Existence of ground state solutions for quasilinear Schrödinger equations with general Choquard type nonlinearity

Abstract

In this paper, we study the following Choquard type quasilinear Schrödinger equation:

$$ -\Delta u+u-\Delta \bigl(u^{2}\bigr)u=\bigl(I_{\alpha }*G(u) \bigr)g(u),\quad x\in {\mathbb{R}}^{N}, $$

where $N\geq 3$, $0<\alpha <N$, and $I_{\alpha }$ is a Riesz potential. By using the minimization method developed by (Tang and Chen in Adv. Nonlinear Anal. 9:413–437, 2020; Willem in Minimax Theorems, 1996), we establish the existence of ground state solutions with general Choquard type nonlinearity. Our results extend the results obtained by (Chen et al. in Appl. Math. Lett. 102:106141, 2020).

Introduction

This article is concerned with the following quasilinear Schrödinger equation:

$$ \begin{aligned}-\Delta u+u-\Delta \bigl(u^{2}\bigr)u= \bigl(I_{\alpha }*G(u)\bigr)g(u),\quad x\in { \mathbb{R}}^{N}, \end{aligned} $$

(1.1)

where $N\geq 3$, $0<\alpha <N$, $I_{\alpha }$ is a Riesz potential (see [16]), and $g: {\mathbb{R}}^{N}\rightarrow {\mathbb{R}}$ satisfies

$(g_{1})$:

$g\in \mathcal{C}({\mathbb{R}},{\mathbb{R}})$;

$(g_{2})$:

there exists $C>0$ such that

$$ \bigl\vert G(t) \bigr\vert \leq C \bigl( \vert t \vert ^{\frac{N+\alpha }{N}}+ \vert t \vert ^{\frac{2(N+\alpha )}{N-2}} \bigr); $$

$(g_{3})$:

$\lim_{t\rightarrow 0}\frac{G(t)}{|t|^{\frac{N+\alpha }{N}}}=0$ and $\lim_{|t|\rightarrow +\infty } \frac{G(t)}{|t|^{\frac{2(N+\alpha )}{N-2}}}=0$;

$(g_{4})$:

there exists $s_{0}\in {\mathbb{R}}$ such that $G(s_{0})>\frac{1}{2}s^{2}_{0}$.

It is well known that the existence of solitary wave solutions for the following quasilinear Schrödinger equation is a hot problem

$$ i\partial _{t} z=-\Delta z+W(x)z-\psi \bigl( \vert z \vert ^{2}\bigr)z-\Delta l\bigl( \vert z \vert ^{2} \bigr)l'\bigl( \vert z \vert ^{2}\bigr)z, $$

(1.2)

where $z : {\mathbb{R}}\times {\mathbb{R}}^{N}\rightarrow \mathbb{C}$, $W : {\mathbb{R}}^{N}\rightarrow {\mathbb{R}}$ is a given potential, $l : {\mathbb{R}}\rightarrow {\mathbb{R}}$ and $\psi : {\mathbb{R}}^{N}\times {\mathbb{R}}\rightarrow {\mathbb{R}}$ are suitable functions. For various types of l and ψ, the quasilinear equation of the form (1.1) has been derived from models of several physical phenomena. For physical background, the readers can refer to [1, 9, 11, 15] and the references therein. If we set the variable $z(t,x)=\exp (-iLt)u(x)$, where $L\in {\mathbb{R}}$ and u is a real function, then so many papers focused on standing wave solutions for (1.2). The readers can refer to [5, 8, 12, 13, 20] and the references therein. As for Choquard type quasilinear Schrödinger equation, there are few papers except for [3, 4, 21]. In [21], a class of quasilinear Choquard equations has been considered via the perturbation method developed by [13], and they showed the existence of positive solution, negative solution, and multiple solutions. Furthermore, the authors [4] established the existence of positive solutions with the periodic potential or bounded potential. In [3], the authors proved the existence of ground state solutions via Jeanjean’s monotonic technique [10].

For the following Choquard equation with a local nonlinear perturbation

$$ \textstyle\begin{cases} -\Delta u+V(x)u=(I_{\alpha }*F(u))f(u)+g(u),\quad x\in \mathbb{R}^{N}; \\ u\in H^{1}(\mathbb{R}^{N}), \end{cases} $$

under some suitable conditions on V, the authors proved the existence of ground state solutions without super-linear conditions near infinity or monotonicity properties on f and g in [6].

To our knowledge, there are no articles to prove the existence of ground state solutions for (1.1) with general Choquard type nonlinearity. In this paper, motivated by [3, 4, 6, 21], we consider the existence of ground state solutions with the Berstycki–Lions conditions, which originated from [2]. To prove our results, we use the minimization method developed by Tang [18] to prove the existence of ground state solutions.

Next, the energy functional associated with (1.1) is given by

$$ J(u)=\frac{1}{2} \int _{{\mathbb{R}}^{N}}\bigl(1+2u^{2}\bigr) \vert \nabla u \vert ^{2}+ \frac{1}{2} \int _{{\mathbb{R}}^{N}}u^{2}-\frac{1}{2} \int _{{\mathbb{R}}^{N}}\bigl(I_{\alpha }*G(u)\bigr)G(u). $$

To our aim, if we choose the variable $u=f(v)$ in [7, 12], then (1.1) reduces to

$$ -\Delta v+f(v)f'(v)=\bigl(I_{\alpha }*G \bigl(f(v)\bigr)\bigr)g\bigl(f(v)\bigr)f'(v),\quad x\in { \mathbb{R}}^{N}, $$

(1.3)

where $f : [0,+\infty )\rightarrow {\mathbb{R}}$ is given by $f'(t)=\frac{1}{\sqrt{1+2f^{2}(t)}}$ on $[0,+\infty )$, $f(0)=0$, and $f(-t)=f(t)$ on $(-\infty ,0]$. Based on the above facts, if v is a weak solution of (1.3), then $u=f(v)$ is a weak solution of (1.1). The energy functional J reduces to the following functional:

$$ \varPhi (v)=\frac{1}{2} \int _{{\mathbb{R}}^{N}} \bigl( \vert \nabla v \vert ^{2}+f^{2}(v) \bigr)-\frac{1}{2} \int _{{\mathbb{R}}^{N}}\bigl(I_{\alpha }*G\bigl(f(v)\bigr)\bigr)G \bigl(f(v)\bigr). $$

(1.4)

Before stating our results, we need to define the set $\mathcal{Q}=\{v\in H^{1}({\mathbb{R}}^{N})\backslash \{0\}: \mathcal{P}(v)=0\}$, where $\mathcal{P}$ is given in Lemma 2.2. Now, we give our result in the following.

Theorem 1.1

Assume that$(g_{1})$–$(g_{4})$are satisfied. Then problem (1.1) has a ground state solution$u=f(v)$such that$\varPhi (v)=\inf_{\mathcal{Q}}\varPhi =\inf_{v\in \varTheta }\max_{t>0} \varPhi (v_{t})>0$, where$v_{t}=v(x/t)$and

$$ \varTheta := \biggl\{ v\in H^{1}\bigl({\mathbb{R}}^{N}\bigr): \int _{{\mathbb{R}}^{N}} \biggl[\frac{1}{2}f^{2}(v) - \frac{N+\alpha }{2} \bigl(I_{\alpha }*G\bigl(f(v)\bigr) \bigr)G\bigl(f(v)\bigr) \biggr]< 0 \biggr\} . $$

Notions

• Let $H^{1}({\mathbb{R}}^{N})= \{ u\in L^{2}({\mathbb{R}}^{N}) : \nabla u\in L^{2}({\mathbb{R}}^{N}) \}$ with the norm $\|u\|= (\int _{{\mathbb{R}}^{N}}(|\nabla u|^{2}+u^{2}) )^{ \frac{1}{2}}$.

• The embedding $H^{1}({\mathbb{R}}^{N})\hookrightarrow L^{s}({\mathbb{R}}^{N})$ is continuous for $s\in [2,2^{*}]$ and $H^{1}_{r}({\mathbb{R}}^{N})\hookrightarrow L^{s}({\mathbb{R}}^{N})$ is compact for $s\in (2,2^{*})$.

• $H^{1}({\mathbb{R}}^{N})\hookrightarrow L^{\frac{2Nq}{N+\alpha }}({ \mathbb{R}}^{N})$ if and only if $\frac{N+\alpha }{N}\leq q\leq \frac{N-2}{N+\alpha }$ (see [16]).

• $L^{p}({\mathbb{R}}^{N})$ denotes the usual Lebesgue space with norms $\|u\|_{p}= (\int _{{\mathbb{R}}^{N}}|u|^{p} )^{ \frac{1}{p}}$, where $1\leq p<\infty$.

• $\int _{{\mathbb{R}}^{N}}\clubsuit$ denotes $\int _{{\mathbb{R}}^{N}}\clubsuit \,\mathrm{d}x$ and C possibly denotes the different constants.

Proof of Theorem 1.1

In this section, we give the proof of Theorem 1.1. Next, let us recall some properties of the variables $f : {\mathbb{R}}\rightarrow {\mathbb{R}}$. These properties have been proved in [7, 12].

Lemma 2.1

([7, 12])

The function$f(t)$and its derivative satisfy the following properties:

(1):

$f(t)/t\rightarrow 1$as$t\rightarrow 0$;

(2):

$f(t)\leq |t|$for any$t\in {\mathbb{R}}$;

(3):

$f(t)\leq 2^{\frac{1}{4}}\sqrt{|t|}$for all$t\in {\mathbb{R}}$;

(4):

$f^{2}(t)/2\leq tf(t)f'(t)\leq f^{2}(t)$for all$t\in {\mathbb{R}}$;

(5):

there exists a constant$C>0$such that

$$ \bigl\vert f(t) \bigr\vert \geq \textstyle\begin{cases} C \vert t \vert , & \textit{if } \vert t \vert \leq 1, \\ C \vert t \vert ^{\frac{1}{2}},& \textit{if } \vert t \vert \geq 1; \end{cases} $$

(6):

$|f(t)f'(t)|\leq \frac{1}{\sqrt{2}}$for all$t\in {\mathbb{R}}$.

By the standard argument in [16, 19], we have the following Pohozaev type identity.

Lemma 2.2

If$v\in H^{1}({\mathbb{R}}^{N})$is a critical point of (1.3), thenvsatisfies

$$ \mathcal{P}(v):=\frac{N-2}{2} \int _{{\mathbb{R}}^{N}} \vert \nabla v \vert ^{2}+ \frac{N}{2} \int _{{\mathbb{R}}^{N}}f^{2}(v)-\frac{N+\alpha }{2} \int _{{ \mathbb{R}}^{N}}\bigl(I_{\alpha }*G\bigl(f(v)\bigr)\bigr)G \bigl(f(v)\bigr)=0. $$

(2.1)

Motivated by [18], by a simple calculation, for any $t\in [0,1)\cup (1,+\infty )$, one has

$$ \begin{gathered} \beta (t):=\alpha +2-(N+\alpha )t^{N-2}+(N-2)t^{N+\alpha }>0 \quad \text{and}\\ \alpha -(N+\alpha )t^{N}-N\bigl(1-t^{N+\alpha } \bigr)>0. \end{gathered} $$

(2.2)

Lemma 2.3

Assume that$(g_{1})$–$(g_{4})$hold. Then, for all$v\in H^{1}({\mathbb{R}}^{N})$and$t>0$,

$$ \varPhi (v)\geq \varPhi (v_{t})+\frac{1-t^{N+\alpha }}{N+\alpha } \mathcal{Q}(v) + \frac{\beta (t)}{2(N+\alpha )} \Vert \nabla v \Vert ^{2}_{2}. $$

Proof

From (1.4), we have

$$ \varPhi (v_{t})=\frac{t^{N-2}}{2} \int _{{\mathbb{R}}^{N}} \vert \nabla v \vert ^{2}+ \frac{t^{N}}{2} \int _{{\mathbb{R}}^{N}}f^{2}(v)- \frac{t^{N+\alpha }}{2} \int _{{\mathbb{R}}^{N}}\bigl(I_{\alpha }*G\bigl(f(v)\bigr)\bigr)G \bigl(f(v)\bigr). $$

Thus, by (2.2), we have

$$ \begin{aligned} &\varPhi (v)-\varPhi (v_{t})\\ &\quad =\frac{1-t^{N-2}}{2} \int _{{\mathbb{R}}^{N}} \vert \nabla v \vert ^{2} + \frac{1-t^{N}}{2} \int _{{\mathbb{R}}^{N}}f^{2}(v)- \frac{1-t^{N+\alpha }}{2} \int _{{\mathbb{R}}^{N}}\bigl(I_{\alpha }*G\bigl(f(v)\bigr)\bigr)G \bigl(f(v)\bigr) \\ &\quad =\frac{1-t^{N+\alpha }}{N+\alpha } \mathcal{Q}(v)+ \frac{\alpha +2-(N+\alpha )t^{N-2}+(N-2)t^{N+\alpha } }{2(N+\alpha )} \Vert \nabla v \Vert ^{2}_{2} \\ &\qquad {}+ \frac{\alpha -(N+\alpha )t^{N}-N(1-t^{N+\alpha })}{N+\alpha } \bigl\Vert f(v) \bigr\Vert ^{2}_{2} \\ &\quad \geq \frac{1-t^{N+\alpha }}{N+\alpha } \mathcal{Q}(v) + \frac{\beta (t)}{2(N+\alpha )} \Vert \nabla v \Vert ^{2}_{2}. \end{aligned} $$

The proof is completed. □

Corollary 2.4

Assume that$(g_{1})$–$(g_{4})$hold. Then, for any$v\in \mathcal{Q}$, $\varPhi (v)=\max_{t>0}\varPhi (v_{t})$.

Lemma 2.5

Assume that$(g_{1})$–$(g_{4})$hold. Then, for any$\varTheta \neq \emptyset$and the set

$$ \bigl\{ v\in H^{1}\bigl({\mathbb{R}}^{N}\bigr)\backslash \{0 \} : \mathcal{P}(v) \leq 0 \bigr\} \subset \varTheta . $$

Proof

By using $(g_{4})$ and the method in [17] and [18], it follows that $\varTheta \neq \emptyset$. Next, for any $v\in H^{1}({\mathbb{R}}^{N})\backslash \{0\}$, it follows from $\mathcal{P}(v)\leq 0$ that

$$ \frac{N}{2} \int _{{\mathbb{R}}^{N}}f^{2}(v)-\frac{N+\alpha }{2} \int _{{ \mathbb{R}}^{N}}\bigl(I_{\alpha }*G\bigl(f(v)\bigr)\bigr)G \bigl(f(v)\bigr) \leq -\frac{N-2}{2} \int _{{ \mathbb{R}}^{N}} \vert \nabla v \vert ^{2}< 0, $$

which shows that $v\in \varTheta$. The proof is completed. □

Lemma 2.6

Assume that$(g_{1})$–$(g_{4})$hold. Then, for any$v\in \varTheta$, there exists unique$t_{v}>0$such that$v_{t_{v}}\in \mathcal{Q}$.

Proof

Let $v\in \varTheta$ be fixed. Set $\varGamma (t):=\varPhi (v_{t})$. Then it follows from $\varGamma '(t)=0$ that

$$ \begin{aligned}\frac{N-2}{2}t^{N-3} \int _{{\mathbb{R}}^{N}} \vert \nabla v \vert ^{2} + \frac{N t^{N-1}}{2} \int _{{\mathbb{R}}^{N}}f^{2}(v) - \frac{(N+\alpha )t^{N+\alpha -1}}{2} \int _{{\mathbb{R}}^{N}}(I_{\alpha }*G\bigl(f(v)\bigr)G\bigl(f(v) \bigr)=0. \end{aligned} $$

Then

$$ \frac{N-2}{2}t^{N-2} \int _{{\mathbb{R}}^{N}} \vert \nabla v \vert ^{2} + \frac{N t^{N}}{2} \int _{{\mathbb{R}}^{N}}f^{2}(v) - \frac{(N+\alpha )t^{N+\alpha }}{2} \int _{{\mathbb{R}}^{N}}(I_{\alpha }*G\bigl(f(v)\bigr)G\bigl(f(v) \bigr)=0, $$

which implies that $\mathcal{P}(v_{t})=0\Leftrightarrow v_{t}\in \mathcal{Q}$. It is easy to check that $\lim_{t\rightarrow 0}\varGamma (t)=0$, $\varGamma (t)>0$ for $t>0$ enough small and $\varGamma (t)<0$ for t large. Thus $\max_{t>0}\varGamma (t)$ is achieved at some $t_{v}>0$ such that $\varGamma '(t_{v})=0$ and $v_{t_{v}}\in \mathcal{Q}$.

Next, we will prove the uniqueness. For any given $v\in \varTheta$, if there exist $t_{1},t_{2}>0$ such that $v_{t_{1}},v_{t_{2}}\in \mathcal{Q}$. Thus $\mathcal{P}(v_{t_{1}})=\mathcal{P} (v_{t_{2}})=0$. Therefore, we have

$$ \begin{aligned}\varPhi (v_{t_{1}})&\geq \varPhi (v_{t_{2}})+ \frac{t^{N}_{1}-t^{N}_{2}}{(N+\alpha )t^{N}_{1}}\mathcal{P} (v_{t_{1}}) +\frac{\beta (t_{2}/t_{1})}{2(N+\alpha )} \Vert \nabla v_{t_{1}} \Vert ^{2}_{2}= \varPhi (v_{t_{2}}) +\frac{\beta (t_{2}/t_{1})}{2(N+\alpha )} \Vert \nabla v_{t_{1}} \Vert ^{2}_{2} \end{aligned} $$

and

$$ \begin{aligned}\varPhi (v_{t_{2}})&\geq \varPhi (v_{t_{1}})+ \frac{t^{N}_{2}-t^{N}_{1}}{(N+\alpha )t^{N}_{2}}\mathcal{P} (v_{t_{2}}) +\frac{\beta (t_{1}/t_{2})}{2(N+\alpha )} \Vert \nabla v_{t_{2}} \Vert ^{2}_{2}= \varPhi (v_{t_{1}}) +\frac{\beta (t_{1}/t_{2})}{2(N+\alpha )} \Vert \nabla v_{t_{2}} \Vert ^{2}_{2}, \end{aligned} $$

which shows that $t_{1}=t_{2}$. Thus $t_{v}>0$ is unique for $v\in \mathcal{Q}$. This completes the proof. □

Lemma 2.7

Assume that$(g_{1})$–$(g_{3})$hold, then$\mathcal{Q}\neq \emptyset$and

$$ \inf_{\mathcal{M}}\varPhi :=c=\inf_{v\in \varTheta }\max _{t>0}\varPhi (v_{t}). $$

Proof

This result is a consequence of Corollary 2.4, Lemma 2.5, and Lemma 2.6. The proof is completed. □

By a standard argument in [19], we can get the following Brezis–Lieb lemma.

Lemma 2.8

Assume that$(g_{1})$–$(g_{4})$hold. If$v_{n}\rightharpoonup v_{0}$in$H^{1}({\mathbb{R}}^{N})$, then

$$ \varPhi (v_{n})=\varPhi (v_{0})+\varPhi (v_{n}-v_{0})+o_{n}(1) $$

and

$$ \mathcal{P}(v_{n})=\mathcal{P}(v_{0})+ \mathcal{P}(v_{n}-v_{0})+o_{n}(1). $$

Lemma 2.9

Assume that$(g_{1})$–$(g_{4})$hold. Then

• (i) there exists$\rho >0$such that$\|\nabla v\|_{2}\geq \rho$for any$v\in \mathcal{Q}$;

• (ii) $c=\inf_{\mathcal{Q}}\varPhi >0$.

Proof

(i) By $(g_{3})$, for any $\varepsilon >0$, there exists $C^{1}_{\varepsilon }>0$ such that

$$ \bigl\vert G\bigl(f(v)\bigr) \bigr\vert ^{\frac{2N}{N+\alpha }}\leq \varepsilon \vert v \vert ^{2} +C^{1}_{\varepsilon } \vert s \vert ^{2^{*}}\quad \text{and}\quad \bigl\vert G\bigl(f(v) \bigr) \bigr\vert ^{ \frac{2N}{N+\alpha }} \leq \varepsilon \vert v \vert ^{2}+\varepsilon \vert s \vert ^{2^{*}}+C^{1}_{\varepsilon } \vert s \vert ^{p}, $$

(2.3)

where $p\in (2,2^{*})$. For any $v\in \mathcal{Q}$, we have that $\mathcal{P}(v)=0$. By the Sobolev embedding theorem, the Hardy–Littlewood–Sobolev inequality in [15], (2.3), and $(g_{1})$, we get

$$\begin{aligned} &\frac{(N-2)}{2} \int _{{\mathbb{R}}^{N}} \vert \nabla v \vert ^{2}+ \frac{N}{2} \int _{{\mathbb{R}}^{N}}f^{2}(v) \\ &\quad =\frac{N+\alpha }{2} \int _{{ \mathbb{R}}^{N}}\bigl(I_{\alpha }*G\bigl(f(v)\bigr)\bigr)G \bigl(f(v)\bigr) \\ &\quad \leq C \biggl(\varepsilon \int _{{\mathbb{R}}^{N}} \bigl\vert f(v) \bigr\vert ^{2}+C^{1}_{\varepsilon } \int _{{\mathbb{R}}^{N}} \vert v \vert ^{2^{*}} \biggr) \leq C \biggl( \varepsilon \int _{{\mathbb{R}}^{N}} \bigl\vert f(v) \bigr\vert ^{2}+C^{1}_{\varepsilon } \biggl( \int _{{\mathbb{R}}^{N}} \vert \nabla v \vert ^{2} \biggr)^{2^{*}/2} \biggr), \end{aligned}$$

which shows that there exists $\rho >0$ such that $\|\nabla v\|^{2}\geq \rho$ for any $v\in \mathcal{Q}$.

(ii) For any $v\in \mathcal{Q}$, from Lemma 2.2, we have

$$ \begin{aligned}\varPhi (v)&=\varPhi (v)-\frac{1}{N+\alpha } \mathcal{P}(v)\geq \frac{\alpha +2}{2(N+\alpha )} \Vert \nabla v \Vert ^{2}_{2}. \end{aligned} $$

(2.4)

This completes the proof. □

Lemma 2.10

Assume that$(g_{1})$–$(g_{3})$hold. Thencis achieved.

Proof

Let $\{v_{n}\}\subset \mathcal{Q}$ be a minimizer for c, that is, $\mathcal{P}(v_{n})=0$ and $\varPhi (v_{n})\rightarrow c$ as $n\rightarrow \infty$. By (2.4), one has

$$ c+o_{n}(1)=\varPhi (v_{n})-\frac{1}{N+\alpha } \mathcal{P}(v_{n}) = \frac{\alpha +2}{2(N+\alpha )} \Vert \nabla v_{n} \Vert ^{2}_{2}+ \frac{N}{2(N+\alpha )} \int _{{\mathbb{R}}^{N}}f^{2}(v_{n}), $$

which shows that $\int _{{\mathbb{R}}^{N}}|\nabla v_{n}|^{2}+\int _{{\mathbb{R}}^{N}}f^{2}(v_{n})$ is bounded and thus $\{v_{n}\}$ is bounded in $D^{1,2}({\mathbb{R}}^{N})$. By the Sobolev inequality, Lemma 2.1-(5), it follows that

$$ \int _{ \vert v_{n} \vert \leq 1}v^{2}_{n}\leq \int _{{\mathbb{R}}^{N}}f^{2}(v_{n}) \quad \text{and} \quad \int _{ \vert v_{n} \vert >1}v^{2}_{n}\leq \int _{ \vert v_{n} \vert >1}v^{2^{*}}_{n} \leq C \biggl( \int _{{\mathbb{R}}^{N}} \vert \nabla v_{n} \vert ^{2} \biggr)^{2^{*}/2}. $$

Therefore

$$ \begin{aligned} \int _{{\mathbb{R}}^{N}}v^{2}_{n}= \int _{ \vert v_{n} \vert \leq 1}v^{2}_{n}+ \int _{ \vert v_{n} \vert >1}v^{2}_{n} \leq \int _{{\mathbb{R}}^{N}}f^{2}(v_{n}) +C \biggl( \int _{{\mathbb{R}}^{N}} \vert \nabla v_{n} \vert ^{2} \biggr)^{2^{*}/2}. \end{aligned} $$

(2.5)

From (2.5), we infer that there exists $C>0$ such that $\int _{{\mathbb{R}}^{N}}v_{n}^{2}\leq C$. Up to a subsequence, there exists $v_{0}\in H^{1}({\mathbb{R}}^{N})$ such that $v_{n}\rightharpoonup v_{0}$ in $H^{1}({\mathbb{R}}^{N})$, $v_{n}\rightarrow v_{0}$ in $L^{r}_{\mathrm{loc}}({\mathbb{R}}^{N})$ for $r\in [2,2^{*})$ and $v_{n}\rightarrow v_{0}$ a.e. on ${\mathbb{R}}^{N}$.

Now, we claim that there exist $\delta >0$ and $\{y_{n}\}\subset {\mathbb{R}}^{N}$ such that $\int _{B_{1}(y_{n})}|v_{n}|^{2}>\delta$. Assume by contradiction, by Lion’s concentration compactness lemma in [19], that $v_{n}\rightarrow 0$ in $L^{r}({\mathbb{R}}^{N})$ for $2< r<2^{*}$. Moreover, by $\mathcal{P}(v_{n})=0$, we know that

$$ \begin{aligned} 0&\leftarrow \int _{{\mathbb{R}}^{N}}(I_{\alpha }*G(f(v_{n})))G(f(v_{n}))\\ &= \frac{N-2}{N+\alpha } \int _{{\mathbb{R}}^{N}} \vert \nabla v_{n} \vert ^{2}+ \frac{N}{N+\alpha } \int _{{\mathbb{R}}^{N}}f^{2}(v_{n}) \geq \frac{N-2}{N+\alpha }\rho ^{2}>0, \end{aligned} $$

as $n\rightarrow +\infty$. This is a contradiction. Thus there exist $\delta >0$ and $\{y_{n}\}\subset {\mathbb{R}}^{N}$ such that $\int _{B_{1}(y_{n})}|v_{n}|^{2}>\delta$. Set $\bar{v}_{n}(x)=v_{n}(x+y_{n})$. Then $\|\bar{v}_{n}\|=\|v_{n}\|$. Thus, up to a subsequence, there exists $\bar{v}_{0}\in H^{1}({\mathbb{R}}^{N})\backslash \{0\}$ such that $\bar{v}_{n}\rightharpoonup \bar{v}_{0}$ in $H^{1}({\mathbb{R}}^{N})$, $\bar{v}_{n}\rightarrow \bar{v}_{0}$ in $L^{r}_{\mathrm{loc}}({\mathbb{R}}^{N})$ for $r\in [2,2^{*})$, and $\bar{v}_{n}\rightarrow \bar{v}_{0}$ a.e. on ${\mathbb{R}}^{N}$. By translation invariance, one has

$$ \varPhi (\bar{v}_{n})\rightarrow c, \qquad \mathcal{P}( \bar{v}_{n}) \rightarrow 0, \quad \text{as } n\rightarrow +\infty $$

(2.6)

and $\int _{B_{1}(0)}|\bar{v}_{n}|^{2}>\delta$. Set $\bar{w}_{n}:=\bar{v}_{n}-\bar{v}_{0}$. Thus Lemma 2.7 yields that

$$ c=\varPhi (\bar{v}_{0})+\varPhi (\bar{w}_{n})+o_{n}(1) \quad \text{and} \quad 0= \mathcal{P}(\bar{v}_{0})+\mathcal{P}( \bar{w}_{n})+o_{n}(1). $$

(2.7)

If there exists a subsequence $\{\bar{w}_{n_{i}}\}$ of $\{\bar{w}_{n}\}$ such that $\bar{w}_{n_{i}}=0$, then up to a subsequence, we have

$$ \varPhi (\bar{v}_{0})=c,\qquad \mathcal{P}( \bar{v}_{0})=0. $$

(2.8)

Next, we assume that $\bar{w}_{n}\neq 0$. We claim that $\mathcal{P}(\bar{v}_{0})\leq 0$. Otherwise, if $\mathcal{P}(\bar{v}_{0})>0$, it follows from (2.7) that $\mathcal{P}(\bar{w}_{n})<0$ for n large. By virtue of Lemma 2.6, there exists $t_{n}>0$ such that $(\bar{w}_{n})_{t_{n}}\in \mathcal{Q}$. By (2.7) and Lemma 2.2, we get

$$ \begin{aligned} &c-\frac{N-2}{N+\alpha } \int _{{\mathbb{R}}^{N}} \vert \nabla \bar{v}_{0} \vert ^{2}- \frac{N}{N+\alpha } \int _{{\mathbb{R}}^{N}}f^{2}(\bar{v}_{0})+o_{n}(1) \\ &\quad = \varPhi (\bar{w}_{n})-\frac{1}{N+\alpha }\mathcal{P}( \bar{w}_{n})\geq \varPhi \bigl((\bar{w}_{n})_{t_{n}}\bigr)- \frac{t^{N+\alpha }_{n}}{N+\alpha } \mathcal{P}(\bar{w}_{n}) \geq c-\frac{t^{N+\alpha }_{n}}{N+\alpha } \mathcal{P}(\bar{w}_{n})\geq c, \end{aligned} $$

which is a contradiction due to $\int _{{\mathbb{R}}^{N}}|\nabla \bar{v}_{0}|^{2}>0$. Thus $\mathcal{P}(\bar{v}_{0})\leq 0$. Since $\bar{v}_{0}\neq 0$, in view of Lemma 2.6, there exists $t_{0}>0$ such that $(\bar{v}_{0})_{t_{0}}\in \mathcal{Q}$. By Lemma 2.3 and the weak semi-continuity of norm, we have

$$ \begin{aligned}c&=\lim_{n\rightarrow \infty } \biggl[\varPhi ( \bar{v}_{n}) - \frac{1}{N+\alpha }\mathcal{P}(\bar{v}_{n}) \biggr] \\ &=\frac{N-2}{N+\alpha }\lim_{n\rightarrow \infty } \int _{{\mathbb{R}}^{N}} \vert \nabla \bar{v}_{n} \vert ^{2} +\frac{N}{N+\alpha }\lim_{n\rightarrow \infty } \int _{{\mathbb{R}}^{N}}f^{2}(\bar{v}_{n}) \\ &\geq \frac{N-2}{N+\alpha } \int _{{\mathbb{R}}^{N}} \vert \nabla \bar{v}_{0} \vert ^{2}+ \frac{N}{N+\alpha } \int _{{\mathbb{R}}^{N}}f^{2}(\bar{v}_{0}) \\ &\geq \varPhi (\bar{v}_{0}) -\frac{t^{N+\alpha }_{0}}{N+\alpha } \mathcal{P}( \bar{v}_{0}) \\ &\geq \varPhi \bigl((\bar{v}_{0})_{t_{0}}\bigr) - \frac{t^{N+\alpha }_{0}}{N+\alpha } \mathcal{P}(\bar{v}_{0}) \\ &\geq c-\frac{t^{N+\alpha }_{0}}{N+\alpha } \mathcal{P}(\bar{v}_{0}) \geq c, \end{aligned} $$

which implies that (2.8) holds. The proof is completed. □

By a standard argument in [14, 18, 19], we can get the following lemma.

Lemma 2.11

Assume that$(g_{1})$–$(g_{4})$hold. If$\tilde{v}\in \mathcal{Q}$and$\varPhi (\tilde{v})=c$, thenṽis a critical point ofΦ.

Proof of Theorem 1.1

By Lemma 2.7, Lemma 2.10, and Lemma 2.11, there exists $v_{0}\in \mathcal{Q}$ such that

$$ \varPhi (v_{0})=c=\inf_{v\in \varTheta }\max_{t>0} \varPhi (v_{t}),\quad \varPhi '(v_{0})=0. $$

This completes the proof. □