\epsilon }_1\right)t^{2^{*}\left(s\right)-1}-\xi t^{{\alpha }_1},\mathrm{for}\left(t,x\right)\in \left[0,\delta \right]\times \overline{\mathrm{\Omega }}.$$

From (8), we know the following:

$$f\left(x,t\right)\ge \left(\eta -{\epsilon }_1\right)t,\mathrm{for}\left(t,x\right)\in \left[M_1,+\infty \right)\times \overline{\mathrm{\Omega }},$$

for ${\alpha }_2\in \left(1,2^{*}\left(s\right)-1\right)$, if we take $\xi \ge max\left\{0,\left(\lambda -{\epsilon }_1\right)M_1^{1-{\alpha }_1}\right\}$, then for any $t\in \left[M_1,+\infty \right)$, we have $\left(\lambda -{\epsilon }_1\right)t-\xi t^{{\alpha }_1}\le 0$; thus,

$$f\left(x,t\right)\ge \left(\lambda -{\epsilon }_1\right)t+\left(\eta -{\epsilon }_1\right)t^{2^{*}\left(s\right)-1}-\xi t^{{\alpha }_1},\left(t,x\right)\in \left[M_1,+\infty \right)\times \overline{\mathrm{\Omega }}.$$

When $t\in \left[\delta ,M_1\right]$, taking $\xi \ge max\left\{0,{\displaystyle \underset{t\in \left[\delta ,M_1\right]}{max}}\{\left(\lambda -{\epsilon }_1\right)t^{1-{\alpha }_1}+\left(\eta -{\epsilon }_1\right)t^{2^{*}\left(s\right)-1-{\alpha }_1}\}\right\},$ we have

$$f\left(x,t\right)\ge \left(\lambda -{\epsilon }_1\right)t+\left(\eta -{\epsilon }_1\right)t^{2^{*}\left(s\right)-1}-\xi t^{{\alpha }_1},\left(t,x\right)\in \left[\delta ,M_1\right]\times \overline{\mathrm{\Omega }}.$$

As mentioned above, if we take

$$\xi \ge max\left\{0,\left(\lambda -{\epsilon }_1\right)M_1^{1-{\alpha }_1},\left(\eta -{\epsilon }_1\right){\delta }^{2^{*}\left(s\right)-1-{\alpha }_1},{\displaystyle \underset{t\in \left[\delta ,M_1\right]}{max}}\{\left(\lambda -{\epsilon }_1\right)t^{1-{\alpha }_1}+\left(\eta -{\epsilon }_1\right)t^{2^{*}\left(s\right)-1-{\alpha }_1}\}\right\},$$

then

$$f\left(x,t\right)\ge \left(\lambda -{\epsilon }_1\right)t+\left(\eta -{\epsilon }_1\right)t^{2^{*}\left(s\right)-1}-\xi t^{{\alpha }_1},\mathrm{for}t\ge 0\mathrm{and}x\in \overline{\mathrm{\Omega }}.$$

Similarly, we may determine that there exists $\xi >0$ such that:

$$f\left(x,t\right)\le \left(\lambda +{\epsilon }_1\right)t+\left(\eta +{\epsilon }_1\right)t^{2^{*}\left(s\right)-1}+\xi t^{{\alpha }_2},\mathrm{for}t\ge 0\mathrm{and}x\in \overline{\mathrm{\Omega }}.$$

The conclusion is proved.

Now, we introduce some notations:

$$C_{\epsilon }={\left(\frac{2\epsilon \left(\overline{\mu }-\mu \right)\left(N-s\right)}{\sqrt{\overline{\mu }}}\right)}^{\sqrt{\overline{\mu }}/\left(2-s\right)},U_{\epsilon }\left(x\right)=\frac{y_{\epsilon }\left(x\right)}{C_{\epsilon }},$$

(9)

where

$$y_{\epsilon }\left(x\right)=\frac{{\left(2\epsilon \left(\overline{\mu }-\mu \right)\left(N-s\right)/\sqrt{\overline{\mu }}\right)}^{\sqrt{\overline{\mu }}/\left(2-s\right)}}{{\left|x\right|}^{\sqrt{\overline{\mu }}-\sqrt{\overline{\mu }-\mu }}{\left(\epsilon +{\left|x\right|}^{\left(2-s\right)\sqrt{\overline{\mu }-\mu }/\sqrt{\overline{\mu }}}\right)}^{\left(N-2\right)/\left(2-s\right)}},\mathrm{for}\mathrm{all}\epsilon >0.$$

From the work of [28], we know the functions $y_{\epsilon }\left(x\right)$ are a positive solution of the equation

$$-\Delta u-\mu \frac{u}{{\left|x\right|}^2}=\frac{{\left|u\right|}^{2^{*}\left(s\right)-2}}{{\left|x\right|}^s}u,x\in \mathrm{\Omega },$$

and satisfy

$${\int }_{{\mathbb{R}}^N}\left({\left|\nabla y_{\epsilon }\left(x\right)\right|}^2-\mu \frac{{\left|y_{\epsilon }\left(x\right)\right|}^2}{{\left|x\right|}^2}\right)dx={\int }_{{\mathbb{R}}^N}\frac{{\left|y_{\epsilon }\left(x\right)\right|}^{2^{*}\left(s\right)}}{{\left|x\right|}^s}dx=A_{\mu ,s}^{\frac{N-s}{2-s}}.$$

(10)

A cut-off function $\phi \in C_0^{\infty }\left(\mathrm{\Omega }\right)$ is defined such that

$$\phi \left(x\right)=\left\{\begin{array}{cc}1,\hfill & \left|x\right|\le R,\hfill \\ 0,\hfill & \left|x\right|\ge 2R,\hfill \end{array}\right.$$

where $B_{2R}\left(0\right)\subset \mathrm{\Omega },0\le \phi \left(x\right)\le 1$, for $R<\left|x\right|<2R$; set

$$u_{\epsilon }\left(x\right)=\phi \left(x\right)U_{\epsilon }\left(x\right),v_{\epsilon }\left(x\right)=\frac{u_{\epsilon }\left(x\right)}{{\left({\int }_{\mathrm{\Omega }}{\left|u_{\epsilon }\left(x\right)\right|}^{2^{*}\left(s\right)}{\left|x\right|}^{-s}dx\right)}^{\frac{1}{2^{*}\left(s\right)}}}.$$

(11)

□

Lemma 2.

Let $v_{\epsilon }\left(x\right)$ be defined as above; then, $v_{\epsilon }\left(x\right)$ satisfies

$${∥v_{\epsilon }\left(x\right)∥}^2=A_{\mu ,s}+O\left({\epsilon }^{\frac{N-2}{2-s}}\right),$$

(12)

$${\int }_{\mathrm{\Omega }}{\left|v_{\epsilon }\left(x\right)\right|}^{q}dx=\left\{\begin{array}{cc}O\left({\epsilon }^{\frac{\sqrt{\overline{\mu }}q}{2-s}}\right),\hfill & 1\le q<\frac{N}{\sqrt{\overline{\mu }}+\sqrt{\overline{\mu }-\mu }},\hfill \\ O\left({\epsilon }^{\frac{\sqrt{\overline{\mu }}q}{2-s}}\left|ln\epsilon \right|\right),\hfill & q=\frac{N}{\sqrt{\overline{\mu }}+\sqrt{\overline{\mu }-\mu }},\hfill \\ O\left({\epsilon }^{\frac{\sqrt{\overline{\mu }}\left(N-q\sqrt{\overline{\mu }}\right)}{\left(2-s\right)\sqrt{\overline{\mu }-\mu }}}\right),\hfill & \frac{N}{\sqrt{\overline{\mu }}+\sqrt{\overline{\mu }-\mu }}<q<2^{*}.\hfill \end{array}\right.$$

(13)

Proof. 

The proof can be found in [28]. □

Lemma 3.

Let $u_{\epsilon }\left(x\right)$, $U_{\epsilon }\left(x\right),$ $A_{\mu ,s}$, $C_{\epsilon }$ be defined as above; then, the exact estimates of ${∥u_{\epsilon }∥}^2$ and ${\int }_{\mathrm{\Omega }}\frac{{\left|u_{\epsilon }\right|}^{2^{*}\left(s\right)}}{{\left|x\right|}^s}dx$ are as follows:

$${∥u_{\epsilon }\left(x\right)∥}^2=C_{\epsilon }^{-2}{A}_{\mu ,s}^{\frac{N-s}{2-s}}+D,{\int }_{\mathrm{\Omega }}\frac{{\left|u_{\epsilon }\left(x\right)\right|}^{2^{*}\left(s\right)}}{{\left|x\right|}^s}dx=C_{\epsilon }^{-2^{*}\left(s\right)}{A}_{\mu ,s}^{\frac{N-s}{2-s}}+E,$$

(14)

where

$$D={\int }_{R\le \left|x\right|\le 2R}\left({\left|\nabla u_{\epsilon }\left(x\right)\right|}^2-\mu \frac{u_{\epsilon }^2\left(x\right)}{{\left|x\right|}^2}\right)dx-{\int }_{\left|x\right|\ge R}\left({\left|\nabla U_{\epsilon }\left(x\right)\right|}^2-\mu \frac{U_{\epsilon }^2\left(x\right)}{{\left|x\right|}^2}\right)dx,$$

$$E=-{\int }_{\left|x\right|\ge R}\frac{{\left|U_{\epsilon }\left(x\right)\right|}^{2^{*}\left(s\right)}}{{\left|x\right|}^s}dx+{\int }_{R\le \left|x\right|\le 2R}\frac{{\left|u_{\epsilon }\left(x\right)\right|}^{2^{*}\left(s\right)}}{{\left|x\right|}^s}dx.$$

Moreover, ∃$R_0>0$ such that, for any $R\le R_0,$

$${\displaystyle \underset{\epsilon \to 0^{+}}{lim}}D<{\int }_{\mathrm{\Omega }}\frac{{\left|\nabla \phi \left(x\right)\right|}^2}{{\left|x\right|}^{2\beta }}dx.$$

(15)

Proof. 

The proof can be found in [33]. We omit it here. □

Lemma 4.

Suppose $\left(f_1\right),$ $\left(f_2\right)$ and $\lambda <{\lambda }_1\left(\mu \right)$ hold. Assume $\left\{u_n\right\}\subset H_0^1\left(\mathrm{\Omega }\right)$ is a (PS)_c sequences; that is,

$$J\left(u_n\right)\to c\in \left(0,\frac{2-s}{2\left(N-s\right)}{A}_{\mu ,s}^{\frac{N-s}{2-s}}\right),$$

and

$$J^{\prime }\left(u_n\right)\to 0,\mathrm{in}{\left(H_0^1\left(\mathrm{\Omega }\right)\right)}^{-1}.$$

Then, there exists $u\in H_0^1\left(\mathrm{\Omega }\right),$ such that $u_n\rightharpoonup u$ weakly in $H_0^1\left(\mathrm{\Omega }\right)$, or a subsequence $u_{n_k}\rightharpoonup u$ weakly in $H_0^1\left(\mathrm{\Omega }\right)$; moreover, $J^{\prime }\left(u\right)=0$ and u is a nontrivial solution of (1).

Proof. 

First, we claim that if $\left(f_1\right),$ $\left(f_2\right)$ and $\lambda <{\lambda }_1\left(\mu \right)$ hold, then any (PS)_c sequence $\left\{u_n\right\}$ is bounded in $H_0^1\left(\mathrm{\Omega }\right)$. Otherwise, suppose that $∥u_n∥\to \infty$; since $J\left(u_n\right)\to c$, there exists $N_1$, such that when $n>N_1,$ $J\left(u_n\right)<c+1$. $J^{\prime }\left(u_n\right)\to 0$ implies $-\frac{1}{\rho }〈J^{\prime }\left(u_n\right),u_n〉<o\left(1\right)∥u_n∥$; thus for any ${\epsilon }_1\in \left(0,{\lambda }_1\left(\mu \right)-\lambda \right)$, when $n>N_1$:

$$\begin{array}{ccc}\hfill c+1+o\left(1\right)∥u_n∥& \ge & J\left(u_n\right)-\frac{1}{\rho }〈J^{\prime }\left(u_n\right),u_n〉\hfill \\ & =& \left(\frac{1}{2}-\frac{1}{\rho }\right){∥u_n∥}^2+{\int }_{\mathrm{\Omega }}\left(\frac{1}{\rho }{f}^{+}\left(x,u_n\right)u_n-F^{+}\left(x,u_n\right)\right)dx\hfill \\ & & +\left(\frac{1}{\rho }-\frac{1}{2^{*}\left(s\right)}\right){\int }_{\mathrm{\Omega }}\frac{{\left(u_n^{+}\right)}^{2^{*}\left(s\right)}}{{\left|x\right|}^s}dx\hfill \\ & \ge & \left(\frac{1}{2}-\frac{1}{\rho }\right){∥u_n∥}^2-\left(\frac{1}{2}-\frac{1}{\rho }\right)\left(\lambda +{\epsilon }_1\right){\int }_{\mathrm{\Omega }}{u}_n^{2}dx\hfill \\ & \ge & \left(\frac{1}{2}-\frac{1}{\rho }\right){∥u_n∥}^2-\left(\frac{1}{2}-\frac{1}{\rho }\right)\frac{\lambda +{\epsilon }_1}{{\lambda }_1\left(\mu \right)}{∥u_n∥}^2\hfill \\ & =& \left(\frac{1}{2}-\frac{1}{\rho }\right)\left(1-\frac{\lambda +{\epsilon }_1}{{\lambda }_1\left(\mu \right)}\right){∥u_n∥}^2.\hfill \end{array}$$

This shows that $\left\{u_n\right\}$ is a bounded sequence in $H_0^1\left(\mathrm{\Omega }\right).$ From the reflexivity of $H_0^1\left(\mathrm{\Omega }\right),$ we know that there exists u, such that $u_n\rightharpoonup u$ (or a consequence of $u_n$ converging to u). Furthermore, $J^{\prime }\left(u\right)=0$ due to the weak continuity of $J^{\prime }$. From $u_n\in H_0^1\left(\mathrm{\Omega }\right)$, $u_n\rightharpoonup u$, due to the compactness of the embedding, we have $u_n\to u$ in $L^{\gamma }\left(\mathrm{\Omega }\right)$ for any $1<\gamma <2^{*}\left(s\right)$. Let $f_1\left(x,u\right)=f\left(x,u\right)u$; from $\left(f_1\right)$, we have $\left|f_1\left(x,u_n\right)\right|\le a+b{\left|u_n\right|}^{2^{*}\left(s\right)}$. From the definition of Uryson operator, we know that $f_1:L^{2^{*}\left(s\right)}\left(\mathrm{\Omega }\right)\to L^1\left(\mathrm{\Omega }\right)$ is a continuous operator. Thus:

$${\displaystyle \underset{n\to \infty }{lim}}{\int }_{\mathrm{\Omega }}\left(f_1\left(x,u_n\right)-f_1\left(x,u\right)\right)dx=0,$$

that is:

$${\displaystyle \underset{n\to \infty }{lim}}{\int }_{\mathrm{\Omega }}f\left(x,u_n\right)u_{n}dx={\int }_{\mathrm{\Omega }}f\left(x,u\right)udx.$$

(16)

Similarly:

$${\displaystyle \underset{n\to \infty }{lim}}{\int }_{\mathrm{\Omega }}F\left(x,u_n\right)dx={\int }_{\mathrm{\Omega }}F\left(x,u\right)dx.$$

In addition, there is the convergence of $∥u_n∥$, $u_n\to u$ in $H_0^1\left(\mathrm{\Omega }\right).$

Assume that $u\equiv 0$ in $\mathrm{\Omega }$; from $〈J^{\prime }\left(u_n\right),u_n〉=o\left(1\right)$ and (16) we know:

$${∥u_n∥}^2-{\int }_{\mathrm{\Omega }}\frac{{\left(u_n^{+}\right)}^{2^{*}\left(s\right)}}{{\left|x\right|}^s}dx=o\left(1\right).$$

(17)

From (4), we can obtain:

$${∥u_n∥}^2\ge A_{\mu ,s}{\left({\int }_{\mathrm{\Omega }}\frac{{\left(u_n^{+}\right)}^{2^{*}\left(s\right)}}{{\left|x\right|}^s}dx\right)}^{\frac{2}{2^{*}\left(s\right)}}.$$

(18)

From (17) and (18), we have:

$$o\left(1\right)\ge {∥u_n∥}^2\left(1-A_{\mu ,s}^{-\frac{2^{*}\left(s\right)}{2}}{∥u_n∥}^{2^{*}\left(s\right)-2}\right).$$

If $∥u_n∥\to 0$, then (17) implies that $J\left(u_n\right)\to 0$, while $J\left(u_n\right)\to c$, which contradicts $c>0$. Hence:

$${∥u_n∥}^2\ge A_{\mu ,s}^{\frac{N-s}{2-s}}+o\left(1\right).$$

(19)

From (13), (17) and (19), we can obtain:

$$\begin{array}{cc}\hfill J\left(u_n\right)& =\frac{1}{2}{∥u_n∥}^2-\frac{1}{2^{*}\left(s\right)}{\int }_{\mathrm{\Omega }}\frac{{\left(u_n^{+}\right)}^{2^{*}\left(s\right)}}{{\left|x\right|}^s}dx+o\left(1\right)\hfill \\ \hfill & =\frac{2-s}{2\left(N-s\right)}{∥u_n∥}^2+o\left(1\right)\hfill \\ \hfill & \ge \frac{2-s}{2\left(N-s\right)}{A}_{\mu ,s}^{\frac{N-s}{2-s}}+o\left(1\right),\hfill \end{array}$$

which contradicts $c<\frac{2-s}{2\left(N-s\right)}{A}_{\mu ,s}^{\frac{N-s}{2-s}}$. Thus, u is not constantly equal to 0 and u is a nontrivial solution of problem (1). □

Lemma 5.

If $\left(f_1\right),$ $\left(f_2\right)$ and $\lambda <{\lambda }_1\left(\mu \right)$ hold, then the functional J admits a $\left(PS\right)$ sequence at level:

$$c={\displaystyle \underset{\gamma \in \mathrm{\Gamma }}{inf}}{\displaystyle \underset{t\in \left[0,1\right]}{max}}J\left(\gamma \left(t\right)\right),$$

where

$$\mathrm{\Gamma }=\left\{\gamma \in C\left(\left[0,1\right],H_0^1\left(\mathrm{\Omega }\right)\right);\gamma \left(0\right)=0,J\left(\gamma \left(1\right)\right)<0\right\}.$$

Proof. 

We need to prove that J satisfies all assumptions of the mountain pass lemma except for the $\left(PS\right)$ condition. Obviously, $J\left(0\right)=0$. Moreover, from the Hardy–Sobolev inequality and the Hardy inequality, we can easily obtain:

$${\int }_{\mathrm{\Omega }}\frac{{\left|u\right|}^{2^{*}\left(s\right)}}{{\left|x\right|}^s}dx\le C_1{∥u∥}^{2^{*}\left(s\right)},{∥u∥}_q^q\le C_2{∥u∥}^q\mathrm{for}1\le q\le 2^{*},u\in H_0^1\left(\mathrm{\Omega }\right).$$

(20)

Then, from (6), (20) and Lemma 1, we have:

$$\begin{array}{cc}\hfill J\left(u\right)& =\frac{1}{2}{∥u∥}^2-\frac{1}{2^{*}\left(s\right)}{\int }_{\mathrm{\Omega }}\frac{{\left(u^{+}\right)}^{2^{*}\left(s\right)}}{{\left|x\right|}^s}dx-{\int }_{\mathrm{\Omega }}{F}^{+}\left(x,u\right)dx\hfill \\ \hfill & \ge \frac{1}{2}{∥u∥}^2-\frac{C_1}{2^{*}\left(s\right)}{∥u∥}^{2^{*}\left(s\right)}-\frac{\eta +{\epsilon }_1}{2^{*}\left(s\right)}{∥u∥}_{2^{*}\left(s\right)}^{2^{*}\left(s\right)}-\frac{\lambda +{\epsilon }_1}{2}{∥u∥}_2^2-\frac{\xi }{{\alpha }_2+1}{∥u∥}_{{\alpha }_2+1}^{{\alpha }_2+1}\hfill \\ \hfill & \ge \frac{1-\left(\lambda +{\epsilon }_1\right)/{\lambda }_1\left(\mu \right)}{2}{∥u∥}^2-\frac{C_1}{2^{*}\left(s\right)}{∥u∥}^{2^{*}\left(s\right)}-\frac{C_2}{2^{*}\left(s\right)}{∥u∥}^{2^{*}\left(s\right)}-\frac{\xi }{{\alpha }_2+1}{∥u∥}^{{\alpha }_2+1},\hfill \end{array}$$

which implies that $\exists \alpha ,\rho >0$ such that:

$$J\left(u\right)\ge \alpha >0,\forall u\in \left\{u\in H_0^1\left(\mathrm{\Omega }\right)|∥u∥=\rho \right\}.$$

Taking $u_0$ $\in H_0^1\left(\mathrm{\Omega }\right)\setminus \left\{0\right\}$, such that ${\int }_{\mathrm{\Omega }}\frac{{\left(u_0^{+}\right)}^{2^{*}\left(s\right)}}{{\left|x\right|}^s}dx\ge C_3>0$, for any $t>0$, we have:

$$\begin{array}{cc}\hfill J\left(t{u}_0\right)& =\frac{t^2}{2}{∥u_0∥}^2-\frac{t^{2^{*}\left(s\right)}}{2^{*}\left(s\right)}{\int }_{\mathrm{\Omega }}\frac{{\left(u_0^{+}\right)}^{2^{*}\left(s\right)}}{{\left|x\right|}^s}dx-{\int }_{\mathrm{\Omega }}{F}^{+}\left(x,t{u}_0\right)dx\hfill \\ \hfill & \le \frac{t^2}{2}{∥u_0∥}^2-\frac{t^{2^{*}\left(s\right)}}{2^{*}\left(s\right)}{C}_3,\hfill \end{array}$$

notice that ${\displaystyle \underset{t\to +\infty }{lim}}J\left(t{u}_0\right)=-\infty$; then, there exists $t_0>0$ such that $∥t_0{u}_0∥>\rho$ and $J\left(t_0{u}_0\right)\le 0$. Using the mountain pass theorem without the (PS) condition (see Theorem 2.2 in [17]), we know that J admits a $PS$ sequence at the c level. □

Lemma 6.

Assume $\left(f_1\right),$ $\left(f_2\right)$ and $0<s<2$, if (H1) or (H2) holds; then,

$$0<c<\frac{2-s}{2\left(N-s\right)}{A}_{\mu ,s}^{\frac{N-s}{2-s}}.$$

(21)

Proof. 

Define

$$g\left(t\right):=J\left(t{v}_{\epsilon }\right)=\frac{t^2}{2}{∥v_{\epsilon }∥}^2-\frac{t^{2^{*}\left(s\right)}}{2^{*}\left(s\right)}-{\int }_{\mathrm{\Omega }}{F}^{+}\left(x,t{v}_{\epsilon }\right)dx,$$

and

$$\overline{g}\left(t\right):=\frac{t^2}{2}{∥v_{\epsilon }∥}^2-\frac{t^{2^{*}\left(s\right)}}{2^{*}\left(s\right)}.$$

It is easy to see that ${\displaystyle \underset{t\to +\infty }{lim}}g\left(t\right)=-\infty ,g\left(0\right)=0$ and $g\left(t\right)>0$ when t is small enough, so some $t_{\epsilon }>0$ exists, such that $g\left(t_{\epsilon }\right)={\displaystyle \underset{t\ge 0}{sup}}g\left(t\right)>0,$ which shows that $c>0.$ Obviously $g^{\prime }\left(t_{\epsilon }\right)=0$; that is,

$$0=g^{\prime }\left(t_{\epsilon }\right)=t_{\epsilon }{∥v_{\epsilon }∥}^2-t_{\epsilon }^{2^{*}\left(s\right)-1}-{\int }_{\mathrm{\Omega }}{f}^{+}\left(x,t_{\epsilon }{v}_{\epsilon }\right)v_{\epsilon }dx,$$

thus,

$${∥v_{\epsilon }∥}^2=t_{\epsilon }^{2^{*}\left(s\right)-2}+\frac{1}{t_{\epsilon }}{\int }_{\mathrm{\Omega }}{f}^{+}\left(x,t_{\epsilon }{v}_{\epsilon }\right)v_{\epsilon }dx\ge t_{\epsilon }^{2^{*}\left(s\right)-2},$$

therefore:

$${\overline{t}}_{\epsilon }:={∥v_{\epsilon }∥}^{\frac{2}{2^{*}\left(s\right)-2}}\ge t_{\epsilon }.$$

(22)

From (7), we know

$${\int }_{\mathrm{\Omega }}f\left(x,t_{\epsilon }{v}_{\epsilon }\right)v_{\epsilon }dx\le \left(\lambda +{\epsilon }_1\right)t_{\epsilon }{\int }_{\mathrm{\Omega }}{v}_{\epsilon }^{2}dx+\left(\eta +{\epsilon }_1\right)t_{\epsilon }^{2^{*}\left(s\right)-1}{\int }_{\mathrm{\Omega }}{v}_{\epsilon }^{2^{*}\left(s\right)}dx+\xi t_{\epsilon }^{{\alpha }_2}{\int }_{\mathrm{\Omega }}{v}_{\epsilon }^{{\alpha }_2+1}dx.$$

Hence,

$${∥v_{\epsilon }∥}^2\le t_{\epsilon }^{2^{*}\left(s\right)-2}+\left(\lambda +{\epsilon }_1\right){\int }_{\mathrm{\Omega }}{\left|v_{\epsilon }\right|}^{2}dx+\left(\eta +{\epsilon }_1\right){\left|t_{\epsilon }\right|}^{2^{*}\left(s\right)-2}{\int }_{\mathrm{\Omega }}{\left|v_{\epsilon }\right|}^{2^{*}\left(s\right)}dx+\xi {\left|t_{\epsilon }\right|}^{{\alpha }_2-1}{\int }_{\mathrm{\Omega }}{\left|v_{\epsilon }\right|}^{{\alpha }_2+1}dx.$$

Moreover, from Lemma 2, we have

$$t_{\epsilon }^{2^{*}\left(s\right)-2}\ge A_{\mu ,s}-{\epsilon }_2.$$

(23)

On the other hand, $\overline{g}\left(t\right)\le$ $\overline{g}\left({\overline{t}}_{\epsilon }\right)$ for any $t\in \left[0,{\overline{t}}_{\epsilon }\right]$. From (5), (12), (13), (22), (23) and Lemma 2, we can obtain:

$$\begin{array}{ccc}\hfill g\left(t_{\epsilon }\right)& =& \overline{g}\left({\overline{t}}_{\epsilon }\right)-{\int }_{\mathrm{\Omega }}F\left(x,t_{\epsilon }{v}_{\epsilon }\right)dx\hfill \\ & \le & \frac{2-s}{2\left(N-s\right)}{∥v_{\epsilon }∥}^{\frac{2\left(N-s\right)}{2-s}}-\frac{\lambda -{\epsilon }_1}{2}{t}_{\epsilon }^2{\int }_{\mathrm{\Omega }}{\left|v_{\epsilon }\right|}^{2}dx+\frac{\xi }{\alpha +1}{\left|t_{\epsilon }\right|}^{{\alpha }_1+1}{\int }_{\mathrm{\Omega }}{\left|v_{\epsilon }\right|}^{{\alpha }_1+1}dx\hfill \\ & & -\frac{\eta -{\epsilon }_1}{2^{*}\left(s\right)}{\left|t_{\epsilon }\right|}^{2^{*}\left(s\right)}{\int }_{\mathrm{\Omega }}{\left|v_{\epsilon }\right|}^{2^{*}\left(s\right)}dx\hfill \\ & \le & \frac{2-s}{2\left(N-s\right)}{∥v_{\epsilon }∥}^{\frac{2\left(N-s\right)}{2-s}}-\frac{\left(\lambda -{\epsilon }_1\right){\left(A_{\mu ,s}-{\epsilon }_2\right)}^{\frac{2}{2^{*}\left(s\right)-2}}}{2}{\int }_{\mathrm{\Omega }}{\left|v_{\epsilon }\right|}^{2}dx\hfill \\ & & +\frac{\xi }{{\alpha }_1+1}{\left(A_{\mu ,s}-{\epsilon }_2\right)}^{\frac{{\alpha }_1+1}{2^{*}\left(s\right)-2}}{\int }_{\mathrm{\Omega }}{\left|v_{\epsilon }\right|}^{{\alpha }_1+1}dx-\frac{\eta -{\epsilon }_1}{2^{*}\left(s\right)}{\left(A_{\mu ,s}-{\epsilon }_2\right)}^{\frac{2^{*}\left(s\right)}{2^{*}\left(s\right)-2}}{\int }_{\mathrm{\Omega }}{\left|v_{\epsilon }\right|}^{2^{*}\left(s\right)}dx.\hfill \end{array}$$

(24)

If (H1) holds, notice that $2\ge \frac{N}{\sqrt{\overline{\mu }}+\sqrt{\overline{\mu }-\mu }}$; then, when $\epsilon$ is sufficiently small, the sign of $-\frac{\lambda -{\epsilon }_1}{2}{t}_{\epsilon }^2{\int }_{\mathrm{\Omega }}{\left|v_{\epsilon }\right|}^{2}dx+\frac{\xi }{\alpha +1}{\left|t_{\epsilon }\right|}^{{\alpha }_1+1}{\int }_{\mathrm{\Omega }}{\left|v_{\epsilon }\right|}^{{\alpha }_1+1}dx-\frac{\eta -{\epsilon }_1}{2^{*}\left(s\right)}{\left|t_{\epsilon }\right|}^{2^{*}\left(s\right)}{\int }_{\mathrm{\Omega }}{\left|v_{\epsilon }\right|}^{2^{*}\left(s\right)}dx$ is decided by the sign of $-{\int }_{\mathrm{\Omega }}{\left|v_{\epsilon }\right|}^{2^{*}\left(s\right)}dx.$ Thus, when $\epsilon$ is small enough, (21) holds true.

If (H2) holds, since ${\alpha }_1>1$ is arbitrary, we can choose ${\alpha }_1>\frac{N}{\sqrt{\overline{\mu }}+\sqrt{\overline{\mu }-\mu }};$ then, using Lemma 2, we know that when $\epsilon$ is sufficiently small, the sign of $\frac{\xi }{\alpha +1}{\left|t_{\epsilon }\right|}^{{\alpha }_1+1}{\int }_{\mathrm{\Omega }}{\left|v_{\epsilon }\right|}^{{\alpha }_1+1}dx-\frac{\eta -{\epsilon }_1}{2^{*}\left(s\right)}{\left|t_{\epsilon }\right|}^{2^{*}\left(s\right)}{\int }_{\mathrm{\Omega }}{\left|v_{\epsilon }\right|}^{2^{*}\left(s\right)}dx$ is decided by the sign of $-{\int }_{\mathrm{\Omega }}{\left|v_{\epsilon }\right|}^{2^{*}\left(s\right)}dx.$ Thus, from (24), when $\epsilon ,$ ${\epsilon }_1$ and ${\epsilon }_2$ are small enough, we have:

$$g\left(t_{\epsilon }\right)<\frac{2-s}{2\left(N-s\right)}{∥v_{\epsilon }∥}^{\frac{2\left(N-s\right)}{2-s}}-\frac{{\lambda }_{*}\left(\mu \right)}{2}{A}_{\mu ,s}^{\frac{2}{2^{*}\left(s\right)-2}}{\int }_{\mathrm{\Omega }}{\left|v_{\epsilon }\right|}^{2}dx.$$

From (14), we know:

$$\begin{array}{ccc}& & {\displaystyle \underset{\epsilon \to 0^{+}}{lim}}\frac{{∥v_{\epsilon }∥}^{\frac{2\left(N-s\right)}{2-s}}-A_{\mu ,s}^{\frac{N-s}{2-s}}}{A_{\mu ,s}^{\frac{N-2}{2-s}}{\int }_{\mathrm{\Omega }}{\left|v_{\epsilon }\right|}^{2}dx}\hfill \\ & =& {\displaystyle \underset{\epsilon \to 0^{+}}{lim}}\frac{{∥u_{\epsilon }∥}^{\frac{2\left(N-s\right)}{2-s}}-A_{\mu ,s}^{\frac{N-s}{2-s}}{\left({\int }_{\mathrm{\Omega }}\frac{{\left|u_{\epsilon }\right|}^{2^{*}\left(s\right)}}{{\left|x\right|}^s}dx\right)}^{\frac{N-s}{2-s}}}{A_{\mu ,s}^{\frac{N-2}{2-s}}{\left({\int }_{\mathrm{\Omega }}\frac{{\left|u_{\epsilon }\right|}^{2^{*}\left(s\right)}}{{\left|x\right|}^s}dx\right)}^{\frac{N-2}{2-s}\frac{2}{2^{*}\left(s\right)}}{\int }_{\mathrm{\Omega }}{\left|u_{\epsilon }\right|}^{2}dx}\hfill \\ & =& {\displaystyle \underset{\epsilon \to 0^{+}}{lim}}\frac{{\left(D+C_{\epsilon }^{-2}{A}_{\mu ,s}^{\frac{N-s}{2-s}}\right)}^{\frac{N-s}{2-s}}-A_{\mu ,s}^{\frac{N-s}{2-s}}{\left(E+C_{\epsilon }^{-2^{*}\left(s\right)}{A}_{\mu ,s}^{\frac{N-s}{2-s}}\right)}^{\frac{N-2}{2-s}}}{A_{\mu ,s}^{\frac{N-2}{2-s}}{\left(E+C_{\epsilon }^{-2^{*}\left(s\right)}{A}_{\mu ,s}^{\frac{N-s}{2-s}}\right)}^{\frac{N-2}{2-s}\frac{2}{2^{*}\left(s\right)}}{\int }_{\mathrm{\Omega }}{\left|u_{\epsilon }\right|}^{2}dx}\hfill \\ & =& {\displaystyle \underset{\epsilon \to 0^{+}}{lim}}\frac{{\left(C_{\epsilon }^{2}D+A_{\mu ,s}^{\frac{N-s}{2-s}}\right)}^{\frac{N-s}{2-s}}-A_{\mu ,s}^{\frac{N-s}{2-s}}{\left(E{C}_{\epsilon }^{2^{*}\left(s\right)}+A_{\mu ,s}^{\frac{N-s}{2-s}}\right)}^{\frac{N-2}{2-s}}}{C_{\epsilon }^2{A}_{\mu ,s}^{\frac{N-2}{2-s}}{\left(E{C}_{\epsilon }^{2^{*}\left(s\right)}+A_{\mu ,s}^{\frac{N-s}{2-s}}\right)}^{\frac{N-2}{2-s}\frac{2}{2^{*}\left(s\right)}}{\int }_{\mathrm{\Omega }}{\left|u_{\epsilon }\right|}^{2}dx}\hfill \\ & =& {\displaystyle \underset{{\epsilon }_0\to 0}{lim}}\frac{{\left(C^{2}D{\epsilon }_0+A_{\mu ,s}^{\frac{N-s}{2-s}}\right)}^{\frac{N-s}{2-s}}-A_{\mu ,s}^{\frac{N-s}{2-s}}{\left(E{C}^{2^{*}\left(s\right)}{\epsilon }_0^{\frac{N-s}{N-2}}+A_{\mu ,s}^{\frac{N-s}{2-s}}\right)}^{\frac{N-2}{2-s}}}{C^2{\epsilon }_0{A}_{\mu ,s}^{\frac{\left(N-s\right)\left(N-2\right)}{{\left(2-s\right)}^2}}{\int }_{\mathrm{\Omega }}{\left|u_{\epsilon }\right|}^{2}dx}\hfill \\ & =& \frac{N-s}{2-s}{\displaystyle \underset{{\epsilon }_0\to 0^{+}}{lim}}\frac{P{C}^2\left(D+{\epsilon }_0\frac{\partial D}{\partial {\epsilon }_0}\right)-A_{\mu ,s}^{\frac{N-s}{2-s}}Q{C}^{2^{*}\left(s\right)}\left(E{\epsilon }_0^{\frac{2-s}{N-2}}+\frac{\partial E}{\partial {\epsilon }_0}{\epsilon }_0^{\frac{N-s}{N-2}}\right)}{C^2{A}_{\mu ,s}^{\frac{\left(N-s\right)\left(N-2\right)}{{\left(2-s\right)}^2}}{\int }_{\mathrm{\Omega }}{\left|u_{\epsilon }\right|}^{2}dx}\hfill \\ & =& \frac{N-s}{2-s}\frac{D}{{\int }_{\mathrm{\Omega }}\frac{{\left|\phi \left(x\right)\right|}^2}{{\left|x\right|}^{2\beta }}dx},\hfill \end{array}$$

(25)

where $C={\left(\frac{2\left(\overline{\mu }-\mu \right)\left(N-s\right)}{\sqrt{\overline{\mu }}}\right)}^{\frac{\sqrt{\mu }}{2-s}},$ ${\epsilon }_0={\epsilon }^{\frac{\sqrt{\overline{\mu }}}{2-s}},$ $P={\left(C^{2}D{\epsilon }_0+A_{\mu ,s}^{\frac{N-s}{2-s}}\right)}^{\frac{N-2}{2-s}},$ $Q={\left(E{C}^{2^{*}\left(s\right)}{\epsilon }_0^{\frac{N-s}{N-2}}+A_{\mu ,s}^{\frac{N-s}{2-s}}\right)}^{\frac{N-4+s}{2-s}}.$

From (15), (14) and (25), we have:

$${\displaystyle \underset{\epsilon \to 0^{+}}{lim}}\frac{{∥v_{\epsilon }∥}^{\frac{2\left(N-s\right)}{2-s}}-A_{\mu ,s}^{\frac{N-s}{2-s}}}{A_{\mu ,s}^{\frac{N-2}{2-s}}{\int }_{\mathrm{\Omega }}{\left|v_{\epsilon }\right|}^{2}dx}<\frac{N-s}{2-s}\frac{{\int }_{\mathrm{\Omega }}\frac{{\left|\nabla \phi \left(x\right)\right|}^2}{{\left|x\right|}^{2\beta }}dx}{{\int }_{\mathrm{\Omega }}\frac{{\left|\phi \left(x\right)\right|}^2}{{\left|x\right|}^{2\beta }}dx};$$

so, if $\epsilon$ is small enough, then $c\le g\left(t_{\epsilon }\right)<\frac{2-s}{2\left(N-s\right)}{A}_{\mu ,s}^{\frac{N-s}{2-s}}.$ □

Proof  of Theorem 1.

By Lemmas 4–6, we can see that Equation (1) has a nonnegative solution $u\in H_0^1\left(\mathrm{\Omega }\right)$; when using the maximum principle, this solution is positive. This completes the proof. □

Proof  of Theorem 2.

Since $\left(f_3\right)$ contains $\left(f_1\right)$, Theorem 1 implies the existence of a positive solution $u_1$ for Equation (1). Let $f\left(x,t\right)=-h\left(x,-t\right)$ for $t\in \mathbb{R}$, $h\left(x,u\right)$ satisfies $\left(f_1\right)$ and $\left(f_2\right)$, then

$$-\Delta u-\mu \frac{u}{{\left|x\right|}^2}=\frac{{\left|u\right|}^{2^{*}\left(s\right)-2}}{{\left|x\right|}^s}u+h\left(x,u\right)$$

has at least one nonnegative solution v. Let $u_2=-v$; then, $u_2$ is a solution of

$$-\Delta u-\mu \frac{u}{{\left|x\right|}^2}=\frac{{\left|u\right|}^{2^{*}\left(s\right)-2}}{{\left|x\right|}^s}u+f\left(x,u\right).$$

Clearly, $u_1,u_2\ne 0$. Therefore, problem (1) has at least two distinct nontrivial solutions. □

3. Conclusions

In this paper, a class of semilinear elliptic equations involving Hardy–Sobolev critical exponents and Hardy terms were investigated. It is worth mentioning that, in order to prove Theorem A, when (H1) holds, the authors [28] used the same analytical techniques as those presented in [26]. In this paper, from the accurate estimates of ${∥u_{\epsilon }∥}^2$ and ${\int }_{\mathrm{\Omega }}{\left|u_{\epsilon }\right|}^{2^{*}\left(s\right)}/{\left|x\right|}^{s}dx$, we can obtain that $c<\frac{2-s}{2\left(N-s\right)}{A}_{\mu ,s}^{\left(N-s\right)/\left(2-s\right)}$; thus, in this case, the mountain pass lemma could also be used. We unify the methods by proving the existence of solutions to Equation (1) for both cases (H1) and (H2). The results in this paper contain all the cases of Theorem A in [28] and the results are also a useful supplement to the research results of Wang [29] and Ding [30].

Now, we consider some extreme cases: $\lambda =0,$ $\lambda =+\infty .$ If $\lambda =+\infty$ and $\eta =0,$ then $f(x,t)$ grows sublinearly at $t=0$ for any $x\in {\mathrm{\Omega }}^0$. From [34], we know that if $0\le \mu <\overline{\mu }-1,$ then the problem (1) has at least one positive solution. It is wort saying that this conclusion is also true for the cases $\lambda =+\infty$ and $0<\eta <+\infty .$ If $\lambda =0$ (in this case, $f(x,t)$ grows superlinearly at $t=0$ for any $x\in {\mathrm{\Omega }}^0$) and $\eta =0,$ from the Ding’s research [30], we know that if $f(x,t)$ satisfies the additional condition

$\left(f_4\right),$ then there exists $\rho >max\{2,\frac{N}{\sqrt{\overline{\mu }}+\sqrt{\overline{\mu }-\mu }},\frac{N-2\sqrt{\overline{\mu }-\mu }}{\sqrt{\overline{\mu }}}\}$ such that $0<\rho F\left(x,t\right)\le f\left(x,t\right)t$ for any $x\in \overline{\mathrm{\Omega }}$, $t\in {\mathbb{R}}^{+}\setminus \left\{0\right\}$.

Then, the problem (1) has at least one positive solution. However, if the function $g(x,t)$ grows linearly at $t=0$ for any $x\in {\mathrm{\Omega }}^0$, from Theorem 1, we know that $\mu =\overline{\mu }-1$ is a critical value. Notice that if $\lambda =0,$ the condition $\left(f_2\right)$ is reduced to $\left(f_4\right)$, we do not need the additional conditions $\rho >max\{2,\frac{N}{\sqrt{\overline{\mu }}+\sqrt{\overline{\mu }-\mu }},\frac{N-2\sqrt{\overline{\mu }-\mu }}{\sqrt{\overline{\mu }}}\}$ anymore, but $2<\rho \le 2^{*}\left(s\right)$ is needed; that is, the condition $\rho >max\{\frac{N}{\sqrt{\overline{\mu }}+\sqrt{\overline{\mu }-\mu }},\frac{N-2\sqrt{\overline{\mu }-\mu }}{\sqrt{\overline{\mu }}}\}$ is not needed. Obviously, $\rho \le 2^{*}\left(s\right)$ is also not needed in this paper. From the above discussions, we know that if the function $f(x,t)$ satisfies the conditions $\left(f_1\right)$ and $\left(f_2\right)$, $0<\mu \le \overline{\mu }-1,$ $0\le \eta <+\infty ,$ then one of the following conjectures is true:

Conjecture 1.

If $0\le \lambda \le +\infty ,$ then the problem (1) has at least one positive solution.

Conjecture 2.

There exist two constants ${\lambda }_2\left(\mu \right)$ and ${\lambda }_3\left(\mu \right),$ such that ${\lambda }_3\left(\mu \right)\ge {\lambda }_2\left(\mu \right)\ge {\lambda }_1\left(\mu \right)$. If $0\le \lambda <{\lambda }_2\left(\mu \right)$ or $\lambda >{\lambda }_3\left(\mu \right),$ then the problem (1) has at least one positive solution. Additionally, there exists a subinterval $(a,b)\subset \left[{\lambda }_2\left(\mu \right),{\lambda }_3\left(\mu \right)\right]$ such that the problem (1) has no solution when $\lambda \in (a,b).$

If the function $f(x,t)$ satisfies the conditions $\left(f_1\right)$ and $\left(f_2\right)$, $\overline{\mu }-1<\mu <\overline{\mu },$ $0\le \eta <+\infty ,$ from [33], we know that when $\lambda \le$ ${\lambda }_{*}\left(\mu \right),$ the problem (1) may have no positive solution; therefore, one of the following conjectures is true:

Conjecture 3.

If ${\lambda }_{*}\left(\mu \right)\le \lambda \le +\infty ,$ then the problem (1) has at least one positive solution.

Conjecture 4.

There exist two constants ${\lambda }_2\left(\mu \right)$ and ${\lambda }_3\left(\mu \right)$, such that ${\lambda }_3\left(\mu \right)\ge {\lambda }_2\left(\mu \right)\ge {\lambda }_1\left(\mu \right)$. If ${\lambda }_{*}\left(\mu \right)<\lambda <{\lambda }_2\left(\mu \right)$ or ${\lambda }_3\left(\mu \right)<\lambda \le +\infty ,$ then the problem (1) has at least one positive solution. Additionally, there exists a subinterval $(a,b)\subset \left[{\lambda }_2\left(\mu \right),{\lambda }_3\left(\mu \right)\right],$ such that the problem (1) has no solution when $\lambda \in (a,b).$

We leave the above four conjectures for further study.