Existence Results for Singular p-Biharmonic Problem with HARDY Potential and Critical Hardy-Sobolev Exponent

Abstract

In this article, we consider the singular $p$-biharmonic problem involving Hardy potential and critical Hardy–Sobolev exponent. Firstly, we study the existence of ground state solutions by using the minimization method on the associated Nehari manifold. Then, we investigate the least-energy sign-changing solutions by considering the Nehari nodal set. In both cases, the critical Sobolev exponent is of great importance as the solutions exists only if we are below the critical Sobolev exponent.

1. Introduction

We study the existence of ground state solutions and least-energy sign-changing solutions for the following problem

$${\Delta }_p^{2}u-{\lambda }_1{\displaystyle \frac{{|u|}^{p-2}}{{|x|}^{2p}}}={\displaystyle \frac{{|u|}^{p_{\ast }\left(\alpha \right)-2}}{{|x|}^{\alpha }}}u+{\lambda }_2\left({|x|}^{-\beta }\ast {|u|}^q\right){|u|}^{q-2}u\mathrm{in}{\mathbb{R}}^N,$$

(1)

where $p>2$, $2<q<p_{\ast }\left(\alpha \right)$, ${\lambda }_1>0$, ${\lambda }_2\in \mathbb{R}$, $\alpha ,\beta \in (0,N)$, and $N\ge 5$. Here, $p_{\ast }\left(\alpha \right)={\displaystyle \frac{p(N-\alpha )}{N-2p}}$ is the critical Sobolev exponent, ${\Delta }_p^{2}u={\Delta (|\Delta u|}^{p-2}\Delta u)$ is the p-biharmonic operator, and ${|x|}^{-\alpha }$ is the Riesz potential of order $\alpha \in (0,N)$.

Problems involving the $p$-biharmonic operator, Hardy potential, and singular nonlinearities are of great importance, as they appear in many applications, such as non-Newtonian fluids, viscous fluids, quantum mechanics, flame propagation, traveling waves in suspension bridges, and many more (see [1,2,3,4]). These type of problems have received a considerable attention recently (see [5,6,7]).

In [8], A. Drissi, A. Ghanmi, and D. D. Repous considered the problem

$${\Delta }_p^{2}u-\lambda {\displaystyle \frac{{|u|}^{p-2}}{{|x|}^{2p}}}+{\Delta }_{p}u={\displaystyle \frac{{|u|}^{p_{\ast }\left(\alpha \right)-2}}{{|x|}^{\alpha }}}u+{\lambda }_{1}f\left(x\right)h\left(u\right)\mathrm{in}{\mathbb{R}}^N,$$

(2)

where $0\le \alpha <2p,$ $1<p<{\displaystyle \frac{N}{2}},$ $\lambda ,{\lambda }_1>0$, and $p_{\ast }\left(\alpha \right)={\displaystyle \frac{p(N-\alpha )}{N-2p}}$. They studied the existence and multiplicity of solutions to (2).

Wang [9] had studied the following problem:

$$\left\{\begin{array}{cccc}\hfill {\Delta }_p^{2}u& =h(x,u)+\lambda {\displaystyle \frac{{|u|}^{r-2}}{{|x|}^{\alpha }}}u\hfill & \hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u& =\Delta u=0\hfill & \hfill & \mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

(3)

where $\Omega$ is a bounded domain. Wang has used the Mountain pass theorem to establish the existence results and Fountain theorem to find the existence of multiple solutions to (3). When $p=2$ and ${\lambda }_2=0$, (1) gives us the following biharmonic Choquard equation:

$${\Delta }^{2}u={\displaystyle \frac{{|u|}^{p_{\ast }\left(\alpha \right)-2}}{{|x|}^{\alpha }}}u\mathrm{in}{\mathbb{R}}^N.$$

(4)

The Choquard equation has also received a lot of attention (see [10,11,12,13,14]) and has appeared in many different contexts. For instance, in 1954, the following Choquard or nonlinear Schrödinger–Newton equation

$$-\Delta u+u={(|x|}^{-2}\ast u^2)u\mathrm{in}{\mathbb{R}}^N,$$

(5)

was first studied by Pekar [15] for $N=3$. Later in 1996, the Equation (5) was used by Penrose as a model in self-gravitating matter(see [16,17]). The following stationary Choquard equation

$$-\Delta u+V\left(x\right)u={(|x|}^{-\alpha }\ast {|u|}^b{)|u|}^{b-2}u\mathrm{in}{\mathbb{R}}^N,$$

arises in quantum theory and in the theory of Bose–Einstein condensation.

Biharmonic equations have been studied widely and have many physical applications such as phase field models of multi-phase systems, micro electro-mechanical systems, in thin film theory, nonlinear surface diffusion on solids, interface dynamics, biophysics, continuum mechanics, differential geometry, the flow in Hele–Shaw cells, the deformation of a nonlinear elastic beam [7], and the Willmore equation (see [18]). As the maximum principle cannot be applied to the biharmonic operator, this makes problems involving this operator even more interesting from a mathematical point of view (see [19,20,21,22,23,24]). In [25], Cao and Dai had considered the following biharmonic equation with Hatree-type nonlinearity:

$${\Delta }^{2}u=\left({|x|}^{-8}\ast {|u|}^2\right){|u|}^b,\mathrm{for}\mathrm{all} x\in {\mathbb{R}}^d,$$

where $0<b\le 1$, and $d\ge 9$. They used the methods of the moving plane and were able to prove that the non-negative classical solutions are radially symmetric. In the subcritical case $0<b<1$, they were also able to obtain results on the nonexistence of nontrivial, non-negative classical solutions.

In [26], Micheletti and Pistoiain studied the following problem:

$$\left\{\begin{array}{cccc}\hfill {\Delta }^{2}u+c\Delta u& =f(x,u)\hfill & \hfill & \mathrm{in}\Omega ,\hfill \\ \hfill u& =\Delta u=0\hfill & \hfill & \mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^N$. They used the mountain pass theorem and obtained multiple nontrivial solutions. Zhao and Xu in [27] studied the existence of infinitely many sign-changing solutions of the above problem by the use of the critical point theorem.

Motivated by the above results, we study the existence of ground state solutions to (1). We also obtained the sign-changing solutions to Equation (1), which is the novelity of this research paper. In the subsection below, we introduce some notations that will be used throughout this paper and present the variational framework.

Notations and Variational Framework

• $H_0^p\left({\mathbb{R}}^N\right)=W_0^{2,p}\left({\mathbb{R}}^N\right)$ is the Hilbert–Sobolev space, and we will be denoting it by E throughout this article.
• The Hardy–Sobolev exponent is related to the following Rellich inequality

  $${\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{|u|}^p}{{|x|}^{2p}}}dx\le {\left({\displaystyle \frac{p^2}{N(p-1)(N-2p)}}\right)}^p{\int }_{{\mathbb{R}}^N}{|\Delta u|}^{p}dx,\mathrm{for}\mathrm{all}u\in E.$$
  As per the Rellich inequality, E can be endowed with the norm

  $${||u||}_E={\left({\int }_{{\mathbb{R}}^N}{|\Delta u|}^p-{\lambda }_1{\displaystyle \frac{{|u|}^p}{{|x|}^{2p}}}\right)}^{{\displaystyle \frac{1}{p}}},$$

  provided that

  $$0<{\lambda }_1<{\left({\displaystyle \frac{N(p-1)(N-2p)}{p^2}}\right)}^p.$$
• $L^s\left({\mathbb{R}}^N\right)$ denotes the Lebesgue space in ${\mathbb{R}}^N$ of order $s\in [1,\infty ]$ with norm ${||.||}_s$.
•  

  $${||u||}_r\le S_r^{{\displaystyle \frac{-1}{p}}}{||u||}_E,$$

  for all $u\in E$ and $p\le r\le p_{\ast }$. Here,

  $$S_r=\underset{u\in E}{inf}{\displaystyle \frac{{\int }_{{\mathbb{R}}^N}\left({|\Delta u|}^p-{\lambda }_1\frac{{|u|}^p}{{|x|}^{2p}}\right)}{{\left({\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }{|u|}^r\right)}^{\frac{p}{r}}}}.$$
• ↪ denotes the continuous embeddings.

Note: The embedding

$$E=H_0^p\left({\mathbb{R}}^N\right)\hookrightarrow L^s\left({\mathbb{R}}^N\right),$$

is continuous for all $p\le s\le p_{\ast }$ and is compact for all $p\le s<p_{\ast }$.

• We will be using the following Hardy–Littlewood–Sobolev inequality

  $$|{\int }_{{\mathbb{R}}^N}\left({|x|}^{-\gamma }\ast u\right)v|\le {C\parallel u\parallel }_r{\parallel v\parallel }_t,$$

  (6)

  for $\gamma \in (0,N)$, $u\in L^r\left({\mathbb{R}}^N\right)$ and $v\in L^t\left({\mathbb{R}}^N\right)$ such that

  $${\displaystyle \frac{1}{r}}+{\displaystyle \frac{1}{t}}+{\displaystyle \frac{\gamma }{N}}=2.$$
• Assume that q satisfies

  $${\displaystyle \frac{p(2N-\beta )}{2N}}<q<{\displaystyle \frac{p(2N-\beta )}{2(N-2p)}}.$$

  (7)
• The Equation (1) has variational structure. We define the energy functional $\mathcal{J}:E\to \mathbb{R}$ by

  $$\begin{array}{cc}\hfill \mathcal{J}\left(u\right)& ={\displaystyle \frac{1}{p}}{\parallel u\parallel }_E^p-{\displaystyle \frac{1}{p_{\ast }\left(\alpha \right)}}{\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }{|u|}^{p_{\ast }\left(\alpha \right)}-{\displaystyle \frac{{\lambda }_2}{2q}}{\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {|u|}^q\right){|u|}^q.\hfill \end{array}$$

  (8)

Using the Hardy–Littlewood–Sobolev inequality (6) with (7), we obtain that the energy functional $\mathcal{J}$ is well defined, and $\mathcal{J}\in C^1\left(E\right)$. Also, a critical point of the energy functional $\mathcal{J}$ is a solution of (1).

Now, in Section 2, we will gather some preliminary results, which will be followed by Section 3, Section 4, Section 5 and Section 6. In these sections, we will present the main results, the proofs of our main results, and the conclusion.

2. Preliminary Results

Lemma 1.

([28], Lemma 1.1, [29], Lemma 2.3) Let $u\in E$ and $e\in [p,p_{\ast }\left(\alpha \right)]$. There exists a constant $C_0>0$ such that

$${\int }_{{\mathbb{R}}^N}{|u|}^e\le C_0||u||{\left(\underset{y\in {\mathbb{R}}^N}{sup}{\int }_{B_1\left(y\right)}{|u|}^e\right)}^{1-{\displaystyle \frac{2}{e}}}.$$

Lemma 2.

([30], Proposition 4.7.12) If a bounded sequence $\left({\mu }_n\right)$ converges to μ almost everywhere in $L^s\left({\mathbb{R}}^N\right)$ for some $s\in (1,\infty )$, then ${\mu }_n$ also converges weakly to μ in $L^s\left({\mathbb{R}}^N\right)$.

Lemma 3.

(Local Brezis–Lieb Lemma) Let $\left({\mu }_n\right)$ be a bounded sequence converging to μ almost everywhere in $L^s\left({\mathbb{R}}^N\right)$ for some $s\in (1,\infty )$. Then, we have

$$\underset{n\to \infty }{lim}{\int }_{{\mathbb{R}}^N}||{\mu }_n{|}^t-|{\mu }_n{-\mu |}^t-{|\mu |}^t{|}^{{\displaystyle \frac{s}{t}}}=0,$$

and

$$\underset{n\to \infty }{lim}{\int }_{{\mathbb{R}}^N}||{\mu }_n{|}^{t-1}{\mu }_n-|{\mu }_n{-\mu |}^{t-1}({\mu }_n-\mu )-{|\mu |}^{t-1}\mu {|}^{{\displaystyle \frac{s}{t}}}=0,$$

for all $1\le t\le s$.

Proof. 

Fix $\epsilon >0$. Then, there exists a constant $C\left(\epsilon \right)>0$ such that

$${||c+d|}^t-{|c|}^t{|}^{{\displaystyle \frac{s}{t}}}\le {\epsilon |c|}^s+C\left(\epsilon \right){|d|}^s,$$

(9)

for all c,$d\in \mathbb{R}$. Next, by (9), we have

$$\begin{array}{cc}\hfill |f_{n,\epsilon }|=& {\left(||{\mu }_n{|}^t-|{\mu }_n{-\mu |}^t-|{\mu }^t|{|}^{{\displaystyle \frac{s}{t}}}-\epsilon {|{\mu }_n-\mu |}^s\right)}^{+}\hfill \\ & \le {(1+C\left(\epsilon \right))|\mu |}^s.\hfill \end{array}$$

Now, using the Lebesgue Dominated Convergence theorem, one could obtain

$$\underset{{\mathbb{R}}^N}{\int }{f}_{n,\epsilon }\to 0\mathrm{as}n\to \infty .$$

(10)

Therefore, we have

$$||{\mu }_n{|}^t-|{\mu }_n{-\mu |}^t-{|\mu |}^t{|}^{{\displaystyle \frac{s}{t}}}\le f_{n,\epsilon }+\epsilon {|{\mu }_n-\mu |}^s,$$

and we obtain

$$\underset{n\to \infty }{lim\; sup}\underset{{\mathbb{R}}^N}{\int }||{\mu }_n{|}^t-|{\mu }_n{-\mu |}^t-{|\mu |}^t{|}^{{\displaystyle \frac{s}{t}}}\le C_0\epsilon ,$$

where $C_0={sup}_n{|{\mu }_n-\mu |}_q^q<\infty$. We finish our proof by letting $\epsilon \to 0$. □

Lemma 4.

(Nonlocal Brezis–Lieb Lemma ([29], Lemma 2.4) Let $N\ge 5$, $q\in [1,{\displaystyle \frac{2N}{2N-\beta }})$ and assume that $\left({\mu }_n\right)$ is a bounded sequence in $L^{{\displaystyle \frac{2Nq}{2N-\beta }}}\left({\mathbb{R}}^N\right)$ such that ${\mu }_n\to \mu$ almost everywhere in ${\mathbb{R}}^N$. Then, we have

$${\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast |{\mu }_n{|}^q\right)|{\mu }_n{|}^{q}dx-{\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast |{\mu }_n{-\mu |}^q\right)|{\mu }_n{-\mu |}^{q}dx\to {\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {|\mu |}^q\right){|\mu |}^{q}dx.$$

Proof. 

Let $s={\displaystyle \frac{2N}{2N-\beta }}$ in Lemma 3; then, one could obtain

$$|{\mu }_n{-\mu |}^q-|{\mu }_n{|}^q\to {|\mu |}^q\mathrm{strongly}\mathrm{in}{L}^{{\displaystyle \frac{2N}{2N-\beta }}}\left({\mathbb{R}}^N\right),$$

(11)

as $n\to \infty$. Using Lemma 2, we obtain

$$|{\mu }_n{-\mu |}^q\rightharpoonup 0\mathrm{weakly} \mathrm{in}{L}^{{\displaystyle \frac{2N}{2N-\beta }}}\left({\mathbb{R}}^N\right).$$

(12)

By use of the Hardy–Littlewood–Sobolev inequality (6), we have

$${|x|}^{-\beta }\ast \left(|{\mu }_n{-\mu |}^q-{|{\mu }_n|}^q\right)\to {|x|}^{-\beta }\ast {|\mu |}^q\mathrm{in}{L}^{{\displaystyle \frac{2N}{\beta }}}\left({\mathbb{R}}^N\right).$$

(13)

Also,

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {|{\mu }_n|}^q\right)|{\mu }_n{|}^{q}dx-{\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {|{\mu }_n-\mu |}^q\right){|{\mu }_n-\mu |}^{q}dx\hfill \\ & =\underset{{\mathbb{R}}^N}{\int }\left[{|x|}^{-\beta }\ast \left(|{\mu }_n{|}^q-{|{\mu }_n-\mu |}^q\right)\right]\left(|{\mu }_n{|}^q-{|{\mu }_n-\mu |}^q\right)dx\hfill \\ \hfill & +2\underset{{\mathbb{R}}^N}{\int }\left[{|x|}^{-\beta }\ast \left(|{\mu }_n{|}^q-{|{\mu }_n-\mu |}^q\right)\right]{|{\mu }_n-\mu |}^{q}dx.\hfill \end{array}$$

(14)

Finally, passing to the limit in (14) and using (11) and (12), the result holds. □

Lemma 5.

Suppose that $N\ge 5$, $\beta \in (0,N)$, and $q\in [1,{\displaystyle \frac{2N}{2N-\beta }})$. Let $\left({\mu }_n\right)$ be a bounded sequence in $L^{{\displaystyle \frac{2Nq}{2N-\beta }}}\left({\mathbb{R}}^N\right)$ such that ${\mu }_n\to \mu$ almost everywhere in ${\mathbb{R}}^N$. Then, for any $h\in L^{{\displaystyle \frac{2Nq}{2N-\beta }}}\left({\mathbb{R}}^N\right)$, we have

$${\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {|{\mu }_n|}^q\right)|{\mu }_n{|}^{q-2}{\mu }_{n}hdx\to {\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {|\mu |}^q\right){|\mu |}^{q-2}\mu hdx.$$

Proof. 

Assume that $h=h^{+}-h^{-}$ and ${\nu }_n={\mu }_n-\mu$. We prove the lemma for $h\ge 0$. Use Lemma 3 by taking $s={\displaystyle \frac{2N}{2N-\beta }}$ together with $(z_n,z)=({\mu }_n,\mu )$ and $(z_n,z)=({\mu }_n{h}^{1/c},\mu h^{1/c})$ in order to obtain

$$\left\{\begin{array}{c}|{\mu }_n{|}^q-|{\nu }_n{|}^q\to {|\mu |}^q\hfill \\ |{\mu }_n{|}^{q-2}{\mu }_{n}h-|{\nu }_n{|}^{q-2}{\nu }_{n}h\to {|\mu |}^{q-2}\mu h\hfill \end{array}\right.\mathrm{strongly}\mathrm{in}{L}^{{\displaystyle \frac{2N}{2N-\beta }}}\left({\mathbb{R}}^N\right).$$

Using the Hardy–Littlewood–Sobolev inequality, we obtain

$$\left\{\begin{array}{c}{|x|}^{-\beta }\ast \left(|{\mu }_n{|}^q-{|{\nu }_n|}^q\right)\to {|x|}^{-\beta }\ast {|\mu |}^q\hfill \\ {|x|}^{-\beta }\ast \left(|{\mu }_n{|}^{q-2}{\mu }_{n}h-{|{\nu }_n|}^{q-2}{\nu }_{n}h\right)\to {|x|}^{-\beta }\ast \left({|\mu |}^{q-2}\mu h\right)\hfill \end{array}\right.\mathrm{strongly}\mathrm{in}{L}^{{\displaystyle \frac{2N}{\beta }}}\left({\mathbb{R}}^N\right).$$

(15)

Next, using Lemma 2, we obtain

$$\left\{\begin{array}{c}|{\mu }_n{|}^{q-2}{\mu }_{n}h\rightharpoonup {|\mu |}^{q-2}\mu h\hfill \\ |{\nu }_n{|}^q\rightharpoonup 0\hfill \\ |{\nu }_n{|}^{q-2}{\nu }_{n}h\rightharpoonup 0\hfill \end{array}\right.\mathrm{weakly}\mathrm{in}{L}^{{\displaystyle \frac{2N}{2N-\beta }}}\left({\mathbb{R}}^N\right)$$

(16)

By Equations (15) and (16), we have

$$\begin{array}{cc}\hfill & \underset{{\mathbb{R}}^N}{\int }\left[{|x|}^{-\beta }\ast \left(|{\mu }_n{|}^q-{|{\nu }_n|}^q\right)\right]\left(|{\mu }_n{|}^{q-2}{\mu }_{n}h-{|{\nu }_n|}^{q-2}{\nu }_{n}h\right)\to {\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {|\mu |}^q\right){|\mu |}^{q-2}\mu h,\hfill \\ \hfill & \underset{{\mathbb{R}}^N}{\int }\left[{|x|}^{-\beta }\ast \left(|{\mu }_n{|}^q-{|{\nu }_n|}^q\right)\right]{|{\nu }_n|}^{q-2}{\nu }_{n}h\to 0,\hfill \\ \hfill & \underset{{\mathbb{R}}^N}{\int }\left[{|x|}^{-\beta }\ast \left(|{\mu }_n{|}^{q-2}{\mu }_{n}h-{|{\nu }_n|}^{q-2}{\nu }_{n}h\right)\right]{|{\nu }_n|}^q\to 0.\hfill \end{array}$$

(17)

Using the Hardy–Littlewood–Sobolev inequality together with Hölder’s inequality, we find

$$\begin{array}{cc}\hfill \left|\underset{{\mathbb{R}}^N}{\int }\left({|x|}^{-\beta }\ast {|{\nu }_n|}^q\right){|{\nu }_n|}^{q-2}{\nu }_{n}h\right|& \le \parallel {\nu }_n{\parallel }_{{\displaystyle \frac{2Nq}{2N-\beta }}}^q\parallel |{\nu }_n{{|}^{q-1}h\parallel }_{{\displaystyle \frac{2N}{2N-\beta }}}\hfill \\ \hfill & \le q\parallel |{\nu }_n{{|}^{q-1}h\parallel }_{{\displaystyle \frac{2N}{2N-\beta }}}.\hfill \end{array}$$

(18)

Next, by Lemma 2, we have ${\nu }_n^{{\displaystyle \frac{2N(q-1)}{2N-\beta }}}\rightharpoonup 0$ weakly in $L^{{\displaystyle \frac{q}{q-1}}}\left({\mathbb{R}}^N\right)$. Therefore,

$$\parallel |{\nu }_n{{|}^{q-1}h\parallel }_{{\displaystyle \frac{2N}{2N-\beta }}}={\left(\underset{{\mathbb{R}}^N}{\int }|{\nu }_n{|}^{{\displaystyle \frac{2N(q-1)}{2N-\beta }}}{|h|}^{{\displaystyle \frac{2N}{2N-\beta }}}\right)}^{{\displaystyle \frac{2N-\beta }{2N}}}\to 0.$$

Hence, using (18), we have

$$\underset{n\to \infty }{lim}\underset{{\mathbb{R}}^N}{\int }\left({|x|}^{-\beta }\ast {|{\nu }_n|}^q\right){|{\nu }_n|}^{q-2}{\nu }_{n}h=0.$$

(19)

Also, one could notice that

$$\begin{array}{cc}\hfill \underset{{\mathbb{R}}^N}{\int }\left({|x|}^{-\beta }\ast {|{\mu }_n|}^q\right){|{\mu }_n|}^{q-2}{\mu }_{n}h=& \underset{{\mathbb{R}}^N}{\int }\left[{|x|}^{-\beta }\ast \left(|{\mu }_n{|}^q-{|{\nu }_n|}^q\right)\right]\left(|{\mu }_n{|}^{q-2}{\mu }_{n}h-{|{\nu }_n|}^{q-2}{\nu }_{n}h\right)\hfill \\ \hfill & +\underset{{\mathbb{R}}^N}{\int }\left[{|x|}^{-\beta }\ast \left(|{\mu }_n{|}^q-{|{\nu }_n|}^q\right)\right]{|{\nu }_n|}^{q-2}{\nu }_{n}h\hfill \\ \hfill & +\underset{{\mathbb{R}}^N}{\int }\left[{|x|}^{-\beta }\ast \left(|{\mu }_n{|}^{q-2}{\mu }_{n}h-{|{\nu }_n|}^{q-2}{\nu }_{n}h\right)\right]{|{\nu }_n|}^q\hfill \\ \hfill & +\underset{{\mathbb{R}}^N}{\int }\left({|x|}^{-\beta }\ast {|{\nu }_n|}^q\right){|{\nu }_n|}^{q-2}{\nu }_{n}h.\hfill \end{array}$$

(20)

We obtain the desired result by passing to the limit in (20), together with (17) and (19). □

3. Main Results

Now, we present our main result on the existence of a ground state solution. We define the Nehari manifold associated with the energy functional J as

$$\mathcal{N}=\{u\in E\setminus \left\{0\right\}:\langle {\mathcal{J}}^{\prime }\left(u\right),u\rangle =0\},$$

(21)

and the ground state solutions will be obtained as minimizers of

$$m=\underset{u\in \mathcal{N}}{inf}\mathcal{J}\left(u\right).$$

Theorem 1.

Let $N\ge 5$, ${\lambda }_2>0$, $p_{\ast }\left(\alpha \right)>2q>p$, and q satisfy (7). Then, the Equation (1) has a ground state solution $u\in E$.

Next, we investigate the existence of least-energy sign-changing solutions for the Equation (1). Now, we use the minimization method on the Nehari nodal set defined as

$$\overline{\mathcal{N}}=\left\{u\in E:u^{\pm }\ne 0\mathrm{and}\langle {\mathcal{J}}^{\prime }\left(u\right),u^{\pm }\rangle =0\right\},$$

and solutions will be obtained as minimizers for

$$\overline{m}=\underset{u\in \overline{\mathcal{N}}}{inf}\mathcal{J}\left(u\right).$$

Here,

$$\begin{array}{cc}\hfill \langle {\mathcal{J}}^{\prime }\left(u\right),u^{\pm }\rangle & =\parallel u^{\pm }{\parallel }_E^p-{\displaystyle \frac{1}{p_{\ast }\left(\alpha \right)}}{\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }{\left(u^{\pm }\right)}^{p_{\ast }\left(\alpha \right)}-{\displaystyle \frac{{\lambda }_2}{2q}}{\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {\left(u^{\pm }\right)}^q\right){\left(u^{\pm }\right)}^q\hfill \\ & -{\displaystyle \frac{{\lambda }_2}{2q}}{\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {\left(u^{\pm }\right)}^q\right){\left(u^{\mp }\right)}^q.\hfill \end{array}$$

Theorem 2.

Let $N\ge 5$, ${\lambda }_2\in \mathbb{R}$, $p_{\ast }\left(\alpha \right)>2q>p$, and q satisfy (7). Then, the Equation (1) has a least-energy sign-changing solution $u\in E$.

We will now examine the ground state solutions to Equation (1) in the following section.

4. Proof of Theorem 1

Here in this section, we will conduct the analysis of the Palais–Smale sequences for $\mathcal{J}{\mid }_{\mathcal{N}}$, and we will take the ideas from [31,32] to prove that any Palais–Smale sequence of $\mathcal{J}{\mid }_{\mathcal{N}}$ is either converging strongly to its weak limit or differs from it by a finite number of sequences, which are the translated solutions of (4). Let us assume that ${\lambda }_2>0$. Then, for any $u,v\in E$, we have

$$\begin{array}{cc}\hfill \langle {\mathcal{J}}^{\prime }\left(tu\right),tu\rangle & =t^p{\parallel u\parallel }_E^p-t^{p_{\ast }\left(\alpha \right)}{\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }{|u|}^{p_{\ast }\left(\alpha \right)}-{\lambda }_2{t}^{2q}{\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {|u|}^q\right){|u|}^q,\hfill \end{array}$$

where $t>0$.

The equation $\langle {\mathcal{J}}^{\prime }\left(tu\right),tu\rangle =0$ has a unique positive solution $t=t\left(u\right)$ as $p_{\ast }\left(\alpha \right)>q>1$. This unique positive solution is also known as the projection of u on $\mathcal{N}$. Next, we will be interested in the main properties of the Nehari manifold $\mathcal{N}$.

Lemma 6.

(i) Energy functional $\mathcal{J}$ is coercive, that is, $\mathcal{J}\ge {c||u||}_E^p$ for some constant $c>0$.

(ii)$\mathcal{J}{\mid }_{\mathcal{N}}$ is bounded from below by a positive constant.

Proof. 

(i)

We have

$$\begin{array}{cc}\hfill \mathcal{J}\left(u\right)& =\mathcal{J}\left(u\right)-{\displaystyle \frac{1}{2q}}\langle {\mathcal{J}}^{\prime }\left(u\right),u\rangle \hfill \\ & =\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2q}}\right){\parallel u\parallel }_E^p+\left({\displaystyle \frac{1}{2q}}-{\displaystyle \frac{1}{p_{\ast }\left(\alpha \right)}}\right){\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }{|u|}^{p_{\ast }\left(\alpha \right)}dx\hfill \\ & \ge \left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2q}}\right){\parallel u\parallel }_E^p.\hfill \end{array}$$

By taking $c={\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2q}}$, we could reach to our conclusion.

(ii)

Here, we use the fact that the embeddings $E\hookrightarrow L^s\left({\mathbb{R}}^N\right)$ and $E\hookrightarrow L^{{\displaystyle \frac{2Nq}{2N-\beta }}}\left({\mathbb{R}}^N\right)$ are continuous for $p\le s\le p^{\ast }$, together with the Hardy–Littlewood–Sobolev inequality. For any $u\in \mathcal{N}$, we have

$$\begin{array}{cc}\hfill 0=\langle {\mathcal{J}}^{\prime }\left(u\right),u\rangle & ={\parallel u\parallel }_E^p-{\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }{|u|}^{p_{\ast }\left(\alpha \right)}-{\lambda }_2{\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {|u|}^q\right){|u|}^q\hfill \\ & \ge {\parallel u\parallel }_E^p-S_{p_{\ast }\left(\alpha \right)}^{-{\displaystyle \frac{p_{\ast }\left(\alpha \right)}{p}}}{\parallel u\parallel }_E^{p_{\ast }\left(\alpha \right)}-C_0{\lambda }_2{\parallel u\parallel }_E^{2q}.\hfill \end{array}$$

Then, there exists some constant $C_1>0$ such that

$${\parallel u\parallel }_E\ge C_1>0\mathrm{for}\mathrm{all}u\in \mathcal{N}.$$

(22)

Next, we use the fact that $\mathcal{J}{\mid }_{\mathcal{N}}$ is coercive, together with (22), in order to obtain

$$\mathcal{J}\left(u\right)\ge \left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2q}}\right)C_1^p>0.$$

□

Lemma 7.

Any critical point u of $\mathcal{J}{\mid }_{\mathcal{N}}$ will be a free critical point.

Proof. 

Let $\mathcal{K}\left(u\right)=\langle {\mathcal{J}}^{\prime }\left(u\right),u\rangle$ for any $u\in E$. By using (22), one could obtain

$$\begin{array}{cc}\hfill \langle {\mathcal{K}}^{\prime }\left(u\right),u\rangle & ={p\parallel u\parallel }^p-p_{\ast }\left(\alpha \right){\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }{|u|}^{p_{\ast }\left(\alpha \right)}-2q{\lambda }_2{\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {|u|}^q\right){|u|}^q\hfill \\ \hfill & ={(p-2q)\parallel u\parallel }_E^p+(2q-p_{\ast }\left(\alpha \right)){\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }{|u|}^{p_{\ast }\left(\alpha \right)}\hfill \\ \hfill & \le -{(2q-p)\parallel u\parallel }_E^p\hfill \\ \hfill & <-(2q-p)C_1,\hfill \end{array}$$

(23)

for any $u\in \mathcal{N}$. Suppose that u is a critical point of $\mathcal{J}$ in $\mathcal{N}$. By the Lagrange multiplier theorem, we obtain that there exists $ϵ\in \mathbb{R}$ such that ${\mathcal{J}}^{\prime }\left(u\right)=ϵ{\mathcal{K}}^{\prime }\left(u\right)$. Therefore, we have $\langle {\mathcal{J}}^{\prime }\left(u\right),u\rangle =ϵ\langle {\mathcal{K}}^{\prime }\left(u\right),u\rangle$. As $\langle {\mathcal{K}}^{\prime }\left(u\right),u\rangle <0$, so $ϵ=0$, and furthermore, we obtain ${\mathcal{J}}^{\prime }\left(u\right)=0$. □

Lemma 8.

Let us suppose the sequence $\left(u_n\right)$ is a $\left(PS\right)$ sequence for $\mathcal{J}{\mid }_{\mathcal{N}}$. Then, it is also a $\left(PS\right)$ sequence for $\mathcal{J}$.

Proof. 

Suppose that $\left(u_n\right)\subset \mathcal{N}$ is a $\left(PS\right)$ sequence for $\mathcal{J}{\mid }_{\mathcal{N}}$. This is due to the fact that

$$\mathcal{J}\left(u_n\right)\ge \left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2q}}\right){\parallel u_n\parallel }_E^p,$$

that is, $\left(u_n\right)$ is bounded in E. Next, we need to prove that ${\mathcal{J}}^{\prime }\left(u_n\right)\to 0$. We could notice that

$${\mathcal{J}}^{\prime }\left(u_n\right)-{ϵ}_n{\mathcal{K}}^{\prime }\left(u_n\right)={\mathcal{J}}^{\prime }{\mid }_{\mathcal{N}}\left(u_n\right)=o\left(1\right),$$

for some ${ϵ}_n\in \mathbb{R}$, which yields

$${ϵ}_n\langle {\mathcal{K}}^{\prime }\left(u_n\right),u_n\rangle =\langle {\mathcal{J}}^{\prime }\left(u_n\right),u_n\rangle +o\left(1\right)=o\left(1\right).$$

Using (23), we obtain ${ϵ}_n\to 0$, and therefore, we obtain ${\mathcal{J}}^{\prime }\left(u_n\right)\to 0$. □

Compactness

We define the energy functional $\mathcal{I}:E\to \mathbb{R}$ by

$$\mathcal{I}\left(u\right)={\displaystyle \frac{1}{p}}{\parallel u\parallel }_E^p-{\displaystyle \frac{1}{p_{\ast }\left(\alpha \right)}}\underset{{\mathbb{R}}^N}{\int }{|x|}^{-\alpha }{|u|}^{p_{\ast }\left(\alpha \right)},$$

and the corresponding Nehari manifold for $\mathcal{I}$ by

$${\mathcal{N}}_{\mathcal{I}}=\{u\in E\setminus \left\{0\right\}:\langle {\mathcal{I}}^{\prime }\left(u\right),u\rangle =0\}.$$

Let

$$m_{\mathcal{I}}=\underset{u\in {\mathcal{N}}_{\mathcal{I}}}{inf}\mathcal{I}\left(u\right).$$

And we have

$$\langle {\mathcal{I}}^{\prime }\left(u\right),u\rangle ={\parallel u\parallel }_E^p-{\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }{|u|}^{p_{\ast }\left(\alpha \right)}.$$

Lemma 9.

Let $\left(u_n\right)\subset {\mathcal{N}}_{\mathcal{I}}$ be a $\left(PS\right)$ sequence of $\mathcal{J}{\mid }_{\mathcal{N}}$, that is, $\left(\mathcal{J}\left(u_n\right)\right)$ is bounded, and ${\mathcal{J}}^{\prime }{\mid }_{\mathcal{N}}\left(u_n\right)\to 0$ strongly in $H_0^{-p}\left({\mathbb{R}}^N\right)$. Then, there exists a solution $u\in E$ of (1) such that, when we replace the sequence $\left(u_n\right)$ with the subsequence, one could have either of the following alternatives:

(i) 

$u_n\to u$ strongly in E;

or

(ii) 

$u_n\rightharpoonup u$ weakly in E. Furthermore, there exists a positive integer $l\ge 1$ and l nontrivial weak solutions to (4), that is, l functions $u_1,u_2,\dots ,u_l\in E$ and l sequences of points $\left(w_{n,1}\right)$, $\left(w_{n,2}\right)$, …, and $\left(w_{n,l}\right)\subset {\mathbb{R}}^N$ such that the following conditions hold:

(a) 

$|w_{n,j}|\to \infty$ and $|w_{n,j}-w_{n,i}|\to \infty$ if $i\ne j$ and $n\to \infty$;

(b) 

$u_n-{\sum }_{j=1}^l{u}_j(·+w_{n,j})\to u$ in E;

(c) 

$\mathcal{J}\left(u_n\right)\to \mathcal{J}\left(u\right)+{\sum }_{j=1}^l\mathcal{I}\left(u_j\right)$.

Proof. 

Since, $\left(u_n\right)\in E$ is a bounded sequence, so there exists $u\in E$ such that, up to a subsequence, we have

$$\left\{\begin{array}{cc}\hfill u_n& \rightharpoonup u\mathrm{weakly}\mathrm{in}E,\hfill \\ \hfill u_n& \rightharpoonup u\mathrm{weakly}\mathrm{in}{L}^s\left({\mathbb{R}}^N\right),p\le s\le p_{\ast },\hfill \\ \hfill u_n& \to u\mathrm{a}.\mathrm{e}.\mathrm{in} {\mathbb{R}}^N.\hfill \end{array}\right.$$

(24)

Using Lemma 5 and (24), we obtain

$${\mathcal{J}}^{\prime }\left(u\right)=0,$$

which gives us that $u\in E$ is a solution of (1). In the case if $u_n\to u$ being strongly in E, then $\left(i\right)$ holds.

Let us assume that $\left(u_n\right)\in E$ does not converge strongly to u and define $f_{n,1}=u_n-u$. Then, $\left(f_{n,1}\right)$ converges weakly (not strongly) to zero in E, and using the Brezis–Lieb Lemma (see [33]), we have

$$\parallel u_n{\parallel }_E^{p_{\ast }\left(\alpha \right)}={\parallel u\parallel }_E^{p_{\ast }\left(\alpha \right)}+{\parallel f_{n,1}\parallel }_E^{p_{\ast }\left(\alpha \right)}+o\left(1\right).$$

(25)

Therefore,

$$\underset{n\to \infty }{lim}{\int }_{{\mathbb{R}}^N}\left({|x|}^{-\alpha }|u_n{|}^{p_{\ast }\left(\alpha \right)}-{|x|}^{-\alpha }{|f_{n,1}|}^{p_{\ast }\left(\alpha \right)}\right)dx=\underset{{\mathbb{R}}^N}{\int }{|x|}^{-\alpha }{|u|}^{p_{\ast }\left(\alpha \right)}dx,$$

which implies that

$${\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }|u_n{|}^{p_{\ast }\left(\alpha \right)}dx=\underset{{\mathbb{R}}^N}{\int }{|x|}^{-\alpha }|f_{n,1}{|}^{p_{\ast }\left(\alpha \right)}dx+\underset{{\mathbb{R}}^N}{\int }{|x|}^{-\alpha }{|u|}^{p_{\ast }\left(\alpha \right)}dx+o\left(1\right),$$

(26)

Using Equations (25) and (26), we have

$$\mathcal{J}\left(u_n\right)=\mathcal{J}\left(u\right)+\mathcal{I}\left(f_{n,1}\right)+o\left(1\right).$$

(27)

Also, for any $h\in E$, we have

$$\langle {\mathcal{I}}^{\prime }\left(f_{n,1}\right),h\rangle =o\left(1\right).$$

(28)

Next, using Lemma 4, we obtain

$$\begin{array}{cc}\hfill 0=\langle {\mathcal{J}}^{\prime }\left(u_n\right),u_n\rangle & =\langle {\mathcal{J}}^{\prime }\left(u\right),u\rangle +\langle {\mathcal{I}}^{\prime }\left(f_{n,1}\right),f_{n,1}\rangle +o\left(1\right)\hfill \\ & =\langle {\mathcal{I}}^{\prime }\left(f_{n,1}\right),f_{n,1}\rangle +o\left(1\right).\hfill \end{array}$$

Hence,

$$\langle {\mathcal{I}}^{\prime }\left(f_{n,1}\right),f_{n,1}\rangle =o\left(1\right).$$

(29)

Also, we have

$$\Gamma :=\underset{n\to \infty }{lim\; sup}\left(\underset{v\in {\mathbb{R}}^N}{sup}{\int }_{B_1\left(v\right)}{|f_{n,1}|}^{p_{\ast }\left(\alpha \right)}\right)>0.$$

Therefore, one could find $w_{n,1}\in {\mathbb{R}}^N$ such that

$${\int }_{B_1\left(w_{n,1}\right)}{|f_{n,1}|}^{p_{\ast }\left(\alpha \right)}>{\displaystyle \frac{\Gamma }{2}}.$$

(30)

So, for any sequence $\left(f_{n,1}(·+w_{n,1})\right)$, there exists $u_1\in E$ such that, up to a subsequence, we have

$$\begin{array}{cc}\hfill f_{n,1}(·+w_{n,1})& \rightharpoonup u_1\mathrm{weakly}\mathrm{in}E,\hfill \\ \hfill f_{n,1}(·+w_{n,1})& \to u_1\mathrm{strongly}\mathrm{in}{L}_{loc}^{p_{\ast }\left(\alpha \right)}\left({\mathbb{R}}^N\right),\hfill \\ \hfill f_{n,1}(·+w_{n,1})& \to u_1\mathrm{a}.\mathrm{e}.\mathrm{in}{\mathbb{R}}^N.\hfill \end{array}$$

Next, by passing to the limit in (30), we obtain

$${\int }_{B_1\left(0\right)}{|u_1|}^{p_{\ast }\left(\alpha \right)}\ge {\displaystyle \frac{\beta }{2}},$$

which gives us $u_1\not\equiv 0$. Since $\left(f_{n,1}\right)\rightharpoonup 0$ weakly in E, we obtain that $\left(w_{n,1}\right)$ is unbounded. Therefore, passing to a subsequence, we obtain $|w_{n,1}|\to \infty$. Next, by using (29), we have ${\mathcal{I}}^{\prime }\left(u_1\right)=0$, which implies that $u_1$ is a nontrivial solution of (4). Next, we define

$$f_{n,2}\left(x\right)=f_{n,1}\left(x\right)-u_1(x-w_{n,1}).$$

Similarly as before, we have

$$\parallel f_{n,1}{\parallel }^{p_{\ast }\left(\alpha \right)}=\parallel u_1{\parallel }^{p_{\ast }\left(\alpha \right)}+{\parallel f_{n,2}\parallel }^{p_{\ast }\left(\alpha \right)}+o\left(1\right).$$

And,

$${\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }|f_{n,1}{|}^{p_{\ast }\left(\alpha \right)}={\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }|u_1{|}^{p_{\ast }\left(\alpha \right)}+\underset{{\mathbb{R}}^N}{\int }{|x|}^{-\alpha }{|f_{n,2}|}^{p_{\ast }\left(\alpha \right)}+o\left(1\right).$$

Therefore,

$$\mathcal{I}\left(f_{n,1}\right)=\mathcal{I}\left(u_1\right)+\mathcal{I}\left(f_{n,2}\right)+o\left(1\right).$$

Using (27), we have

$$\mathcal{J}\left(u_n\right)=\mathcal{J}\left(u\right)+\mathcal{I}\left(u_1\right)+\mathcal{I}\left(f_{n,2}\right)+o\left(1\right).$$

By using the same approach as above, we have

$$\langle {\mathcal{I}}^{\prime }\left(f_{n,2}\right),h\rangle =o\left(1\right)\mathrm{for}\mathrm{any}h\in E$$

and

$$\langle {\mathcal{I}}^{\prime }\left(f_{n,2}\right),f_{n,2}\rangle =o\left(1\right).$$

Now, if $\left(f_{n,2}\right)\to 0$ strongly, then we could take $l=1$ in the Lemma 9 to conclude the proof.

Suppose that $f_{n,2}\rightharpoonup 0$ weakly (not strongly) in E and we iterate the process l times; we could find a set of sequences $\left(w_{n,j}\right)\subset {\mathbb{R}}^N$, where $1\le j\le l$, with

$$|w_{n,j}|\to \infty \mathrm{and}|w_{n,i}-w_{n,j}|\to \infty \mathrm{as}n\to \infty ,i\ne j$$

and l nontrivial solutions $u_1$, $u_2$, …, $u_l\in E$ of (4) such that, by letting

$$f_{n,j}\left(x\right):=f_{n,j-1}\left(x\right)-u_{j-1}(x-w_{n,j-1}),2\le j\le l,$$

we obtain

$$f_{n,j}(x+w_{n,j})\rightharpoonup u_j\mathrm{weakly}\mathrm{in}E$$

and

$$\mathcal{J}\left(u_n\right)=\mathcal{J}\left(u\right)+\sum _{j=1}^l\mathcal{I}\left(u_j\right)+\mathcal{I}\left(y_{n,l}\right)+o\left(1\right).$$

As $\mathcal{J}\left(u_n\right)$ is bounded and $\mathcal{I}\left(u_j\right)\ge b_{\mathcal{I}}$, we obtain the desired result by iterating the process a finite number of times. □

Lemma 10.

For any $e\in (0,m_{\mathcal{I}})$, any sequence $\left(u_n\right)$ which is a ${\left(PS\right)}_e$ sequence of $\mathcal{J}{\mid }_{\mathcal{N}}$, is also relatively compact.

Proof. 

We assume $\left(u_n\right)$ is a ${\left(PS\right)}_e$ sequence of $\mathcal{J}$ in $\mathcal{N}$. Using Lemma 9, we have $\mathcal{I}\left(u_j\right)\ge m_{\mathcal{I}}$. Therefore, up to a subsequence $u_n\to u$ strongly in E, this gives us that u is a solution of (1). □

In order to finish the proof of Theorem 1, we need the following result.

Lemma 11.

$$m<m_{\mathcal{I}}.$$

Proof. 

Let us denote the ground state solution of (4) by $R\in E$, and such a solution exists (see [34] and references therein). Suppose that the projection of R on $\mathcal{N}$ is $tR$, that is, $t=t\left(R\right)>0$ is the unique real number such that $tR\in \mathcal{N}$. Since $R\in {\mathcal{N}}_{\mathcal{I}}$ and $tR\in \mathcal{N}$, we obtain

$${||R||}^p={\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }{|R|}^{p_{\ast }\left(\alpha \right)}$$

(31)

and

$$t^p{\parallel R\parallel }^p=t^{p_{\ast }\left(\alpha \right)}{\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }{|R|}^{p_{\ast }\left(\alpha \right)}+{\lambda }_2{t}^{2q}{\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {|R|}^q\right){|R|}^q.$$

We could notice that $t<1$ from the above two equalities. Therefore, we obtain

$$\begin{array}{cc}\hfill m\le \mathcal{J}\left(tR\right)& ={\displaystyle \frac{1}{p}}{t}^p{\parallel R\parallel }^p-{\displaystyle \frac{1}{p_{\ast }\left(\alpha \right)}}{t}^{p_{\ast }\left(\alpha \right)}{\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }{|R|}^{p_{\ast }\left(\alpha \right)}-{\displaystyle \frac{{\lambda }_2}{2q}}{t}^{2q}{\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {|R|}^q\right){|R|}^q\hfill \\ \hfill & =\left({\displaystyle \frac{t^p}{p}}-{\displaystyle \frac{t^{p_{\ast }\left(\alpha \right)}}{p_{\ast }\left(\alpha \right)}}\right){\parallel R\parallel }^p-{\displaystyle \frac{1}{2q}}\left(t^p{||R||}^p-t^{p_{\ast }\left(\alpha \right)}{\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }{|R|}^{p_{\ast }\left(\alpha \right)}\right)\hfill \\ \hfill & =t^p\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2q}}\right){\parallel R\parallel }^p+t^{p_{\ast }\left(\alpha \right)}\left({\displaystyle \frac{1}{2q}}-{\displaystyle \frac{1}{p_{\ast }\left(\alpha \right)}}\right){\parallel R\parallel }^p\hfill \\ \hfill & <\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2q}}\right){\parallel R\parallel }^p+\left({\displaystyle \frac{1}{2q}}-{\displaystyle \frac{1}{p_{\ast }\left(\alpha \right)}}\right){\parallel R\parallel }^p\hfill \\ \hfill & <\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{p_{\ast }\left(\alpha \right)}}\right){\parallel R\parallel }^p=\mathcal{I}\left(R\right)=m_{\mathcal{I}}.\hfill \end{array}$$

Hence, we reach our conclusion. □

Furthermore, using the Ekeland variational principle, for any $n\ge 1$, there exists $\left(u_n\right)\in \mathcal{N}$ such that

$$\begin{array}{cccc}\hfill \mathcal{J}\left(u_n\right)& \le m+{\displaystyle \frac{1}{n}}\hfill & \hfill & \mathrm{for}\mathrm{all}n\ge 1,\hfill \\ \hfill \mathcal{J}\left(u_n\right)& \le \mathcal{J}\left(\tilde{u}\right)+{\displaystyle \frac{1}{n}}\parallel \tilde{u}-u_n\parallel \hfill & \hfill & \mathrm{for}\mathrm{all}\tilde{u}\in \mathcal{N},n\ge 1.\hfill \end{array}$$

Next, we could easily deduce that $\left(u_n\right)\in \mathcal{N}$ is a ${\left(PS\right)}_e$ sequence for $\mathcal{J}$ on $\mathcal{N}$. Using Lemmas 11 and 10, we obtain that, up to a subsequence $u_n\to u$ strongly in E, this is a ground state solution of the $\mathcal{J}$.

5. Proof of Theorem 2

In this section, we are concerned with the existence of a least-energy sign-changing solution of (1).

Proof of Theorem

Lemma 12.

Let us assume that $N\ge 5$, $p_{\ast }\left(\alpha \right)>2q>p$ and ${\lambda }_2\in \mathbb{R}$. There exists a unique pair $(\overline{{\theta }_1},\overline{{\theta }_2})\in (0,\infty )\times (0,\infty )$ such that, for any $u\in E$ and $u^{\pm }\ne 0$, $\overline{{\theta }_1}{u}^{+}+\overline{{\theta }_2}{u}^{-}\in \overline{\mathcal{N}}$. Also, for any $u\in \overline{\mathcal{N}}$, $\mathcal{J}\left(u\right)\ge \mathcal{J}({\theta }_1{u}^{+}+{\theta }_2{u}^{-})$ for all ${\theta }_1,{\theta }_2\ge 0$.

Proof. 

We will be following the idea of [35] in order to prove this result. Let us define the function $\phi :[0,\infty )\times [0,\infty )\to \mathbb{R}$ by

$$\begin{array}{cc}\hfill \phi ({\theta }_1,{\theta }_2)& =\mathcal{J}({\theta }_1^{{\displaystyle \frac{1}{2p_{\ast }\left(\alpha \right)}}}{u}^{+}+{\theta }_2^{{\displaystyle \frac{1}{2p_{\ast }\left(\alpha \right)}}}{u}^{-})\hfill \\ \hfill & ={\displaystyle \frac{{\theta }_1^{\frac{p}{2p_{\ast }\left(\alpha \right)}}}{p}}\parallel u^{+}{\parallel }_E^p+{\displaystyle \frac{{\theta }_2^{\frac{p}{2p_{\ast }\left(\alpha \right)}}}{p}}{\parallel u^{-}\parallel }_E^p-{\lambda }_2{\displaystyle \frac{{\theta }_1^{\frac{q}{p_{\ast }\left(\alpha \right)}}}{2q}}{\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {\left(u^{+}\right)}^q\right){\left(u^{+}\right)}^q\hfill \\ & -{\lambda }_2{\displaystyle \frac{{\theta }_2^{\frac{q}{p_{\ast }\left(\alpha \right)}}}{2q}}{\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {\left(u^{-}\right)}^q\right){\left(u^{-}\right)}^q-{\lambda }_2{\displaystyle \frac{{\theta }_1^{\frac{q}{2p_{\ast }\left(\alpha \right)}}{\theta }_2^{\frac{q}{2p_{\ast }\left(\alpha \right)}}}{2q}}{\int }_{{\mathbb{R}}^N}({|x|}^{-\beta }\ast {\left(u^{+}\right)}^q{\left(u^{-}\right)}^q\hfill \\ & -{\displaystyle \frac{{\theta }_1^{\frac{1}{2}}}{p_{\ast }\left(\alpha \right)}}{\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }{\left(u^{+}\right)}^{p_{\ast }\left(\alpha \right)}-{\displaystyle \frac{{\theta }_2^{\frac{1}{2}}}{p_{\ast }\left(\alpha \right)}}{\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }{\left(u^{-}\right)}^{p_{\ast }\left(\alpha \right)}.\hfill \end{array}$$

Note that $\phi$ is strictly concave, and therefore, $\phi$ has at most one maximum point. We also have

$$\underset{{\theta }_1\to \infty }{lim}\phi ({\theta }_1,{\theta }_2)=-\infty \mathrm{for}\mathrm{all} {\theta }_2\ge 0\mathrm{and}\underset{{\theta }_2\to \infty }{lim}\phi ({\theta }_1,{\theta }_2)=-\infty \mathrm{for}\mathrm{all} {\theta }_1\ge 0,$$

(32)

and one could notice that

$$\underset{{\theta }_1\searrow 0}{lim}{\displaystyle \frac{\partial \phi }{\partial {\theta }_1}}({\theta }_1,{\theta }_2)=\infty \mathrm{for}\mathrm{all} {\theta }_2>0\mathrm{and}\underset{{\theta }_2\searrow 0}{lim}{\displaystyle \frac{\partial \phi }{\partial {\theta }_2}}({\theta }_1,{\theta }_2)=\infty \mathrm{for}\mathrm{all} {\theta }_1>0.$$

(33)

Using (32) and (33), it could be seen that the maximum cannot be achieved at the boundary. Hence, $\phi$ has exactly one maximum point $(\overline{{\theta }_1},\overline{{\theta }_2})\in (0,\infty )\times (0,\infty )$. □

We divide our proof into two steps.

Step 1. The energy level $\overline{m}>0$ is achieved by some $\sigma \in \overline{\mathcal{N}}$.

Let $\left(u_n\right)\subset \overline{\mathcal{N}}$ be a minimizing sequence for $\overline{m}$. We have

$$\begin{array}{cc}\hfill \mathcal{J}\left(u_n\right)& =\mathcal{J}\left(u_n\right)-{\displaystyle \frac{1}{2q}}\langle {\mathcal{J}}^{\prime }\left(u_n\right),u_n\rangle \hfill \\ \hfill & =\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2q}}\right)\parallel u_n{\parallel }_E^p+\left({\displaystyle \frac{1}{2q}}-{\displaystyle \frac{1}{p_{\ast }\left(\alpha \right)}}\right){\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }{|u_n|}^{p_{\ast }\left(\alpha \right)}\hfill \\ \hfill & \ge \left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{2q}}\right){\parallel u_n\parallel }_E^p\hfill \\ \hfill & \ge C_1{\parallel u_n\parallel }_E^p,\hfill \end{array}$$

for some positive constant $C_1>0$. Therefore, for $C_2>0$, one has

$$\parallel u_n{\parallel }_E^p\le C_2\mathcal{J}\left(u_n\right)\le M,$$

that is, $\left(u_n\right)$ is bounded in E. Hence, $\left(u_n^{+}\right)$ and $\left(u_n^{-}\right)$ are also bounded in E. By passing to a subsequence, there exists $u^{+}$, $u^{-}\in E$ such that

$$u_n^{+}\rightharpoonup u^{+} \mathrm{and} u_n^{-}\rightharpoonup u^{-}\mathrm{weakly}\mathrm{in}E.$$

Since q satisfies (7), we have that the embeddings $E\hookrightarrow L^s\left({\mathbb{R}}^N\right)$ and $E\hookrightarrow L^{{\displaystyle \frac{2Nq}{2N-\beta }}}\left({\mathbb{R}}^N\right)$ are compact for $p<s<p_{\ast }$. Hence,

$$u_n^{\pm }\to u^{\pm }\mathrm{strongly}\mathrm{in}{L}^{p_{\ast }\left(\alpha \right)}\left({\mathbb{R}}^N\right)\cap L^{{\displaystyle \frac{2Nq}{2N-\beta }}}\left({\mathbb{R}}^N\right).$$

(34)

Next, by using the Hardy–Littlewood–Sobolev inequality, we obtain

$$\begin{array}{cc}\hfill C\left(\parallel u_n^{\pm }{\parallel }_{L^{p_{\ast }\left(\alpha \right)}\left({\mathbb{R}}^N\right)}^p+{\parallel u_n^{\pm }\parallel }_{L^{{\displaystyle \frac{2Nq}{2N-\beta }}}}^p\right)& \le \parallel u_n^{\pm }{\parallel }_E^p\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }|u_n^{\pm }{|}^{p_{\ast }\left(\alpha \right)}+|{\lambda }_2|{\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {|u_n^{\pm }|}^q\right){|u_n^{\pm }|}^q\hfill \\ \hfill & \le C\left(\parallel u_n^{\pm }{\parallel }_{L^{p_{\ast }\left(\alpha \right)}\left({\mathbb{R}}^N\right)}^{p_{\ast }\left(\alpha \right)}+{\parallel u_n^{\pm }\parallel }_{L^{{\displaystyle \frac{2Nq}{2N-\beta }}}}^q\right)\hfill \\ \hfill & \le C\left(\parallel u_n^{\pm }{\parallel }_{L^{p_{\ast }\left(\alpha \right)}\left({\mathbb{R}}^N\right)}^p+{\parallel u_n^{\pm }\parallel }_{L^{{\displaystyle \frac{2Nq}{2N-\beta }}}}^p\right)\left(\parallel u_n^{\pm }{\parallel }_{L^{p_{\ast }\left(\alpha \right)}\left({\mathbb{R}}^N\right)}^{p_{\ast }\left(\alpha \right)-p}+{\parallel u_n^{\pm }\parallel }_{L^{{\displaystyle \frac{2Nq}{2N-\beta }}}}^{q-p}\right).\hfill \end{array}$$

Since $u_n^{\pm }\ne 0$, we obtain

$$\left(\parallel u_n^{\pm }{\parallel }_{L^{p_{\ast }\left(\alpha \right)}\left({\mathbb{R}}^N\right)}^{p_{\ast }\left(\alpha \right)-p}+{\parallel u_n^{\pm }\parallel }_{L^{{\displaystyle \frac{2Nq}{2N-\beta }}}}^{q-p}\right)\ge C>0\mathrm{for}\mathrm{all}n\ge 1.$$

(35)

Using (34) and (35), we have that $u^{\pm }\ne 0$. Next, we use (34) together with the Hardy–Littlewood–Sobolev inequality and deduce

$$\begin{array}{cccc}\hfill & {\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }{\left(u_n^{\pm }\right)}^{p_{\ast }\left(\alpha \right)}\hfill & \hfill & {\int }_{{\mathbb{R}}^N}{|x|}^{-\alpha }{\left(u^{\pm }\right)}^{p_{\ast }\left(\alpha \right)},\hfill \\ \hfill & {\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {\left(u_n^{\pm }\right)}^q\right){\left(u_n^{\pm }\right)}^q\hfill & \hfill & \to {\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {\left(u^{\pm }\right)}^q\right){\left(u^{\pm }\right)}^q,\hfill \end{array}$$

and

$${\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {\left(u_n^{+}\right)}^q\right){\left(u_n^{-}\right)}^q\to {\int }_{{\mathbb{R}}^N}\left({|x|}^{-\beta }\ast {\left(u^{+}\right)}^q\right){\left(u^{-}\right)}^q.$$

Also, by Lemma 12, there exists a unique pair $(\overline{{\theta }_1},\overline{{\theta }_2})$ such that $\overline{{\theta }_1}{u}^{+}+\overline{{\theta }_2}{u}^{-}\in \overline{\mathcal{N}}$. Since the norm ${\parallel .\parallel }_E$ is weakly lower semi-continuous, we deduce that

$$\begin{array}{cc}\hfill \overline{m}\le \mathcal{J}(\overline{{\theta }_1}{u}^{+}+\overline{{\theta }_2}{u}^{-})& \le \underset{n\to \infty }{lim\; inf}\mathcal{J}(\overline{{\theta }_1}{u}^{+}+\overline{{\theta }_2}{u}^{-})\hfill \\ \hfill & \le \underset{n\to \infty }{lim\; sup}\mathcal{J}(\overline{{\theta }_1}{u}^{+}+\overline{{\theta }_2}{u}^{-})\hfill \\ \hfill & \le \underset{n\to \infty }{lim}\mathcal{J}\left(u_n\right)\hfill \\ \hfill & =\overline{m}.\hfill \end{array}$$

Finally, we take $\sigma =\overline{{\theta }_1}{u}^{+}+\overline{{\theta }_2}{u}^{-}{u}^{-}\in \overline{\mathcal{N}}$ in order to conclude the proof.

Step 2. $\sigma \in \overline{\mathcal{N}}$ is the critical point of $\mathcal{J}:E\to \mathbb{R}$.

Let us assume that the $\sigma$ is not a critical point of $\mathcal{J}$. Then, there exists $\tau \in C_c^{\infty }\left({\mathbb{R}}^N\right)$ such that $\langle {\mathcal{J}}^{\prime }\left(\sigma \right),\tau \rangle =-2.$ Since, $\mathcal{J}$ is continuous and differentiable, there exists small $\Xi >0$ such that

$$\langle {\mathcal{J}}^{\prime }({\theta }_1{u}^{+}+{\theta }_2{u}^{-}+\omega \overline{\sigma }),\overline{\sigma }\rangle \le -1\mathrm{if}{({\theta }_1-\overline{{\theta }_1})}^2+{({\theta }_2-\overline{{\theta }_2})}^2\le {\Xi }^2\mathrm{and}0\le \omega \le \Xi .$$

(36)

Let $D\subset {\mathbb{R}}^2$ be an open disc of radius $\Xi >0$ centered at $(\overline{{\theta }_1},\overline{{\theta }_2})$. Also, define a continuous function $\Phi :D\to [0,1]$ by

$$\Phi ({\theta }_1,{\theta }_2)=\left\{\begin{array}{l}1\mathrm{if}{({\theta }_1-\overline{{\theta }_1})}^2+{({\theta }_2-\overline{{\theta }_2})}^2\le {\displaystyle \frac{{\Xi }^2}{16}},\\ 0\mathrm{if}{({\theta }_1-\overline{{\theta }_1})}^2+{({\theta }_2-\overline{{\theta }_2})}^2\ge {\displaystyle \frac{{\Xi }^2}{4}}.\end{array}\right.$$

Next, we define a continuous map $T:D\to E$ by

$$T({\theta }_1,{\theta }_2)={\theta }_1{u}^{+}+{\theta }_2{u}^{-}+\Xi \Phi ({\theta }_1,{\theta }_2)\overline{\sigma }\mathrm{for}\mathrm{all}({\theta }_1,{\theta }_2)\in D$$

and $Q:D\to {\mathbb{R}}^2$ by

$$Q({\theta }_1,{\theta }_2)=(\langle {\mathcal{J}}^{\prime }\left(T({\theta }_1,{\theta }_2)\right),T{({\theta }_1,{\theta }_2)}^{+}\rangle ,\langle {\mathcal{J}}^{\prime }\left(T({\theta }_1,{\theta }_2)\right),T{({\theta }_1,{\theta }_2)}^{-}\rangle )\mathrm{for}\mathrm{all}({\theta }_1,{\theta }_2)\in D.$$

One could deduce that Q is continuous, as the mapping $u\mapsto u^{+}$ is continuous in E. In such a case, we are on the boundary of D, that is, ${({\theta }_1-\overline{{\theta }_1})}^2+{({\theta }_2-\overline{{\theta }_2})}^2={\Xi }^2$, and then $\Phi =0$. Hence, one could obtain $T({\theta }_1,{\theta }_2)={\theta }_1{u}^{+}+{\theta }_2{u}^{-}$ and by Lemma 12, we have

$$Q({\theta }_1,{\theta }_2)\ne 0\mathrm{on}\partial D.$$

Therefore, the Brouwer degree is well defined, $deg(Q,\mathrm{int}(D),(0,0\left)\right)=1$, and there exists $({\theta }_{11},{\theta }_{21})\in \mathrm{int}\left(D\right)$ such that $Q({\theta }_{11},{\theta }_{21})=(0,0)$. This further implies that $T({\theta }_{11},{\theta }_{21})\in \overline{\mathcal{N}}$, and by the definition of $\overline{m}$, one could deduce

$$\mathcal{J}\left(T({\theta }_{11},{\theta }_{21})\right)\ge \overline{m}.$$

(37)

Using Equation (36), we obtain

$$\begin{array}{cc}\hfill \mathcal{J}\left(T({\theta }_{11},{\theta }_{21})\right)& =\mathcal{J}({\theta }_{11}{u}^{+}+{\theta }_{21}{u}^{-})+{\int }_0^1{\displaystyle \frac{d}{dt}}\mathcal{J}({\theta }_{11}{u}^{+}+{\theta }_{21}{u}^{-}+\Xi t\Phi ({\theta }_{11},{\theta }_{21})\overline{\sigma })dt\hfill \\ \hfill & =\mathcal{J}({\theta }_{11}{u}^{+}+{\theta }_{21}{u}^{-})-\Xi \Phi ({\theta }_{11},{\theta }_{21}).\hfill \end{array}$$

(38)

Next, by definition of $\Phi$, we obtain $\Phi ({\theta }_{11},{\theta }_1)=1$ when $({\theta }_{11},{\theta }_{21})=(\overline{{\theta }_1},\overline{{\theta }_2})$. Therefore, we have

$$\mathcal{J}\left(T({\theta }_{11},{\theta }_{21})\right)\le \mathcal{J}({\theta }_{11}{u}^{+}+{\theta }_{21}{u}^{-})-\Xi \le \overline{m}-\Xi <\overline{m}.$$

When $({\theta }_{11},{\theta }_{21})\ne (\overline{{\theta }_1},\overline{{\theta }_2})$, then by Lemma 12, we have

$$\mathcal{J}({\theta }_{11}{u}^{+}+{\theta }_{21}{u}^{-})<\mathcal{J}(\overline{{\theta }_1}{u}^{+}+\overline{{\theta }_2}{u}^{-})=\overline{m},$$

which yields

$$\mathcal{J}\left(T({\theta }_{11},{\theta }_{21})\right)\le \mathcal{J}({\theta }_{11}{u}^{+}+{\theta }_{21}{u}^{-})<\overline{m},$$

which is a contradiction to Equation (37). Hence, we conclude our proof.

6. Conclusions

We have used the variational method, which has a very rich history, as it has given rise to the functional energy. In the proof of the first main theorem, the mountain pass theorem has been used with the Ekeland variational principle to prove the existence of a ground state solution for a p-biharmonic problem involving the critical Hardy–Sobolev exponent. In the second theorem, the existence of the least-energy sign-changing solution has been derived by using the minimization method on the Nehari nodal set. The critical Hardy nonlinearity is quite complicated, and the method we used to manipulate the critical Hardy nonlinearity is an application of the Brezis–Lieb lemma. In the future, our aim is obtain even stronger results for problems which involve such nonlinearities.