Existence and Asymptotic Behavior of Ground State Solutions to Kirchhoff-Type Equations of General Convolution Nonlinearity with a Steep Potential Well

Abstract

In this paper, we consider a new kind of Kirchhoff-type equation which is stated in the introduction. Under certain assumptions on potentials, we prove by variational methods that the equation has at least a ground state solution and investigate the asymptotic behavior of solutions.

1. Introduction

In this article, we investigate the existence and asymptotic behavior of ground state solutions in the following Kirchhoff-type equation:

$$\left\{\begin{array}{c}-(a+b{\int }_{{\mathbb{R}}^N}{|\nabla u|}^2\mathrm{d}x)\Delta u+\lambda V\left(x\right)u=(I_{\alpha }\ast F\left(u\right))f\left(u\right)+g\left(u\right),\mathrm{in}{\mathbb{R}}^N,\hfill \\ u\in H^1\left({\mathbb{R}}^N\right),\hfill \end{array}\right.$$

(1)

where $a>0$, $b\ge 0$ are constants, $\lambda$ is a positive parameter, $N\ge 3$, $F\left(t\right)={\int }_0^{t}f\left(s\right)\mathrm{d}s$, and $I_{\alpha }$ is a Riesz potential whose order is $\alpha \in (N-2,N)$. Here, $I_{\alpha }$ is defined by $I_{\alpha }$=$\frac{\Gamma \left(\frac{N-\alpha }{2}\right)}{\Gamma \left(\frac{\alpha }{2}\right){\pi }^{\frac{N}{2}}{2}^{\alpha }{|x|}^{N-\alpha }}$. Moreover, $V\left(x\right):{\mathbb{R}}^N\to \mathbb{R}$ is a potential function that satisfies the following:

(V1) $V\left(x\right)\in C({\mathbb{R}}^N,\mathbb{R})$, and $V\left(x\right)\ge 0$ for all $x\in {\mathbb{R}}^N$;

(V2) There exists $V_0>0$, such that ${\mathcal{V}}_0:=\{x\in {\mathbb{R}}^N:V\left(x\right)\le V_0\}$ is nonempty and has a finite measure;

(V3) $\Omega :=int{V}^{-1}\left(0\right)$ is a nonempty open set that has a local Lipschitz boundary and $\overline{\Omega }=V^{-1}\left(0\right)$.

Additionally, we suppose that the function $f\in C^1(\mathbb{R},\mathbb{R})$ verifies the following:

$\left(f1\right)f\left(t\right)=o\left(t^{\frac{\alpha }{N}}\right)$ as $t\to 0;$

$\left(f2\right)\underset{|t|\to +\infty }{lim}\frac{f\left(t\right)}{t^{\frac{\alpha +2}{N-2}}}=0;$

$\left(f3\right)\frac{f\left(t\right)}{t}$ is increasing on $(0,+\infty )$ and decreasing on $(-\infty ,0)$;

$\left(f4\right)f\left(t\right)$ is increasing on $\mathbb{R}$.

Furthermore, we assume that the function $g\in C(\mathbb{R},\mathbb{R})$ satisfies the following:

$\left(g1\right)$ there exist constants $C_0>0$ and $p\in (2,2^{\ast }]$, such that $|g\left(t\right)|\le C_0{(1+|t|}^{p-1}),$$\forall t\in \mathbb{R}$;

$\left(g2\right)g\left(t\right)=o\left(t\right)$ as $t\to 0;$

$\left(g3\right)g\left(t\right)t\ge 4G\left(t\right)\ge 0$ for all $t\in \mathbb{R}$, where $G\left(t\right)={\int }_0^{t}g\left(s\right)\mathrm{d}s;$

$\left(g4\right)\underset{|t|\to +\infty }{lim}\frac{G\left(t\right)}{t^2}=\infty .$

These hypotheses of $V\left(x\right)$ were first put forward by Bartsch and Wang [1] in their research on the nonlinear Schrödinger equations and have attracted the attention of several researchers; e.g., see [2,3,4]. We note that the conditions $\left(V1\right)$–$\left(V3\right)$ imply that $\lambda V$ represents a potential well, which has the bottom $V^{-1}\left(0\right)$, and its steepness is controlled by the positive parameter, $\lambda$. In consideration of this condition, $\lambda V$ is often referred to as the steep potential well if $\lambda$ is sufficiently large.

In the past decades, many scholars have studied the existence of nontrivial solutions for the Kirchhoff-type problem:

$$\left\{\begin{array}{c}-(a+b{\int }_{{\mathbb{R}}^3}{|\nabla u|}^2\mathrm{d}x)\Delta u+V\left(x\right)u=g(x,u),\mathrm{in}{\mathbb{R}}^3,\hfill \\ u\in H^1\left({\mathbb{R}}^3\right),\hfill \end{array}\right.$$

(2)

where $a>0$, $b\ge 0$, $V:{\mathbb{R}}^3\to \mathbb{R}$ is a potential function, and $g\in C({\mathbb{R}}^3\times \mathbb{R},\mathbb{R})$. Equation (2) is often referred to as a nonlocal problem on account of the presence of the term $b{\int }_{{\mathbb{R}}^3}{|\nabla u|}^2\mathrm{d}x$, which implies that (2) is no longer a pointwise identity. This phenomenon causes some mathematical difficulties, but at the same time, it makes the research for such a problem particularly interesting. This problem has a profound and interesting physical context. In fact, as long as one sets $V\left(x\right)=0$ and replaces ${\mathbb{R}}^3$ by a bounded domain $\Omega \subset {\mathbb{R}}^3$ in (2), then one can get the following Kirchhoff Dirichlet problem:

$$\left\{\begin{array}{cc}-(a+b{\int }_{\Omega }|\nabla u{|}^2\mathrm{d}x)\Delta u=g(x,u),\hfill & x\in \Omega ,\hfill \\ u=0\hfill & x\in \partial \Omega .\hfill \end{array}\right.$$

(3)

It is closely related to the stationary analogue of the equation below:

$$u_{tt}-(a+b{\int }_{\Omega }{|\nabla u|}^2\mathrm{d}x)\Delta u=g(x,u),$$

where u denotes the displacement, $g(x,u)$ is the external force, a is the initial tension, while b is related to the inherent characteristics of the string (such as Young’s modulus). This hyperbolic equation generalizes the following equation:

$$\rho \frac{{\partial }^{2}u}{\partial t^2}-\left(\frac{{\rho }_0}{h}+\frac{E}{2L}{\int }_0^L|\frac{\partial u}{\partial x}|\mathrm{d}x\right)\frac{{\partial }^{2}u}{\partial x^2}=0.$$

(4)

G. Kirchhoff first proposed this equation as an extension of classical D’Alembert’s wave equations for the free vibration of elastic strings. His model takes into account the changes in length of the string produced by transverse vibrations. In (4), L is the length of the string, h is the area of cross-section, E denotes the Young modulus of the material, $\rho$ is the mass density, and ${\rho }_0$ denotes the initial tension. As a matter of fact, nonlocal problems also appear in other fields as biological systems, where u describes a process which depends on the average of itself (for example, population density). Soon after, J. L. Lions [5] finished the pioneer work. He introduced the functional analysis approach. Since then, Kirchhoff equations have increasingly attracted the attention of researchers. In [6,7], the authors considered (3) in the case where $g(x,u):=g\left(u\right)$ and proved the existence and asymptotic behavior of least-energy sign-changing solutions.

Moreover, many researchers have focused on the Kirchhoff-type problem defined in the whole space, ${\mathbb{R}}^3$ (even ${\mathbb{R}}^N$), i.e., problem (2). For example, in [8], Li et al. obtained a positive solution for (2) by using the cut-off technique and monotone method. Li and Ye in [9] proved that (2) had a ground state solution in the case of $g(x.u)={|u|}^{p-1}u$ and $2<p\le 3$. However, unfortunately, most of those results needed to assume that g satisfies the classical Ambrosetti–Rabinowitz condition.

$(A-R)$ condition: there exists $\mu >2$, such that:

$$0<\mu G(x,s)\le sg(x,s)$$

for all $s>0$, where $G(x,s)={\int }_0^{s}g(x,t)\mathrm{d}t$. On the bright side, later, Ye [10] obtained a positive, high-energy solution with superlinear nonlinearity. However, there are still few results on the existence of a ground state solution to (2) without an $(A-R)$ condition (see [11,12,13]). It is worth mentioning that in [13], Guo studied the following Kirchhoff-type problem:

$$\left\{\begin{array}{c}-(a+b{\int }_{{\mathbb{R}}^3}{|\nabla u|}^2\mathrm{d}x)\Delta u+V\left(x\right)u=f\left(u\right),\mathrm{in}{\mathbb{R}}^3,\hfill \\ u\in H^1\left({\mathbb{R}}^3\right).\hfill \end{array}\right.$$

(5)

He proved the existence of a positive ground state solution to (5) without any $(A-R)$-type conditions. Furthermore, in [14], the authors obtained the existence of a nontrivial solution for the following Kirchhoff-type equation:

$$\left\{\begin{array}{c}-(a+b{\int }_{{\mathbb{R}}^3}{|\nabla u|}^2{\mathrm{d}x)\Delta u+V\left(x\right)u=|u|}^{p-2}u+\lambda f\left(u\right),\mathrm{in}{\mathbb{R}}^3,\hfill \\ u\in H^1\left({\mathbb{R}}^3\right).\hfill \end{array}\right.$$

(6)

In [14], there was no Ambrosetti–Rabinowtiz condition and no growth condition. Furthermore, their conclusion holds for general supercritical nonlinearity. Readers can see [8,9,10,13,14,15,16,17,18] and the references therein for more results on Kirchhoff-type problems.

Very recently, Sun and Wu [19] considered the following Kirchhoff-type problem with the steep potential well V:

$$\left\{\begin{array}{c}-(a+b{\int }_{{\mathbb{R}}^N}{|\nabla u|}^2\mathrm{d}x)\Delta u+\lambda V\left(x\right)u=g(x,u),\mathrm{in}{\mathbb{R}}^N,\hfill \\ u\in H^1\left({\mathbb{R}}^N\right),\hfill \end{array}\right.$$

(7)

where $N\ge 3$, $a>0$, $b>0$ are constants, and $\lambda$ is a positive parameter. The potential V satisfies the conditions $\left(V1\right)--\left(V3\right)$, and nonlinearity $g(x,s)$ is asymptotically k-linear (k = 1,3,4) with respect to s at infinity. They proved the existence and nonexistence of nontrivial solutions by variational methods. Additionally, the authors also explored the asymptotic behavior of nontrivial solutions for (7). Subsequently, with the help of the variational framework developed by [20], Du et al. [21] studied (7) when $N=3$ and $g(x,u)$ behaved similar to ${|u|}^{p-2}u$ with $4<p<6$, and subsequently proved the existence and asymptotic behavior of ground state solutions. In [22], the authors obtained the existence of nontrivial solutions for the case of $N=3$ and $g(x,u)={|u|}^{p-2}u$ with $4\le p<6$. Furthermore, in [23], Zhang and Du obtained the positive solutions for b small and $\lambda$ large with the use of the truncation technique and the parameter-dependent compactness lemma, when $N=3$ and $2<p<4$. In [24], Luo and Tang proved the existence and asymptotic behavior of ground state solutions for (7), with critical nonlinearities for the case of $N=3$.

On the other hand, when $a=1$, $b=0$, $g=(I_{\alpha }{\ast |u|}^p{)|u|}^{p-2}u$, Equation (2) is reduced to the following:

$$-\Delta u+V\left(x\right)u=(I_{\alpha }{\ast |u|}^p{)|u|}^{p-2}u,$$

(8)

which is called a nonlinear Choquard-type equation. The origin of this problem, especially its physical background, can be found in [25] and references therein. Moreover, readers can see [17,26,27,28,29,30,31,32,33] for recent achievements.

Enlightened by the works we mentioned above, especially by [23,24], we the studied the Kirchhoff-type problem (7) with a general convolution in ${\mathbb{R}}^N$, i.e., problem (1). Note that we considered the critical growth case. More precisely, we aimed to get the existence of the ground state solutions of Equation (1) and explored their asymptotic behavior as $\lambda \to \infty$.

The main outcome of our investigation is shown below:

Theorem 1.

If $\left(V1\right)$–$\left(V3\right)$, $\left(f1\right)$–$\left(f4\right)$, and $\left(g1\right)$–$\left(g4\right)$ hold, then Equation (1) has at least one ground state solution.

Theorem 2.

Assume that $u_{{\lambda }_n}$ are solutions for problem (1), and Ω is defined by (V2), then $u_{{\lambda }_n}\to \overline{u}$ in $E_{\lambda }$ as ${\lambda }_n\to \infty$, where $\overline{u}\in H_0^1(\Omega )$, is a nontrivial solution of the equation below:

$$\left\{\begin{array}{c}-(a+b{\int }_{\Omega }{|\nabla u|}^2\mathrm{d}x)\Delta u=(I_{\alpha }\ast F\left(u\right))f\left(u\right)+g\left(u\right),x\in \Omega ,\hfill \\ u=0on\partial \Omega .\hfill \end{array}\right.$$

(9)

2. Preliminaries

In this section, we will establish the variational framework for Equation (1) and give some useful lemmas.

For the convenience of expression, from now on, we use the following notations:

• $H^1\left({\mathbb{R}}^N\right)$, in which the norm ${\parallel u\parallel }_{H^1\left({\mathbb{R}}^N\right)}=[{\int }_{{\mathbb{R}}^N}{(|\nabla u|}^2+V\left(x\right)u^2{\left)\mathrm{d}x\right]}^{\frac{1}{2}}$;
• $E:=\{u\in H^1\left({\mathbb{R}}^N\right):{\int }_{{\mathbb{R}}^N}V\left(x\right)u^2\mathrm{d}x<\infty \}$ is equipped with an equivalent norm:

  $$\parallel u\parallel =[{\int }_{{\mathbb{R}}^N}{(a|\nabla u|}^2+V\left(x\right)u^2{\left)\mathrm{d}x\right]}^{\frac{1}{2}};$$
• For $\lambda >0$, we define the norm ${\parallel u\parallel }_{\lambda }=[{\int }_{{\mathbb{R}}^N}{(a|\nabla u|}^2+\lambda V\left(x\right)u^2{\left)\mathrm{d}x\right]}^{\frac{1}{2}}$ and $E_{\lambda }:={(E,\parallel u\parallel }_{\lambda })$;
• $L^s\left({\mathbb{R}}^N\right)(1\le s<\infty )$ denotes the Lebesgue space, with the norm ${|u|}_s=({\int }_{{\mathbb{R}}^N}{|u|}^s{\mathrm{d}x)}^{1/s};$
• For any $u\in H^1\left({\mathbb{R}}^N\right)\setminus \left\{0\right\},u_t$ is denoted as:

  $$u_t=\left\{\begin{array}{c}0,t=0,\hfill \\ \sqrt{t}u\left(\frac{x}{t}\right),t>0;\hfill \end{array}\right.$$
• For any $x\in {\mathbb{R}}^N$ and $r>0$,$B_r\left(x\right):=\{y\in {\mathbb{R}}^N:|y-x|<r\};$
• $C,C_1,C_2,...$ represent positive constants possibly different in different lines.

Remark 1.

It is obvious that for $\forall \lambda \ge 1$, ${\parallel u\parallel }_{\lambda }\ge \parallel u\parallel$.

Remark 2.

From (V1)–(V2), we can get the following equations:

$$\begin{gathered} {\int }_{{\mathbb{R}}^N}{(|\nabla u|}^2+V\left(x\right)u^2)\mathrm{d}x={\int }_{{\mathbb{R}}^N}{|\nabla u|}^2\mathrm{d}x+{\int }_{{\mathcal{V}}_0}{u}^2\mathrm{d}x+{\int }_{{\mathbb{R}}^N\setminus {\mathcal{V}}_0}{u}^2\mathrm{d}x \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le {\int }_{{\mathbb{R}}^N}{|\nabla u|}^2\mathrm{d}x+|{\mathcal{V}}_0{|}^{\frac{2}{N}}{\int }_{{\mathcal{V}}_0}{|u|}^{2^{\ast }}\mathrm{d}x+V_0^{-1}{\int }_{{\mathbb{R}}^N\setminus {\mathcal{V}}_0}V\left(x\right)u^2\mathrm{d}x \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le C_1{\int }_{{\mathbb{R}}^N}{(a|\nabla u|}^2+V\left(x\right)u^2)\mathrm{d}x. \end{gathered}$$

(10)

Thus, by (10), the Hölder and Sobolev inequalities, there exist constants (independent of λ), such that:

$${\int }_{{\mathbb{R}}^N}{|u|}^s\mathrm{d}x={|u|}_s^s\le C_2{\parallel u\parallel }_{H^1\left({\mathbb{R}}^N\right)}^s\le C_3{\parallel u\parallel }_{\lambda }^s,$$

(11)

as $\lambda \ge 1$, for any $s\in [2,2^{\ast }]$. It implies that the embedding $E_{\lambda }\hookrightarrow H^1\left({\mathbb{R}}^N\right)$ is continuous. As a consequence, the functional $I_{\lambda }:E_{\lambda }\to \mathbb{R}$ given by the equations below:

$$\begin{gathered} I_{\lambda }\left(u\right)=\frac{1}{2}{\int }_{{\mathbb{R}}^N}{[a|\nabla u|}^2+\lambda V\left(x\right)u^2]\mathrm{d}x+\frac{b}{4}{\left({\int }_{{\mathbb{R}}^N}{|\nabla u|}^2\mathrm{d}x\right)}^2 \\ \hspace{1em}\hspace{1em}\hspace{1em}-\frac{1}{2}{\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast F\left(u\right))F\left(u\right)\mathrm{d}x-{\int }_{{\mathbb{R}}^N}G\left(u\right)\mathrm{d}x,u\in E_{\lambda } \end{gathered}$$

(12)

which is well defined, and it is of class $C^1$ for $\lambda \ge 1$ with the following derivative:

$$\begin{gathered} \langle I_{\lambda }^{\prime }\left(u\right),v\rangle ={\int }_{{\mathbb{R}}^N}[a\nabla u\nabla v+\lambda V\left(x\right)uv]\mathrm{d}x+b{\int }_{{\mathbb{R}}^N}{|\nabla u|}^2\mathrm{d}x{\int }_{{\mathbb{R}}^N}\nabla u\nabla v\mathrm{d}x \\ \hspace{1em}\hspace{1em}{\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast F\left(u\right))f\left(u\right)v\mathrm{d}x-{\int }_{{\mathbb{R}}^N}g\left(u\right)v\mathrm{d}x. \end{gathered}$$

(13)

for all $u,v\in E_{\lambda }$. Thus, the critical points of the functional $I_{\lambda }$ are the weak solutions of problem (1).

Lemma 1.

Assume (f1)–(f4) are fulfilled, then we have the following:

(1)

$Forall\epsilon >0,thereisa{C}_{\epsilon }>0suchthat|f\left(t\right)|\le {\epsilon |t|}^{\frac{\alpha }{N}}+C_{\epsilon }{|t|}^{\frac{\alpha +2}{N-2}}and|F\left(t\right)|\le {\epsilon |t|}^{\frac{N+\alpha }{N}}+C_{\epsilon }{|t|}^{\frac{N+\alpha }{N-2}};$

(2)

$Forall\epsilon >0,thereisa{C}_{\epsilon }>0suchthatforeveryp\in (2,2^{\ast }),|F\left(t\right){|\le \epsilon (|t|}^{\frac{N+\alpha }{N}}+{|t|}^{\frac{N+\alpha }{N-2}})+C_{\epsilon }{|t|}^{\frac{p(N+\alpha )}{2N}},and|F\left(t\right){|}^{\frac{2N}{N+\alpha }}\le {\epsilon (|t|}^2+{|t|}^{\frac{2N}{N-2}})+C_{\epsilon }{|t|}^p;$

(3)

$Foranys\ne 0,sf\left(s\right)>2F\left(s\right)andF\left(s\right)>0.$

Proof. 

One can easily obtain the results by elementary calculation.  □

Lemma 2

(The Hardy–Littlewood–Sobolev inequality [34]). $Let0<\alpha <N,p,q>1and1\le r<s<\infty besuchthat:$

$$\frac{1}{p}+\frac{1}{q}=1+\frac{\alpha }{N},\frac{1}{r}-\frac{1}{s}=\frac{\alpha }{N}.$$

(1)

$Foranyf\in L^p\left({\mathbb{R}}^N\right)andg\in L^q\left({\mathbb{R}}^N\right),onehas$

$$|{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{f\left(x\right)g\left(y\right)}{{|x-y|}^{N-\alpha }}\mathrm{d}x\mathrm{d}y|\le {C(N,\alpha ,p)\parallel f\parallel }_{L^p\left({\mathbb{R}}^N\right)}{\parallel g\parallel }_{L^q\left({\mathbb{R}}^N\right)};$$

(2)

$Foranyf\in L^r\left({\mathbb{R}}^N\right)onehas$

$$\parallel \frac{1}{{|·|}^{N-\alpha }}\ast f{\parallel }_{L^s\left({\mathbb{R}}^N\right)}\le C(N,\alpha ,r){\parallel f\parallel }_{L^r\left({\mathbb{R}}^N\right)}.$$

Remark 3.

By Lemma 1(1), Lemma 2(1) and the Sobolev embedding theorem, we can get the following equations:

$$\begin{gathered} |{\int }_{{\mathbb{R}}^N}\left(I_{\alpha }\ast F\left(u\right)\right)F\left(u\right)\mathrm{d}x|\le {C|F\left(u\right)|}_{\frac{2N}{N+\alpha }}^2 \\ \hspace{1em}\hspace{1em}\le C{\left[{\int }_{{\mathbb{R}}^N}{\left({|u|}^{\frac{N+\alpha }{N}}+{|u|}^{\frac{N+\alpha }{N-2}}\right)}^{\frac{\left(2N\right)}{N+\alpha }}\mathrm{d}x\right]}^{\frac{N+\alpha }{N}} \\ \hspace{1em}\hspace{1em}\hspace{1em}\le C{\left[{\int }_{{\mathbb{R}}^N}\left({|u|}^2+{|u|}^{\frac{2N}{N-2}}\right)\mathrm{d}x\right]}^{\frac{N+\alpha }{N}} \\ \hspace{1em}\le {C(\parallel u\parallel }^{\frac{2N+2\alpha }{N}}+{\parallel u\parallel }^{\frac{2N+2\alpha }{N-2}}). \end{gathered}$$

(14)

3. Ground State Solution for Problem (1)

In this section, we will prove the existence of ground state solutions for problem (1).

We first consider the mountain path geometry.

Lemma 3.

The functional $I_{\lambda }$ possesses the mountain-pass geometry, i.e.,

 1. 

There exist $\rho ,\delta >0$ such that $I_{\lambda }\ge \delta$ for all $\parallel u\parallel =\rho$;

 2. 

There exist $e\in E_{\lambda }\setminus \left\{0\right\}$ such that $\parallel e\parallel >\rho$ and $I_{\lambda }\left(e\right)<0$.

Proof. 

(1) By $\left(g1\right)$ and Lemma 1, we have the following:

$$I_{\lambda }\left(u\right)\ge C_4{\parallel u\parallel }^2-C_5{(\parallel u\parallel }^{\frac{2N+2\alpha }{N}}+{\parallel u\parallel }^{\frac{2N+2\alpha }{N-2}})-C_{\epsilon }{\parallel u\parallel }^p.$$

(15)

Thus, there exist $\rho ,\delta >0$ such that $I_{\lambda }\ge \delta$ for all ${\parallel u\parallel }_{\lambda }=\rho >0$ that is small enough.

(2) We freely choose $u\in C_0^{\infty }\left({\mathbb{R}}^N\right)$, then we can get the following equations:

$$\begin{gathered} I_{\lambda }\left(u_t\right)=\frac{a{t}^{N-1}}{2}{\int }_{{\mathbb{R}}^N}{|\nabla u|}^2\mathrm{d}x+\frac{t^{N+1}}{2}{\int }_{{\mathbb{R}}^N}\lambda V\left(x\right)u^2\mathrm{d}x+\frac{b{t}^{2N-2}}{4}{\left({\int }_{{\mathbb{R}}^N}{|\nabla u|}^2\mathrm{d}x\right)}^2 \\ -\frac{t^{N+\alpha }}{2}{\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast F\left(\sqrt{t}u\right))F\left(\sqrt{t}u\right)\mathrm{d}x-t^N{\int }_{{\mathbb{R}}^N}G\left(\sqrt{t}u\right)\mathrm{d}x\to -\infty , \end{gathered}$$

(16)

as $t\to +\infty$, since $\alpha >N-2$.

Note that

$$\parallel u_t{\parallel }^2=a{t}^{N-1}{\int }_{{\mathbb{R}}^N}{|\nabla u|}^2\mathrm{d}x+t^{N+1}\lambda {\int }_{{\mathbb{R}}^N}V\left(x\right)u^2\mathrm{d}x.$$

Hence, taking $e=u_{t_0}$, with $t_0>0$ large, we have $\parallel e\parallel >\rho$ and $I_{\lambda }\left(e\right)<0.$ □

Remark 4.

Now, we can define the mountain-pass level of $I_{\lambda }$:

$$c_{\lambda }=\underset{\gamma \in \Gamma }{inf}\underset{t\in [0,1]}{max}{I}_{\lambda }\left(\gamma \left(t\right)\right)>0,$$

where $\Gamma =\{\gamma \in C([0,1],E_{\lambda }):\gamma \left(0\right)=0,I_{\lambda }\left(\gamma \left(1\right)\right)<0\}$.

Lemma 4.

$I_{\lambda }\left(u\right)$ satisfies the $\left(PS\right)$ condition.

Proof. 

Let $\left\{u_n\right\}$ be a ${\left(PS\right)}_{c_{\lambda }}$ sequence of $I_{\lambda }\left(u\right)$, i.e., $I_{\lambda }\left(u_n\right)\to c_{\lambda }$ and $I_{\lambda }^{\prime }\left(u_n\right)\to 0$. Then, by $\left(g3\right)$, we have the following:

$$\begin{array}{ll}{c}_{\lambda }+& o\left(1\right)\parallel u_n{\parallel }_{\lambda }\ge 4I_{\lambda }\left(u_n\right)-\langle (I_{\lambda }^{\prime }\left(u_n\right),u_n\rangle \\ & ={\int }_{{\mathbb{R}}^N}[a|\nabla u_n{|}^2+\lambda V\left(x\right)|u_n{|}^2]\mathrm{d}x+{\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast F\left(u_n\right))[f\left(u_n\right)u_n-2F\left(u_n\right)]\mathrm{d}x+{\int }_{{\mathbb{R}}^N}[g\left(u_n\right)u_n-4G\left(u_n\right)]\mathrm{d}x\\ & \ge {\int }_{{\mathbb{R}}^N}[a|\nabla u_n{|}^2+\lambda V\left(x\right)|u_n{|}^2]\mathrm{d}x.\end{array}$$

(17)

Therefore, $\left\{u_n\right\}$ is bounded in $E_{\lambda }$. Hence, up to a subsequence, we may assume that there exists a u, such that:

$$\left\{\begin{array}{c}{u}_n\rightharpoonup u\mathrm{in}{E}_{\lambda },\hfill \\ u_n\to u\mathrm{in}{L}_{loc}^s\left({\mathbb{R}}^N\right),\forall s\in [2,2^{\ast }),\hfill \\ u_n\to ua.e.\mathrm{on}{\mathbb{R}}^N.\hfill \end{array}\right.$$

Let

$$A^2=\underset{n\to \infty }{lim}{\int }_{{\mathbb{R}}^N}|\nabla u_n{|}^2\mathrm{d}x\ge {\int }_{{\mathbb{R}}^N}{|\nabla u|}^2\mathrm{d}x,$$

(18)

and define

$$J_{\lambda }\left(u\right)=\frac{1}{2}{\parallel u\parallel }_{\lambda }^2+\frac{b}{2}{A}^2{\int }_{{\mathbb{R}}^N}{|\nabla u|}^2\mathrm{d}x-\frac{1}{2}{\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast F\left(u\right))F\left(u\right)\mathrm{d}x-{\int }_{{\mathbb{R}}^N}G\left(u\right)\mathrm{d}x.$$

(19)

Now, we claim that $J_{\lambda }^{\prime }\left(u\right)=0$. Indeed, from $I_{\lambda }^{\prime }\left(u_n\right)\to 0$, we obtain the following equation:

$$(a+b{A}^2){\int }_{{\mathbb{R}}^N}a\nabla u\nabla v\mathrm{d}x+{\int }_{{\mathbb{R}}^N}\lambda V\left(x\right)uv]\mathrm{d}x-{\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast F\left(u\right))f\left(u\right)v\mathrm{d}x-{\int }_{{\mathbb{R}}^N}g\left(u\right)v\mathrm{d}x=0$$

(20)

for any $v\in E_{\lambda }$, which implies $J_{\lambda }^{\prime }\left(u\right)=0$. Next, we set $w_n:=u_n-u$ and prove that $w_n\to 0$ in $E_{\lambda }$. It follows from (13) that:

$$\begin{array}{ll}o\left(1\right)=& \langle I_{\lambda }^{\prime }\left(u_n\right),u_n\rangle \\ & ={\int }_{{\mathbb{R}}^N}[a|\nabla u_n{|}^2+\lambda V\left(x\right)u_n^2]\mathrm{d}x+b{\left({\int }_{{\mathbb{R}}^N}{|\nabla u_n|}^2\mathrm{d}x\right)}^2\\ & -{\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast F\left(u_n\right))f\left(u_n\right)u_n\mathrm{d}x-{\int }_{{\mathbb{R}}^N}g\left(u_n\right)u_n\mathrm{d}x-\langle J_{\lambda }^{\prime }\left(u\right),u\rangle \\ & ={\int }_{{\mathbb{R}}^N}[a|\nabla u_n{|}^2+\lambda V\left(x\right)u_n^2]\mathrm{d}x+b{\left({\int }_{{\mathbb{R}}^N}{|\nabla u_n|}^2\mathrm{d}x\right)}^2\\ & -{\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast F\left(u_n\right))f\left(u_n\right)u_n\mathrm{d}x-{\int }_{{\mathbb{R}}^N}g\left(u_n\right)u_n\mathrm{d}x\\ & -{\parallel u\parallel }_{\lambda }^2-b{A}^2{|\nabla u|}_2^2+{\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast F\left(u\right))f\left(u\right)u\mathrm{d}x+{\int }_{{\mathbb{R}}^N}g\left(u\right)u\mathrm{d}x\\ & =\parallel w_n{\parallel }_{\lambda }^2+b{A}^4-b{A}^2{|\nabla u|}_2^2+o\left(1\right)\\ & \ge {\parallel w_n\parallel }_{\lambda }^2+o\left(1\right).\end{array}$$

(21)

Hence, $w_n\to 0$ in $E_{\lambda }$, which implies that $u_n\to u$ in $E_{\lambda }$, i.e., $I_{\lambda }\left(u\right)$ satisfies the $\left(PS\right)$ condition. This completes the proof. □

Remark 5.

Then according to [35] and Lemma 3, $I_{\lambda }$ has a critical point, $u_{\lambda }$, with $I_{\lambda }\left(u_{\lambda }\right)=c_{\lambda }$. Now, we recall the Nehari manifold:

$$\mathcal{M}:=\{u\in E_{\lambda }\setminus \left\{0\right\}:\langle I_{\lambda }^{\prime }\left(u\right),u\rangle =0\}.$$

Let $m_{\lambda }=\underset{u\in \mathcal{M}}{inf}{I}_{\lambda }\left(u\right),$ then for any $u\in \mathcal{M}$, we have:

$$I_{\lambda }\left(u\right)=I_{\lambda }\left(u\right)-\frac{1}{4}\langle I_{\lambda }^{\prime }\left(u\right),u\rangle \ge C_3{\parallel u_n\parallel }_{\lambda }^2\ge 0.$$

Hence, $m_{\lambda }$ is well-defined. Moreover, by the argument similar to that of Chapter 4 [35], we have the following characterization:

$$c_{\lambda }=\underset{\gamma \in \Gamma }{inf}\underset{t\in [0,1]}{max}{I}_{\lambda }\left(\gamma \left(t\right)\right)=m_{\lambda }=\underset{u\in \mathcal{M}}{inf}{I}_{\lambda }\left(u\right).$$

Proof of Theorem 1.

From Lemmas 3 and 4, we know that there exists a bounded ${\left(PS\right)}_{c_{\lambda }}$ sequence $\left\{u_n\right\}$, that is, $I_{\lambda }\left(u_n\right)\to c_{\lambda }=m_{\lambda }$, $I^{\prime }\left(u_n\right)\to 0$. Next, let $\delta :=\underset{n\to \infty }{lim\; sup}$$\underset{y\in {\mathbb{R}}^N}{sup}{\int }_{B_1\left(y\right)}{|u_n|}^2\mathrm{d}x$. We claim that $\delta >0$. On the contrary, by the Lions’ concentration compactness principle, we have $u_n\to 0$ in $L^p\left({\mathbb{R}}^N\right)$ for $2<p<2^{\ast }$. By Lemma 1(2), for any $\epsilon >0$, there exists a constant $C_{\epsilon }>0$, such that:

$$\begin{array}{l}\underset{n\to \infty }{lim\; sup}{\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast F\left(u_n\right))f\left(u_n\right)u_n\mathrm{d}x\\ \le C\underset{n\to \infty }{lim\; sup}{\left[\epsilon ({\int }_{{\mathbb{R}}^N}|u_n{|}^2\mathrm{d}x+{\int }_{{\mathbb{R}}^N}|u_n{|}^{\frac{2N}{N-2}}\mathrm{d}x)+C_{\epsilon }{\int }_{{\mathbb{R}}^N}{|u_n|}^p\mathrm{d}x\right]}^{\frac{N+\alpha }{N}}\\ \le C{\left[\epsilon C_4+C_{\epsilon }\underset{n\to \infty }{lim\; sup}{\int }_{{\mathbb{R}}^N}{|u_n|}^p\mathrm{d}x\right]}^{\frac{N+\alpha }{N}}\\ =C{\left(\epsilon C_6\right)}^{\frac{N+\alpha }{N}}.\end{array}$$

Note that $\epsilon$ is arbitrary, and we thus get:

$${\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast F\left(u_n\right))f\left(u_n\right)u_n\mathrm{d}x=o\left(1\right).$$

Combining this with $I_{\lambda }^{\prime }\left(u_n\right)\to 0$, we can get the following equations:

$$\begin{array}{ll}o\left(1\right)& =\langle I^{\prime }\left(u_n\right),u_n\rangle \\ & ={\int }_{{\mathbb{R}}^N}[a|\nabla u_n{|}^2+\lambda V\left(x\right)u_n^2]\mathrm{d}x+b{\left({\int }_{{\mathbb{R}}^N}{|\nabla u_n|}^2\mathrm{d}x\right)}^2\\ & -{\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast F\left(u_n\right))f\left(u_n\right)u_n\mathrm{d}x-{\int }_{{\mathbb{R}}^N}g\left(u_n\right)u_n\mathrm{d}x\\ & \ge C_7{\parallel u_n\parallel }_{\lambda }^2-{\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast F\left(u_n\right))f\left(u_n\right)u_n\mathrm{d}x-{\int }_{{\mathbb{R}}^N}g\left(u_n\right)u_n\mathrm{d}x,\end{array}$$

(22)

which implies that

$$C_7{\parallel u_n\parallel }_{\lambda }^2\le {\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast F\left(u_n\right))f\left(u_n\right)u_n\mathrm{d}x+{\int }_{{\mathbb{R}}^N}g\left(u_n\right)u_n\mathrm{d}x+o\left(1\right)$$

(23)

Then, we have $\parallel u_n{\parallel }_{\lambda }\to 0$, which implies $u_n\to 0$ in $E_{\lambda }$. We deduce that $c_{\lambda }=0$, which contradicts the fact that $c_{\lambda }>0$. Hence, $\delta >0$, and there exists $\left\{y_n\right\}\subset {\mathbb{R}}^N$, such that ${\int }_{B_1\left(y_n\right)}{|u_n|}^p\mathrm{d}x\ge \frac{\delta }{2}>0$. We set $v_n\left(x\right)=u_n(x+y_n)$, then $\parallel u_n{\parallel }_{\lambda }=\parallel v_n{\parallel }_{\lambda },{\int }_{B_1\left(0\right)}{|v_n|}^p\mathrm{d}x>\frac{\delta }{2}$ and $I_{\lambda }\left(v_n\right)\to c_{\lambda }=m_{\lambda },I_{\lambda }^{\prime }\left(v_n\right)\to 0$. Thus, there exists a $v_0\ne 0$, such that:

$$\left\{\begin{array}{c}{v}_n\rightharpoonup v_0\mathrm{in}{E}_{\lambda },\hfill \\ v_n\to v_0\mathrm{in}{L}_{loc}^s\left({\mathbb{R}}^N\right),\forall s\in [2,2^{\ast })\hfill \\ v_n\to v_{0}a.e.\mathrm{on}{\mathbb{R}}^N.\hfill \end{array}\right.$$

Then, for any $\phi \in C_0^{\infty }\left({\mathbb{R}}^N\right)$, we have $0=\langle I_{\lambda }^{\prime }\left(v_n\right),\phi \rangle +o\left(1\right)=\langle I_{\lambda }^{\prime }\left(v_0\right),\phi \rangle$, which means that $v_0$ is a solition of Equation (1).

On the other hand, combining this with the Fatou Lemma, we can obtain the following:

$$\begin{array}{ll}{m}_{\lambda }& =I_{\lambda }\left(v_n\right)-\frac{1}{4}\langle I_{\lambda }^{\prime }\left(v_n\right),v_n\rangle +o\left(1\right)\\ & =\frac{1}{4}{\int }_{{\mathbb{R}}^N}[a|\nabla v_n{|}^2+\lambda V\left(x\right)|v_n{|}^2]\mathrm{d}x+\frac{1}{4}{\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast F\left(v_n\right))[f\left(v_n\right)v_n-2F\left(v_n\right)]\mathrm{d}x\\ & +\frac{1}{4}{\int }_{{\mathbb{R}}^N}[g\left(u_n\right)u_n-4G\left(u_n\right)]\mathrm{d}x+o\left(1\right)\\ & \ge \frac{1}{4}{\int }_{{\mathbb{R}}^N}[a|\nabla v_0{|}^2+\lambda V\left(x\right)v_0^2]\mathrm{d}x+\frac{1}{4}{\int }_{{\mathbb{R}}^N}(I_{\alpha }\ast F\left(v_0\right))[f\left(v_0\right)v_0-2F\left(v_0\right)]\mathrm{d}x\\ & +\frac{1}{4}{\int }_{{\mathbb{R}}^N}[g\left(u_n\right)u_n-4G\left(u_n\right)]\mathrm{d}x+o\left(1\right)\\ & =I_{\lambda }\left(v_0\right)-\frac{1}{4}\langle I_{\lambda }^{\prime }\left(v_0\right),v_0\rangle +o\left(1\right)\\ & =I_{\lambda }\left(v_0\right)+o\left(1\right).\end{array}$$

(24)

At the same time, we know that $m_{\lambda }=c\le I_{\lambda }\left(v_0\right)$ by the definition of m. Then, we can deduce that $v_0$ is a ground state solution of Equation (1). Thus, we complete the proof of Theorem 1. □

4. Asymptotic Behavior of Solutions for Equation (1)

In this section, we will investigate the asymptotic behavior of solutions for (1).

Proof of Theorem 2.

Let $u_{\lambda }$ be the ground state solution of (1) obtained in Theorem 1, we can get that $I_{\lambda }\left(u_{\lambda }\right)=m_{\lambda }$ and $I_{\lambda }^{\prime }\left(u_{\lambda }\right)=0$. Define $u_n:=u_{{\lambda }_n}$, then there exists a sequence $\left\{u_n\right\}$, such that $I_{{\lambda }_n}\left(u_n\right)=m_{{\lambda }_n}$ and $I_{{\lambda }_n}^{\prime }\left(u_n\right)=0$. It follows from (17) that $\left\{u_n\right\}$ is bounded in $E_{{\lambda }_n}$, that is, there exists $T>0$, such that:

$$\parallel u_n{\parallel }_{{\lambda }_n}\le T.$$

(25)

Thus, up to a subsequence, we may assume that there exists a $u_0$, such that:

$$\left\{\begin{array}{c}{u}_n\rightharpoonup u_0\mathrm{in}{E}_{{\lambda }_n},\hfill \\ u_n\to u_0\mathrm{in}{L}_{loc}^s\left({\mathbb{R}}^N\right),\forall s\in [2,2^{\ast }),\hfill \\ u_n\to u_{0}a.e.\mathrm{on}{\mathbb{R}}^N.\hfill \end{array}\right.$$

Now, we show that $u_n\to u_0$ in $L^s\left({\mathbb{R}}^N\right)$ for $s\in (2,2^{\ast })$. We then define the following equations:

$$D_R:=\{x\in {\mathbb{R}}^N\setminus B_R:V\left(x\right)\ge V_0\}$$

(26)

and

$$A_R:=\{x\in {\mathbb{R}}^N\setminus B_R:V\left(x\right)<V_0\}.$$

(27)

Then, we have $meas\left(A_R\right)\to 0$ as $R\to \infty$ by (V2) and

$${\int }_{D_R}{u}_n^2\mathrm{d}x\le \frac{1}{{\lambda }_n{V}_0}{\int }_{D_R}{\lambda }_{n}V\left(x\right)u_n^2\mathrm{d}x\le \frac{C_8}{{\lambda }_n{V}_0}\to 0$$

(28)

as ${\lambda }_n\to \infty$. Combing these with the Hölder and Sobolev inequality, for any $s\in (2,2^{\ast })$, we get the following equation:

$${\int }_{A_R}{u}_n^2\mathrm{d}x\le {\left({\int }_{A_R}{u}_n^s\mathrm{d}x\right)}^{\frac{2}{s}}{\left({\int }_{A_R}1\mathrm{d}x\right)}^{\frac{s-2}{s}}\le {\parallel u_n\parallel }_{{\lambda }_n}^2{\left(meas\left(A_R\right)\right)}^{\frac{s-2}{s}}.$$

(29)

Thus, we can obtain the following:

$$\begin{array}{ll}{\int }_{B_R^c}{u}_n^s\mathrm{d}x=& {\left({\int }_{B_R^c}{|u_n|}^2\mathrm{d}x\right)}^{\frac{2^{\ast }-s}{2^{\ast }-2}}{\left({\int }_{B_R^c}{|u_n|}^{2^{\ast }}\mathrm{d}x\right)}^{\frac{s-2}{2^{\ast }-2}}\\ & \le C_9{\left({\int }_{D_R}{u}_n^2\mathrm{d}x+{\int }_{A_R}{u}_n^2\mathrm{d}x\right)}^{\frac{2^{\ast }-s}{2^{\ast }-2}}\\ & \le C_9{\left(\frac{C_8}{{\lambda }_n{V}_0}+C_{10}{\left(meas\left(A_R\right)\right)}^{\frac{s-2}{s}}\right)}^{\frac{2^{\ast }-s}{2^{\ast }-2}}\to 0\end{array}$$

(30)

as ${\lambda }_n\to \infty$, where $B_R^c:=\{x\in {\mathbb{R}}^N:|x|\ge R\}$. Then,

$${\int }_{B_R^c}||u_n{|}^s-|u_0{|}^s|\mathrm{d}x\le {\int }_{B_R^c}|u_n{|}^s\mathrm{d}x+{\int }_{B_R^c}{|u_0|}^s\mathrm{d}x\to 0,$$

(31)

as $R\to \infty$. Since $u_n\to u_0$ in $L_{loc}^s\left({\mathbb{R}}^N\right)$, with $s\in (2,2^{\ast })$, we derive the following equation:

$${\int }_{|x|<R}|u_n{|}^s\mathrm{d}x\to {\int }_{|x|<R}{|u_0|}^s\mathrm{d}x.$$

(32)

Therefore, $u_n\to u_0$ in $L^s\left({\mathbb{R}}^N\right)$ for $s\in (2,2^{\ast })$ as ${\lambda }_n\to \infty$.

Next, we set $z_n:=u_n-u_0$, and we can prove that $z_n\to 0$ in $E_{\lambda }$ as the proof of Lemma 4.

Thus, together with Fatou’s Lemma and (25), we have the equation below:

$${\int }_{{\mathbb{R}}^N}V\left(x\right)u_0^2\mathrm{d}x\le \underset{n\to \infty }{lim\; inf}{\int }_{{\mathbb{R}}^N}V\left(x\right)u_n^2\mathrm{d}x\le \frac{\underset{n\to \infty }{lim\; inf}{\parallel u_n\parallel }_{{\lambda }_n}^2}{{\lambda }_n}\to 0.$$

(33)

Hence, by $\left(V3\right)$, we deduce that $u_0=0$ a.e. $x\in {\mathbb{R}}^N\setminus \Omega$ and $u_0\in H_0^1(\Omega )$. Then, we obtain the following equation:

$$(a+b{\int }_{\Omega }|\nabla u_0{|}^2\mathrm{d}x){\int }_{\Omega }\nabla u_0\nabla v\mathrm{d}x={\int }_{\Omega }(I_{\alpha }\ast F\left(u_0\right))f\left(u_0\right)v\mathrm{d}x+{\int }_{\Omega }g\left(u_0\right)v\mathrm{d}x$$

(34)

for any $v\in H_0^1(\Omega )$. This completes the proof. □