Existence and Multiplicity of Solutions for Semilinear Elliptic Systems with Periodic Potential

Abstract

In this paper, we consider the following semilinear elliptic systems:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\Delta u+V(x)u=F_{u}(x, u, v),\quad \text{ in } \mathbb {R}^{N},\\ -\Delta v+V(x)v=F_{v}(x, u, v),\quad \text{ in } \mathbb {R}^{N},\\ \end{array} \right. \end{aligned}$$

where $V:\mathbb {R}^{N}\rightarrow \mathbb {R},~F_{u}(x,u,v)$ and $F_{v}(x,u,v)$ are periodic in x. We assume that 0 is a right boundary point of the essential spectrum of $-\triangle +V$. Under appropriate assumptions on $F_{u}(x, u, v)$ and $F_{v}(x, u, v)$, we prove the above system has a ground-state solution by using the Nehari-type technique in a strongly indefinite setting. Furthermore, the existence of infinitely many geometrically distinct solutions is obtained via variational methods. Recent results from the literature are improved and extended.

1 Introduction

In this paper, we consider the existence and multiplicity of nontrivial solutions to the following semilinear elliptic systems:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\Delta u+V(x)u=F_{u}(x, u, v), \quad \text{ in } \mathbb {R}^{N},\\ -\Delta v+V(x)v=F_{v}(x, u, v), \quad \text{ in } \mathbb {R}^{N}.\\ \end{array} \right. \end{aligned}$$

(1.1)

When $\Omega$ is a bounded domain of $\mathbb {R}^{N}$, the problem

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\Delta u=\lambda \big (a(x)u+b(x)v\big )+F_{u}(x, u, v), \quad \text{ in } \Omega ,\\ -\Delta v=\lambda \big (b(x)u+c(x)v\big )+F_{v}(x, u, v), \quad \text{ in } \Omega ,\\ u(x)=v(x)= 0, \quad \text{ on } \partial \Omega , \end{array} \right. \end{aligned}$$

(1.2)

which is related to reaction–diffusion systems that appear in chemical and biological phenomena, including the steady- and unsteady-state situation (see [1]), has been extensively investigated in recent years. For the results on existence, multiple solutions, and positive solutions to problem (1.2), we refer the reader to [1,2,3,4,5] and the references therein. In [3], Qu and Tang obtained the existence and multiplicity of weak solutions for problem (1.2) by using the Ekeland variational principle together with variational methods, and some new existence theorems of weak solutions were obtained in Duan et al. [2]. Lots of work has been done when $\Omega$ is an unbounded domain of $\mathbb R^{N}$, and we refer the reader to [6,7,8,9,10,11,12,13,14,15,16] and the references therein.

Recall that the spectrum $\sigma (-\Delta +V)$ of $-\Delta +V$ is purely continuous and may contain gaps, i.e., open intervals free of spectrum (see [17]). In [18], Szulkin and Weth considered the following Schrödinger equation:

$$\begin{aligned} -\Delta u+V(x)u=f(x,u), \quad \text{ in } \mathbb {R}^{N}. \end{aligned}$$

(1.3)

Assuming that $0\not \in \sigma (-\Delta +V)$, they proved that problem (1.3) possesses a ground-state solution, which is just a minimizer of the energy functional associated with problem (1.3) on the Nehari–Pankov manifold [19]. Later, Mederski [20] considered the ground-state solutions to the system of coupled Schrödinger equations as follows:

$$\begin{aligned} -\triangle u_{i}+V_{i}(x)u_{i}=\partial _{u_{i}}F(x,u),~~\text{ on }~\mathbb {R}^{N},~~i=1,2,\ldots ,K, \end{aligned}$$

(1.4)

where F and $V_{i}$ are periodic in x, $0\not \in \sigma (-\Delta +V_{i}),~i=1,2,\ldots ,K$. Moreover, they made use of a new linking-type result involving the Nehari–Pankov manifold and assumed that F satisfies the following conditions:

• 1. $f_{i}:\mathbb {R}^{N}\times \mathbb {R}^{K}\rightarrow \mathbb {R}$ is measurable, $\mathbb {Z}^{N}$-periodic in $x\in \mathbb {Z}^{N}$ and continuous in $u\in \mathbb {R}^{K}$ for $a.e.~x\in \mathbb {R}^{N}$. Moreover, $f=(f_{1},f_{2},\ldots ,f_{K})=\partial _{u}F$, where $F:\mathbb {R}^{N}\times \mathbb {R}^{K}\rightarrow \mathbb {R}$ is differentiable with respect to the second variable $u\in \mathbb {R}^{K}$ and $F(x,0)=0$ for a.e. $x\in \mathbb {R}^{N}$.

• 2. There are $a>0$ and $2<p<2^{*}$ such that

  $$\begin{aligned} |f(x,u)|\le a\left( 1+|u|^{p-1}\right) ,~~\text{ for } \text{ all }~u\in \mathbb {R}^{K}\text{ and } \text{ a.e. }~ x\in \mathbb {R}^{N}. \end{aligned}$$

• 3. $f(x,u)=o(u)$ uniformly with respect to x as $|u|\rightarrow 0$.

  In [21], Guo and Mederski considered the existence and nonexistence of ground-state solutions of system (1.4) with $K=1$ and $V(x)=V_{1}(x)-\frac{\mu }{|x|^{2}}$, where $V_{1}\in L^{\infty }(\mathbb {R}^{N})$, $V_{1}$ is $\mathbb {Z}^{N}$-periodic in $x\in \mathbb {R}^{N}$, and $0\not \in \sigma (-\triangle +V)$. Moreover, they assumed that f(x, u) satisfies (1)–(3) and (4)–(5) as follows:

• 4. $\frac{F(x,u)}{u^{2}}\rightarrow \infty$ uniformly in x as $|u|\rightarrow \infty$, where F is the primitive of f with respect to u, that is, $F(x,u)=\int _{0}^{u}f(x,s)\mathrm {d}s$.

• 5. $u\mapsto \frac{f(x,u)}{|u|}$ is nondecreasing on $(-\infty ,0)$ and $(0,+\infty )$.

When 0 is a right boundary point of the essential spectrum of $-\Delta +V$ and f(x, u) is superlinear and subcritical, Mederski [22] obtained the existence of ground-state solutions and multiple solutions of system (1.3) with $u(x)\rightarrow 0,~\text{ as } ~|x|\rightarrow \infty$.

Inspired by the above facts, more precisely by [21,22,23], the aim of this paper is to study the existence and multiplicity of nontrivial solutions to problem (1.1) via variational methods. To the best of our knowledge, there have been few works concerning this case up to now.

We assume that V(x) and F(x, u, v) satisfy the following hypotheses:

(V) $V\in C(\mathbb {R}^{N},\mathbb {R})$, V is 1-periodic in $x_{i},~i=1,2,\ldots ,N$, $0\in \sigma (-\Delta +V)$, and there exists $\alpha >0$ such that $(0,\alpha ]\cap \sigma (-\Delta +V)=\emptyset$.

$(F_{1})$$F\in C^{1}(\mathbb R^{N}\times \mathbb R^{2})$, $F_{u}(x,u,v)$ and $F_{v}(x,u,v)$ are measurable, 1-periodic in $x_{i},~i=1,2,\ldots ,N$.

$(F_{2})$ There exist $c>0$ and $2<\gamma \le p<2^{*}$ such that $|F_{u}(x,u,v)|\le c(|(u,v)|^{\gamma -1}+|(u,v)|^{p-1})$ and $|F_{v}(x,u,v)|\le c(|(u,v)|^{\gamma -1}+|(u,v)|^{p-1})$ for all $(u,v)\in \mathbb R\times \mathbb R$ and $x\in \mathbb R^{N}$, where $|(u,v)|=(u^{2}+v^{2})^{\frac{1}{2}}$.

$(F_{3})$ There exists $d>0$ such that

$$\begin{aligned} F(x,u,v)\ge d|(u,v)|^{\gamma },~~\text{ for }~|(u,v)|\le 1,~x\in \mathbb R^{N}. \end{aligned}$$

$(F_{4})$$\frac{F(x,u,v)}{|(u,v)|^{2}}\rightarrow \infty$ uniformly in $x\in \mathbb R^{N}$ as $|(u,v)|\rightarrow \infty$.

$(F_{5})$ If $F_{u}(x,u_{2},v_{2})u_{1}>0$ or $F_{v}(x,u_{2},v_{2})v_{1}>0$ for any $(u_{2},v_{2}),~(u_{1},v_{1})\in \mathbb {R}^{2}$, then we have

$$\begin{aligned} \begin{aligned} F\left( x,u_{2},v_{2}\right) -F\left( x,u_{1},v_{1}\right)&\le \frac{(F_{u}(x,u_{2},v_{2})u_{2})^{2}-(F_{u}(x,u_{2},v_{2})u_{1})^{2}}{2F_{u}(x,u_{2},v_{2})u_{2}}\\&\quad +\frac{(F_{v}(x,u_{2},v_{2})v_{2})^{2}-(F_{v}(x,u_{2},v_{2})v_{1})^{2}}{2F_{v}(x,u_{2},v_{2})v_{2}}. \end{aligned} \end{aligned}$$

$(F_{6})$$F_{u}(x,u,v)u\ge 0$ and $F_{v}(x,u,v)v\ge 0$ and $F_{u}(x,u,v)u+F_{v}(x,u,v)v\ge 2F(x,u,v)$ for any $(x,u,v)\in \mathbb R^{N}\times \mathbb R^{2}$.

$(F_{7})$$F(x,-u,-v)=F(x,u,v)$ for any $(x,u,v)\in \mathbb {R}^{N}\times \mathbb {R}^{2}$.

Assumptions (V) and $(F_{1}){-}(F_{6})$ allow us to find a function space $E_{2,\gamma }$ (see Sect. 2) on which the energy functional associated with (1.1) is given by

$$\begin{aligned} \begin{aligned} J(u,v)&=\frac{1}{2}\int _{\mathbb {R}^{N}}\left( |\nabla u|^{2}+V(x)|u|^{2}\right) \mathrm {d}x +\frac{1}{2}\int _{\mathbb {R}^{N}}\left( |\nabla v|^{2}+V(x)|v|^{2}\right) \mathrm {d}x\\&\quad -\int _{\mathbb {R}^{N}}F(x, u, v)\mathrm {d}x, \end{aligned} \end{aligned}$$

(1.5)

and

$$\begin{aligned} \begin{aligned} \langle J'(u,v), (\varphi ,\psi )\rangle&=\int _{\mathbb {R}^{N}}\nabla u\nabla \varphi \mathrm {d}x +\int _{\mathbb {R}^{N}}V(x)u\varphi \mathrm {d}x-\int _{\mathbb {R}^{N}}F_{u}(x,u,v)\varphi \mathrm {d}x\\&\quad +\int _{\mathbb {R}^{N}}\nabla v\nabla \psi \mathrm {d}x +\int _{\mathbb {R}^{N}}V(x)v\psi \mathrm {d}x-\int _{\mathbb {R}^{N}}F_{v}(x,u,v)\psi \mathrm {d}x. \end{aligned} \end{aligned}$$

(1.6)

Now we state our main results.

Theorem 1.1

Suppose that (V) and $(F_{1}){-}(F_{6})$ hold, then problem (1.1) has a ground-state solution $(u,v)\in \mathcal {N}$ such that $J(u,v)=\inf \limits _{\mathcal {N}}J>0$. Furthermore, $(u,v)\in \big (H^{2}_{loc}(\mathbb {R}^{N})\times H^{2}_{loc}(\mathbb {R}^{N})\big )\cap \big (L^{t}(\mathbb {R}^{N})\times L^{t}(\mathbb {R}^{N})\big )$ for $\gamma \le t\le 2^{*}$.

Theorem 1.2

Suppose that (V) and $(F_{1}){-}(F_{7})$ hold, then problem (1.1) has infinitely many geometrically distinct solutions which lie in $\big (H^{2}_{loc}(\mathbb {R}^{N})\times H^{2}_{loc}(\mathbb {R}^{N})\big )\cap \big (L^{t}(\mathbb {R}^{N})\times L^{t}(\mathbb {R}^{N})\big )$ for $\gamma \le t\le 2^{*}$.

Notation Throughout this paper, we shall denote by $\Vert \cdot \Vert _{r}$ the $L^{r}$-norm and C various positive generic constants, which may vary from line to line. $2^{*}=\frac{2N}{N-2}$ is the critical Sobolev exponent. Also if we take a subsequence of a sequence $\{(u_{n},v_{n})\}$ we shall denote it again by $\{(u_{n},v_{n})\}$.

This paper is organized as follows: In Sect. 2, some preliminary results are presented. In Sect. 3, we introduce the Nehari–Pankov manifold $\mathcal {N}\subset E_{2,\gamma }$ on which we minimize J to find a ground state and we prove Theorem 1.1. Eventually, in Sect. 4, the multiplicity result is obtained and we prove Theorem  1.2.

2 Variational Setting and Preliminaries

In this section we outline the variational framework for problem (1.1) and give some preliminary lemmas.

It follows from condition (V) that there exists a decomposition of $H^{1}(\mathbb {R}^{N})=X^{+}\oplus X^{-}$ corresponding to the decomposition of the spectrum of $\sigma (S)$ into $\sigma (S)\cap [\alpha ,+\infty )$ and $\sigma (S)\cap (-\infty ,0]$, where $S=-\triangle +V$ with the domain $D(S)=H^{2}(\mathbb {R}^{N})$. We introduce a new norm $\Vert \cdot \Vert _{X}$ on $X^{+}$ (resp. $X^{-}$) by setting

$$\begin{aligned} \Vert u^{+}\Vert ^{2}_{X}=\int _{\mathbb {R}^{N}}\left( |\nabla u^{+}|^{2}+V(x)|u^{+}|^{2}\right) \mathrm {d}x \end{aligned}$$

and

$$\begin{aligned} \Vert u^{-}\Vert ^{2}_{X}=-\int _{\mathbb {R}^{N}}\left( |\nabla u^{-}|^{2}+V(x)|u^{-}|^{2}\right) \mathrm {d}x \end{aligned}$$

for $u^{+}\in X^{+}$ and $u^{-}\in X^{-}$. Then, $\Vert \cdot \Vert _{X}$ is equivalent to $\Vert \cdot \Vert _{H^{1}}$ on $X^{+}$ and is weaker than $\Vert \cdot \Vert _{H^{1}}$ on $X^{-}$ (see [24])

$$\begin{aligned} \langle u,v\rangle _{X}=\int _{\mathbb {R}^{N}}\left( \nabla u^{+}\nabla v^{+}+V(x)u^{+}v^{+}\right) \mathrm {d}x -\int _{\mathbb {R}^{N}}\left( \nabla u^{-}\nabla v^{-}+V(x)u^{-}v^{-}\right) \mathrm {d}x \end{aligned}$$

and a norm given by

$$\begin{aligned} \Vert u\Vert _{X}^{2}=\Vert u^{+}\Vert _{X}^{2}+\Vert u^{-}\Vert _{X}^{2}, \end{aligned}$$

which is equivalent to the usual Sobolev norm in $H^{1}(\mathbb {R}^{N})$, that is

$$\begin{aligned} \Vert u\Vert _{H^{1}}=\bigg (\int _{\mathbb {R}^{N}}(|\nabla u|^{2}+|u|^{2})\mathrm {d}x\bigg )^{\frac{1}{2}}. \end{aligned}$$

Therefore, $X^{+}$ and $X^{-}$ are orthogonal with respect to the inner product $\langle \cdot ,\cdot \rangle _{X}$ as well.

As usual, for $1\le p<+\infty$, we let

$$\begin{aligned} \Vert u\Vert _{p}=\left( \int _{\mathbb {R}^{N}}|u(x)|^{p}\mathrm {d}x\right) ^{\frac{1}{p}},~~u\in L^{p}\left( \mathbb {R}^{N}\right) , \end{aligned}$$

and

$$\begin{aligned} \Vert u\Vert _{\infty }=ess ~\sup \limits _{x\in \mathbb {R}^{N}}|u(x)|, ~~u\in L^{\infty }\left( \mathbb {R}^{N}\right) . \end{aligned}$$

Then, $E=X\times X=E^{+}\oplus E^{-}$, where $E^{+}=X^{+}\times X^{+},~E^{-}=X^{-}\times X^{-}$, is a Hilbert space with the following inner product

$$\begin{aligned} \langle (u,v),(\varphi ,\psi )\rangle =\langle u,\varphi \rangle _{X}+\langle v,\psi \rangle _{X} ~~(u,v),~(\varphi ,\psi )\in X\times X, \end{aligned}$$

and the norm

$$\begin{aligned} \Vert (u,v)\Vert ^{2}=\langle (u,v),(u,v)\rangle =\Vert u\Vert _{X}^{2}+\Vert v\Vert _{X}^{2},~~(u,v)\in X\times X. \end{aligned}$$

Note that the energy functional J can be written as follows:

$$\begin{aligned} \begin{aligned} J(u,v)&=\frac{1}{2}\big (\Vert (u^{+},v^{+})\Vert ^{2}-\Vert (u^{-},v^{-})\Vert ^{2}\big ) -\int _{\mathbb {R}^{N}}F(x, u, v)\mathrm {d}x\\&=\frac{1}{2}\Vert (u^{+},v^{+})\Vert ^{2}-I(u,v), \end{aligned} \end{aligned}$$

where

$$\begin{aligned} I(u,v)=\frac{1}{2}\Vert (u^{-},v^{-})\Vert ^{2} +\int _{\mathbb {R}^{N}}F(x, u, v)\mathrm {d}x \end{aligned}$$

for any $(u,v)=(u^{+},v^{+})+(u^{-},v^{-})\in E^{+}\oplus E^{-}$. We do not know whether J has critical points in $H^{1}(\mathbb {R}^{N})\times H^{1}(\mathbb {R}^{N})$ since $0\in \sigma (S)$. Furthermore, I is not defined on E owing to our assumptions on F(x, u, v). Hence, similar to [24], we are going to define a space $E_{2,\lambda }$ such that there are continuous embeddings

$$\begin{aligned} H^{1}\left( \mathbb {R}^{N}\right) \times H^{1}\left( \mathbb {R}^{N}\right) \hookrightarrow E_{2,\lambda }\hookrightarrow E, \end{aligned}$$

where I is well defined on $E_{2,\lambda }$ and J admits critical points on $E_{2,\lambda }$.

Let $\big (P_{\lambda }:L^{2}(\mathbb {R}^{N})\rightarrow L^{2}(\mathbb {R}^{N})\big )_{\lambda \in \mathbb {R}}$ denote the spectral family of S. Let $L^{-}=P_{0}\big (L^{2}(\mathbb {R}^{N})\big )$ and $L^{+}=(id-P_{0})\big (L^{2}(\mathbb {R}^{N})\big )$. Thus, we have the orthogonal decomposition $L^{2}(\mathbb {R}^{N})=L^{+}+L^{-}$, $E^{+}=\big (H^{1}(\mathbb {R}^{N}\cap L^{+})\big )\times \big (H^{1}(\mathbb {R}^{N}\cap L^{+}\big ))$, and $E^{-}=\big (H^{1}(\mathbb {R}^{N}\cap L^{-}\big ))\times \big (H^{1}(\mathbb {R}^{N} L^{-}\big )$ (see[17, 25]). Furthermore,

$$\begin{aligned} \Vert u\Vert _{X}^{2}=\int _{-\infty }^{+\infty }|\lambda |d\Vert P_{\lambda }u\Vert _{2}^{2}. \end{aligned}$$

Let us suppose that $2\le \mu \le \gamma$. $L^{\mu ,\gamma }=L^{\mu }(\mathbb {R}^{N})+L^{\gamma }(\mathbb {R}^{N})$ denotes the Banach space of all vector fields of the form $u=u_{1}+u_{2}$, where $u_{1}\in L^{\mu }(\mathbb {R}^{N})$ and $u_{2}\in L^{\gamma }(\mathbb {R}^{N})$, endowed with the following norm

$$\begin{aligned} \Vert u\Vert _{\mu ,\gamma }=\inf \left\{ \Vert u_{1}\Vert _{\mu }+\Vert u_{2}\Vert _{\gamma }\right\} . \end{aligned}$$

It follows from Proposition 2.5 in [26] that the infimum in $\Vert \cdot \Vert _{\mu ,\gamma }$ is attained. Furthermore, there is a continuous embedding

$$\begin{aligned} L^{t}\left( \mathbb {R}^{N}\right) \hookrightarrow L^{\mu ,\gamma }\left( \mathbb {R}^{N}\right) \end{aligned}$$

for any $\mu \le t\le \gamma$, and if $\mu =\gamma$, then the norms $\Vert \cdot \Vert _{\mu ,\gamma }$ and $\Vert \cdot \Vert _{\gamma }$ are equivalent. Let $X^{-}_{\mu ,\gamma }$ and $X^{-}_{\gamma }$ be the completions of $X^{-}$ with the respect to the norms

$$\begin{aligned} \Vert |\cdot |\Vert _{\mu ,\gamma }=\bigg (\Vert \cdot \Vert _{X}^{2}+\Vert \cdot \Vert _{\mu ,\gamma }^{}\bigg )^{\frac{1}{2}} \end{aligned}$$

and

$$\begin{aligned} \Vert |\cdot |\Vert _{\gamma }=\bigg (\Vert \cdot \Vert _{X}^{2}+\Vert \cdot \Vert _{\gamma }^{}\bigg )^{\frac{1}{2}}, \end{aligned}$$

respectively. Therefore, we have the following continuous embeddings

$$\begin{aligned} X^{-}\hookrightarrow X^{-}_{\gamma }\hookrightarrow X^{-}_{\mu ,\gamma }\hookrightarrow X. \end{aligned}$$

Space $X^{-}_{\gamma }$ has been introduced in [24], and note that if $\mu =\gamma$, then $X^{-}_{\mu ,\gamma }=X^{-}_{\gamma }$ and the norms $\Vert |\cdot |\Vert _{\mu ,\gamma }$ and $\Vert |\cdot |\Vert _{\gamma }$ are equivalent. In this paper, space $E^{-}_{\mu ,\gamma }=X^{-}_{\mu ,\gamma }\times X^{-}_{\mu ,\gamma }$ with $\mu =2$ is crucial due to the superlinear growth conditions $(F_{3})$ and $(F_{4})$. Moreover, we have the following lemmas from [22, 24].

Lemma 2.1

$E^{-}_{\mu ,\gamma }=E^{-}_{\gamma }$ and norms $\Vert \cdot \Vert _{\mu ,\gamma }$, $\Vert \cdot \Vert _{\gamma }$ are equivalent for any $2\le \mu \le \gamma \le 2^{*}$.

Lemma 2.2

If $2\le \mu \le \gamma \le 2^{*}$, then $E^{-}_{\mu ,\gamma }$ embeds continuously into $H^{2}_{loc}(\mathbb {R}^{N})\times H^{2}_{loc}(\mathbb {R}^{N})$ and $L^{t}(\mathbb {R}^{N})\times L^{t}(\mathbb {R}^{N})$ for $\gamma \le t\le 2^{*}$ and compactly embeds into $L^{t}_{loc}(\mathbb {R}^{N})\times L^{t}_{loc}(\mathbb {R}^{N})$ for $2\le t\le 2^{*}$.

Note that we have the following continuous embeddings

$$\begin{aligned} H^{1}\left( \mathbb {R}^{N}\right) \times H^{1}\left( \mathbb {R}^{N}\right) \hookrightarrow E_{\mu ,\gamma }=E^{+}\oplus E^{-}_{\mu ,\gamma }\hookrightarrow E, \end{aligned}$$

where $E_{\mu ,\gamma }$ is endowed with the norm

$$\begin{aligned} \Vert (u,v)\Vert =\bigg (\Vert (u^{+},v^{+})\Vert ^{2}+\Vert |(u^{-},v^{-})|\Vert _{\mu ,\gamma }^{2}\bigg )^{\frac{1}{2}} \end{aligned}$$

for $(u,v)=(u^{+},v^{+})+(u^{-},v^{-})\in E^{+}\oplus E^{-}_{\mu ,\gamma }$. Since $\Vert \cdot \Vert _{\mu ,\gamma }$ is uniformly convex (see Proposition 2.6 in [26]), then $E_{\mu ,\gamma }$ is reflective and bounded sequences in $E_{\mu ,\gamma }$ are relatively weakly compact. Then, it follows from the Sobolev embedding that Lemma 2.2 holds and $J:E_{\mu ,\gamma }\rightarrow \mathbb {R}$ given by (1.5) is a well defined $C^{1}$-map. Furthermore, it follows from Lemma 2.1 and Corollary 2.3 in [24] that a solution to problem (1.1) in $E_{\mu ,\gamma }$ vanishes at infinity. Analogously, we have the following corollary.

Corollary 2.3

If $(u,v)\in E_{\mu ,\gamma }$ solves problem (1.1), then $(u(x),v(x))\rightarrow (0,0)$ as $|x|\rightarrow \infty$.

In order to ensure that a unit sphere in $E^{+}$

$$\begin{aligned} S^{+}=\big \{(u,v)\in E^{+}\big |\Vert (u,v)\Vert =1\big \} \end{aligned}$$

is a $C^{1}$-submanifold of $E^{+}$, we suppose that $E^{+}$ is a Hilbert space with the scalar product $\langle \cdot ,\cdot \rangle$ such that $\langle (u,v),(u,v)\rangle =\Vert (u,v)\Vert ^{2}$ for any $(u,v)\in E^{+}$. In addition to the norm topology, we need the topology $\mathcal {T}$ on E, which is the product of the norm topology in $E^{+}$ and the weak topology in $E^{-}$. In particular, $(u_{n},v_{n})\overset{\mathcal {T}}{\rightarrow } (u,v)$ provided that $(u^{+}_{n},v^{+}_{n})\rightarrow (u^{+},v^{+})$ and $(u^{-}_{n},v^{-}_{n})\rightharpoonup (u^{-},v^{-})$.

We define Nehari–Pankov manifold [22] as follows:

$$\begin{aligned} \mathcal {N}=\big \{(u,v)\in E{\setminus }\{(0,0)\}:\langle J'(u,v),(u,v)\rangle =0,~\langle J'(u,v),(\omega ,s)\rangle =0,~\forall ~(\omega ,s)\in E^{-}\big \}. \end{aligned}$$

We say that J satisfies the $(PS)_{c}^{\mathcal {T}}$-condition in $\mathcal {N}$ if every $(PS)_{c}$ sequence in $\mathcal {N}$ has a subsequence which converges in $\mathcal {T}$:

$$\begin{aligned} (u_{n},v_{n})\in \mathcal {N},~J'(u_{n},v_{n})\rightarrow 0,~J(u_{n},v_{n})\rightarrow c\Rightarrow (u_{n},v_{n})\overset{\mathcal {T}}{\rightarrow }(u,v)\in E. \end{aligned}$$

Lemma 2.4

([22, 27]) Let $J\in C^{1}(E,\mathbb {R})$ be a map of the form

$$\begin{aligned} J(u,v)=\frac{1}{2}\Vert (u^{+},v^{+})\Vert ^{2}-I(u,v) \end{aligned}$$

for any $(u,v)=(u^{+},v^{+})+(u^{-},v^{-}) \in E^{+}\oplus E^{-}$ such that

• $(J_{1})$$I(u,v)\ge I(0,0)$ for any $(u,v)\in E$ and I is $\mathcal {T}$-sequentially lower semicontinuous, i.e., if $(u_{n},v_{n})\overset{\mathcal {T}}{\rightarrow }(u_{0},v_{0})$, then $\liminf \limits _{n\rightarrow \infty }I(u_{n},v_{n})\ge I(u_{0},v_{0})$.

• $(J_{2})$ If $(u_{n},v_{n})\overset{\mathcal {T}}{\rightarrow }(u_{0},v_{0})$ and $I(u_{n},v_{n})\rightarrow I(u_{0},v_{0})$, then $(u_{n},v_{n})\rightarrow (u_{0},v_{0})$.

• $(J_{3})$ If $(u,v)\in \mathcal {N}$, then $J(u,v)> J(tu+\omega ,tv+s)$ for $t\ge 0$, $(\omega ,s)\in E^{-}$ such that $(tu+\omega ,tv+s)\ne (u,v)$.

• $(J_{4})$$0<\inf \limits _{(u,v)\in E^{+},\Vert (u,v)\Vert =r}J(u,v)$.

• $(J_{5})$$\Vert (u^{+},v^{+})\Vert + I(u,v)\rightarrow \infty$ as $\Vert (u,v)\Vert \rightarrow \infty$.

• $(J_{6})$$I(t_{n}u_{n},t_{n}v_{n})/t^{2}_{n}\rightarrow \infty$ if $t_{n}\rightarrow \infty$ and $(u^{+}_{n},v_{n}^{+})\rightarrow (u^{+}_{0},v_{0}^{+})$ for some $(u^{+}_{0},v_{0}^{+})\ne (0,0)$,  $n\rightarrow \infty$.

Then,

(a):

$c=\inf \limits _{\mathcal {\mathcal {N}}}J>0$ and there exists a $(PS)_{c}$-sequence $(u_{n},v_{n})_{n\in \mathbb {N}}$, i.e., $J(u_{n},v_{n})\rightarrow c$ and $J'(u_{n},v_{n})\rightarrow 0,~n\rightarrow \infty$. If J satisfies the $(PS)_{c}^{\mathcal {T}}$-condition in $\mathcal {N}$, then c is achieved by a critical point of J.

(b):

There is a homeomorphism $m: S^{+}\rightarrow \mathcal {N}$ such that $m^{-1}(u,v)=(\frac{u^{+}}{\Vert (u^{+},v^{+})\Vert },\frac{v^{+}}{\Vert (u^{+},v^{+})\Vert })$, m(u, v) is the unique maximum point of J on $(\mathbb {R}^{+}u,\mathbb {R}^{+}v)\oplus E^{-}$ for $(u,v)\in E$, and $J\circ m: S^{+}\rightarrow \mathbb {R}$ is of class $C^{1}$. Furthermore, a sequence $(u_{n},v_{n})\subset S^{+}$ is $J \circ m$, if and only if $m(u_{n}, v_{n})$ is a Palais-Smale sequence for J, and $(u,v)\in S^{+}$ is a critical point of $J\circ m$ if and only if m(u, v) is a critical point of J.

3 Ground-State Solutions

In order to look for critical points of $J:E_{2,\gamma }\rightarrow \mathbb {R}$, we define the following Nehari–Pankov manifold:

$$\begin{aligned}&\mathcal {N}=\left\{ (u,v)\in E_{2,\gamma }{\setminus } E^{-}_{2,\gamma }:\langle J'(u,v),(u,v)\rangle \right. \nonumber \\&\quad \quad \left. =0,\langle J'(u,v),(\omega ,s)\rangle =0,~\forall ~(\omega ,s)\in E^{-}_{2,\gamma }\right\} . \end{aligned}$$

(3.1)

Lemma 3.1

There exists a constant $a>0$ such that for any $(u,v)\in E_{2,\gamma }$

$$\begin{aligned} \int _{\mathbb {R}^{N}}F(x,u,v)\mathrm {d}x\ge a \min \left\{ \Vert (u,v)\Vert _{2,\gamma }^{2},\Vert (u,v)\Vert _{2,\gamma }^{\gamma }\right\} . \end{aligned}$$

(3.2)

Proof

It follows from $(F_{3})$ and $(F_{4})$ that there exists $b>0$ such that

$$\begin{aligned} F(x,u,v)\ge b \min \left\{ |(u,v)|^{2}, |(u,v)|^{\gamma }\right\} \end{aligned}$$

(3.3)

for all $(u,v)\in \mathbb {R}^{2}$ and $x\in \mathbb {R}^{N}$. Therefore,

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^{N}}F(x,u,v)\mathrm {d}x&\ge b \left( \int _{\Omega _{(u,v)}}|(u,v)|^{2}\mathrm {d}x+ \int _{\Omega _{(u,v)}^{c}}|(u,v)|^{\gamma }\mathrm {d}x\right) \\&=b \left( \Vert (u,v)\chi _{\Omega _{(u,v)}}\Vert _{2}^{2}+\Vert (u,v)\chi _{\Omega _{(u,v)}^{c}}\Vert _{\gamma }^{\gamma }\right) \\&\ge a \min \left\{ \Vert (u,v)\Vert ^{2}_{2,\gamma ,\infty },\Vert (u,v)\Vert ^{\gamma }_{2,\gamma ,\infty } \right\} , \end{aligned} \end{aligned}$$

for some constant $a>0$, where $\chi$ denotes the characteristic function, $\Omega _{(u,v)}=\{x\in \mathbb {R}^{N}\big \Vert (u,v)\Vert >1\}$, and

$$\begin{aligned} \begin{aligned} \Vert (u,v)\Vert _{2,\gamma ,\infty }&=\inf \left\{ \max \left\{ \Vert (u_{1},v_{1})\Vert _{2}, \Vert (u_{2},v_{2})\Vert _{\gamma }\right\} \big | (u,v)=(u_{1},v_{1})+(u_{2},v_{2}),\right. \\&\left. \quad (u_{1},v_{1})\in L^{2}\left( \mathbb {R}^{N}\right) \times L^{2}\left( \mathbb {R}^{N}\right) , ~(u_{2},v_{2})\in L^{\gamma }\left( \mathbb {R}^{N}\right) \times L^{\gamma }\left( \mathbb {R}^{N}\right) \right\} \end{aligned} \end{aligned}$$

defines a norm on $L^{2,\gamma }(\mathbb {R}^{N}) \times L^{2,\gamma }(\mathbb {R}^{N})$, which is equivalent to $\Vert .\Vert _{2,\gamma }$ (see Proposition 2.4 in [26]). The proof is complete. $\square$

The following lemma shows that $(J_{4})-(J_{6})$ hold for J.

Lemma 3.2

The following conditions hold:

(a):

$\inf \limits _{(u,v)\in E^{+},\Vert (u,v)\Vert =r}J(u,v)>0.$

(b):

$\Vert (u^{+},v^{+})\Vert +I(u,v)\rightarrow \infty ~\text{ as }~\Vert (u,v)\Vert \rightarrow \infty .$

(c):

$\frac{I(t_{n}u_{n},t_{n}v_{n})}{t_{n}^{2}}\rightarrow \infty$ if $(u^{+}_{n},v^{+}_{n})\rightarrow (u^{+}_{0},v^{+}_{0})\ne (0,0)$ and $t_{n}\rightarrow \infty ,~as~n\rightarrow \infty$.

Proof

(a) If $(u,v)\in E^{+}$, then it follows from $(F_{2})$ that

$$\begin{aligned} J(u,v)\ge \frac{1}{2}\Vert (u,v)\Vert ^{2}-\frac{c}{\gamma }\Vert (u,v)\Vert _{\gamma }^{\gamma }-\frac{c}{p}\Vert (u,v)\Vert _{p}^{p}. \end{aligned}$$

Since the embeddings from $E^{+}$ into $L^{\gamma }(\mathbb {R}^{N})\times L^{\gamma }(\mathbb {R}^{N})$ and $L^{p}(\mathbb {R}^{N})\times L^{p}(\mathbb {R}^{N})$ are continuous, then we have

$$\begin{aligned} J(u,v)\ge \frac{1}{2}\Vert (u,v)\Vert ^{2}-C\big (\Vert (u,v)\Vert ^{\gamma }+\Vert (u,v)\Vert ^{p}\big ). \end{aligned}$$

Therefore, the inequality in (a) holds.

(b) Suppose that $\Vert (u_{n},v_{n})\Vert \rightarrow \infty ,~n\rightarrow \infty$, and $(\Vert (u^{+}_{n},v^{+}_{n})\Vert )_{n\in \mathbb {N}}$ is bounded. Then, $(\Vert (u^{+}_{n},v^{+}_{n})\Vert _{2,\gamma })_{n\in \mathbb {N}}$ is bounded and

$$\begin{aligned} \Vert (u^{-}_{n},v^{-}_{n})\Vert ^{2}_{2,\gamma }=\Vert (u^{-}_{n},v^{-}_{n})\Vert ^{2} +\Vert (u^{-}_{n},v^{-}_{n})\Vert ^{2}_{2,\gamma }\rightarrow \infty , ~n\rightarrow \infty . \end{aligned}$$

If $\Vert (u^{-}_{n},v^{-}_{n})\Vert \rightarrow \infty$, then $I(u_{n},v_{n})\rightarrow \infty$. Suppose that $(\Vert (u^{-}_{n},v^{-}_{n})\Vert )_{n\in \mathbb {N}}\rightarrow \infty ~\text{ as } n\rightarrow \infty$, then $\Vert (u^{-}_{n},v^{-}_{n})\Vert _{2,\gamma }\rightarrow \infty$ and by (3.2), we have $I(u_{n},v_{n})\rightarrow \infty ,~n\rightarrow \infty$.

(c) Suppose that, up to a subsequence, $I(t_{n}u_{n},t_{n}v_{n})/t^{2}_{n}$ is bounded, $(u^{+}_{n},v^{+}_{n})\rightarrow (u^{+}_{0},v^{+}_{0})\ne (0,0)$, and $t_{n}\rightarrow \infty$, then by (3.2), we have

$$\begin{aligned} \frac{I(t_{n}u_{n},t_{n}v_{n})}{t^{2}_{n}}\ge \frac{1}{2}\Vert (u^{-}_{n},v^{-}_{n})\Vert ^{2}+ c \min \left\{ \Vert (u_{n},v_{n})\Vert ^{2}_{2,\gamma },t_{n}^{\gamma -2}\Vert (u_{n},v_{n})\Vert ^{\gamma }_{2,\gamma }\right\} \end{aligned}$$

and then $(||(u^{-}_{n},v^{-}_{n})||_{2,\gamma })_{n\in \mathbb {N}}$ is bounded. It follows from Lemma 2.2 that $(u^{-}_{n},v^{-}_{n})\rightharpoonup (u^{-}_{0},v^{-}_{0})$ in $E^{-}_{2,\gamma }$ and $(u^{-}_{n}(x),v^{-}_{n}(x))\rightarrow (u^{-}_{0}(x),v^{-}_{0}(x))$ a.e. on $\mathbb {R}^{N}$. If the Lebesgue measure $|\Omega |>0$, where $\Omega =\big \{x\in \mathbb {R}^{N} |(u^{+}_{0}(x)+u^{-}_{0}(x),v^{+}_{0}(x)+v^{-}_{0}(x))\ne (0,0)\big \}$, then it follows from $(F_{4})$ and Fatou’s lemma that

$$\begin{aligned} \int _{\mathbb {R}^{N}}\frac{F(x,t_{n}u_{n},t_{n}v_{n})}{t_{n}^{2}}\mathrm {d}x\rightarrow \infty ,~n\rightarrow \infty . \end{aligned}$$

Therefore, we obtain $I(t_{n}u_{n},t_{n}v_{n})/t_{n}^{2}\rightarrow \infty$ as $n\rightarrow \infty$, which is a contradiction. Hence, $|\Omega |=0$ and $(u^{-}_{0},v^{-}_{0})=-(u^{+}_{0},v^{+}_{0})$. Since $\langle (u^{-}_{0},v^{-}_{0}),(u^{+}_{0},v^{+}_{0})\rangle =0$, then $(u^{+}_{0},v^{+}_{0})=(0,0)$. This is a contradiction; then, $I(t_{n}u_{n},t_{n}v_{n})/t_{n}^{2}\rightarrow \infty$, as $n\rightarrow \infty$.

Lemma 3.3

The following conditions hold:

(a):

$I(u,v)\ge 0$ for any $(u,v)\in E_{2,\gamma }$ and I is $\mathcal {T}$-sequentially lower semicontinuous.

(b):

If $(u_{n},v_{n})\overset{\mathcal {T}}{\rightarrow }(u_{0},v_{0})$ and $I(u_{n},v_{n})\rightarrow I(u_{0},v_{0})$, then $(u_{n},v_{n})\rightarrow (u_{0},v_{0})$.

(c):

If $(u,v)\in \mathcal {N}$, then $J(u,v)> J(tu+\omega ,tv+s)$ for $t\ge 0$, $(\omega ,s)\in E^{-}$ such that $(tu+\omega ,tv+s)\ne (u,v)$.

Proof

(a) Let $(u_{n},v_{n})\overset{\mathcal {T}}{\rightarrow }(u_{0},v_{0})$, since $E_{2,\gamma }$ is compactly embedded in $L^{2}_{loc}(\mathbb {R}^{N})\times L^{2}_{loc}(\mathbb {R}^{N})$ and $(u_{n}(x),v_{n}(x))\rightarrow (u_{0}(x),v_{0}(x))$ a.e. in $\mathbb {R}^{N}$. Then, it follows from Fatou’s lemma and the weakly sequentially lower semicontinuity of the map $E^{-}\ni (u^{-},v^{-})\rightarrow \frac{1}{2}\Vert (u^{-},v^{-})\Vert ^{2}$ that $\liminf \limits _{n\rightarrow \infty } I(u_{n},v_{n})\ge I(u_{0},v_{0}).$

(b) Let $(u_{n},v_{n})\overset{\mathcal {T}}{\rightarrow }(u_{0},v_{0})$ and $I(u_{n},v_{n})\rightarrow I(u_{0},v_{0})$. Since $E_{2,\gamma }\ni (u,v)\rightarrow \int _{\mathbb {R}^{N}} F(x,u,v)\mathrm {d}x$ is $\mathcal {T}$-sequentially lower semicontinuous, then

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\Vert \left( u^{-}_{n},v^{-}_{n}\right) \Vert ^{2}=\Vert \left( u^{-}_{0},v^{-}_{0}\right) \Vert ^{2} \end{aligned}$$

and

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }\int _{\mathbb {R}^{N}} F(x,u_{n},v_{n})\mathrm {d}x=\int _{\mathbb {R}^{N}} F(x,u_{0},v_{0})\mathrm {d}x. \end{aligned}$$

(3.4)

Observe that

$$\begin{aligned} \begin{aligned} \Vert (u^{-}_{n}-u^{-}_{0},v^{-}_{n}-v^{-}_{0})\Vert ^{2}&=\Vert (u^{-}_{n},v^{-}_{n})\Vert ^{2}-\Vert (u^{-}_{0},v^{-}_{0})\Vert ^{2}\\&\quad -2\langle (u^{-}_{n}-u^{-}_{0},v^{-}_{n}-v^{-}_{0}),(u^{-}_{0},v^{-}_{0})\rangle \\&\rightarrow 0,~n\rightarrow \infty .\\ \end{aligned} \end{aligned}$$

Therefore, $(u_{n},v_{n})=(u^{+}_{n}+u^{-}_{n},v^{+}_{n}+v^{-}_{n})\rightarrow (u_{0},v_{0})=(u^{+}_{0}+u^{-}_{0},v^{+}_{0}+v^{-}_{0})$. Since we need to prove that $(u^{-}_{n},v^{-}_{n})\rightarrow (u^{-}_{0},v^{-}_{0})$ a.e. in $\mathbb {R}^{N}$, let us consider the function $L:\mathbb {R}^{N}\times [0,1]\rightarrow \mathbb {R}$ given by $L(x,t)=F(x,u_{n}-tu_{0},v_{n}-tv_{0})$. Then,

$$\begin{aligned} F(x,u_{n}-u_{0},v_{n}-v_{0})-F(x,u_{n},v_{n})=L(x,1)-L(x,0)=\int _{0}^{1}\frac{\partial L}{\partial s}(x,s)\mathrm {d}s. \end{aligned}$$

Furthermore,

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^{N}}&\bigg [F(x,u_{n}-u_{0},v_{n}-v_{0})+F(x,u_{0},v_{0})-F(x,u_{n},v_{n})\bigg ]\mathrm {d}x\\&=\int _{\mathbb {R}^{N}}\bigg [\int _{0}^{1}\frac{\partial L}{\partial s}(x,s)\mathrm {d}s+F(x,u_{0},v_{0})\bigg ]\mathrm {d}x\\&=\int _{\mathbb {R}^{N}}\int _{0}^{1}\frac{\partial L}{\partial s}(x,s)\mathrm {d}s\mathrm {d}x+ \int _{\mathbb {R}^{N}}F(x,u_{0},v_{0})\mathrm {d}x\\&=\int _{0}^{1}\int _{\mathbb {R}^{N}}\bigg [-F_{u}(x,u_{n}-su_{0},v_{n}-sv_{0})u_{0}\\&\quad - F_{v}(x,u_{n}-su_{0},v_{n}-sv_{0})v_{0}\bigg ]\mathrm {d}x\mathrm {d}s+ \int _{\mathbb {R}^{N}}F(x,u_{0},v_{0})\mathrm {d}x. \end{aligned} \end{aligned}$$

(3.5)

Let $E\subset \mathbb {R}^{N}$ be a measurable set, then it follows from the Hölder inequality that

$$\begin{aligned} \begin{aligned} \int _{E}&|F_{u}(x,u_{n}-su_{0},v_{n}-sv_{0})u_{0}|\mathrm {d}x\\&\le \varepsilon \int _{E}|(u_{n}-su_{0},v_{n}-sv_{0})||u_{0}|\mathrm {d}x\\&\quad +C(\varepsilon ) \int _{E}|(u_{n}-su_{0},v_{n}-sv_{0})|^{p-1}|u_{0}|\mathrm {d}x\\&\quad +C \int _{E}|(u_{n}-su_{0},v_{n}-sv_{0})|^{\gamma -1}|u_{0}|\mathrm {d}x\\&\le \varepsilon \Vert (u_{n}-su_{0},v_{n}-sv_{0})\chi _{E}\Vert _{2}\Vert u_{0}\chi _{E}\Vert _{2}\\&\quad +C(\varepsilon ) \Vert (u_{n}-su_{0},v_{n}-sv_{0})\chi _{E}\Vert _{p}^{p-1}\Vert u_{0}\chi _{E}\Vert _{p}\\ {}&\quad +C \Vert (u_{n}-su_{0},v_{n}-sv_{0})\chi _{E}\Vert _{\gamma }^{\gamma -1}\Vert u_{0}\chi _{E}\Vert _{\gamma }. \end{aligned} \end{aligned}$$

Therefore, $F_{u}(x,u_{n}-su_{0},v_{n}-sv_{0})u_{0}$ is uniformly integrable and by the Vitali convergence theorem, we derive

$$\begin{aligned}&\int _{0}^{1}\int _{\mathbb {R}^{N}}-F_{u}(x,u_{n}-su_{0},v_{n}-sv_{0})u_{0}\mathrm {d}x\mathrm {d}s\rightarrow \\&\int _{0}^{1}\int _{\mathbb {R}^{N}}-F_{u}(x,u_{0}-su_{0},v_{0}-sv_{0})u_{0}\mathrm {d}x\mathrm {d}s \end{aligned}$$

as $n\rightarrow \infty$. Analogously,

$$\begin{aligned}&\int _{0}^{1}\int _{\mathbb {R}^{N}}-F_{v}(x,u_{n}-su_{0},v_{n}-sv_{0})v_{0}\mathrm {d}x\mathrm {d}s\rightarrow \\&\int _{0}^{1}\int _{\mathbb {R}^{N}}-F_{v}(x,u_{0}-su_{0},v_{0}-sv_{0})v_{0}\mathrm {d}x\mathrm {d}s \end{aligned}$$

as $n\rightarrow \infty$. Furthermore,

$$\begin{aligned} \begin{aligned}&\int _{0}^{1}\int _{\mathbb {R}^{N}}\bigg [-F_{u}(x,u_{n}-su_{0},v_{n}-sv_{0})u_{0}- F_{v}(x,u_{n}-su_{0},v_{n}-sv_{0})v_{0}\bigg ] \mathrm {d}x\mathrm {d}s\\&\quad =\int _{\mathbb {R}^{N}}\int _{0}^{1}\bigg [- F_{u}(x,u_{n}-su_{0},v_{n}-sv_{0})u_{0}- F_{v}(x,u_{n}-su_{0},v_{n}-sv_{0})v_{0}\bigg ]\mathrm {d}s\mathrm {d}x\\&\quad =\int _{\mathbb {R}^{N}}\int _{0}^{1}- \frac{\partial }{\partial s}\bigg [F(x,u_{n}-su_{0},v_{n}-sv_{0})\bigg ]\mathrm {d}s\mathrm {d}x\\&\quad =\int _{\mathbb {R}^{N}}\big (F(x,0,0)-F(x,u_{0},v_{0})\big )\mathrm {d}x\\&\quad =\int _{\mathbb {R}^{N}}-F(x,u_{0},v_{0})\mathrm {d}x. \end{aligned} \end{aligned}$$

Then, from (3.4) and (3.5), we derive

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }F(x,u_{n}-u_{0},v_{n}-v_{0})\mathrm {d}x=0. \end{aligned}$$

It follows from (3.2) that $(u_{n},v_{n})\rightarrow (u_{0},v_{0})$ in $L^{2,\gamma }(\mathbb {R}^{N})\times L^{2,\gamma }(\mathbb {R}^{N})$.

(c) Let $(u,v)\in \mathcal {N}$. Observe that for any $t\ge 0$ and $(\omega ,s)\in E^{-}_{2,\gamma }$

$$\begin{aligned} \begin{aligned} J(tu+\omega ,tv+s)-J(u,v)&=-\frac{1}{2}\Vert (\omega ,s)\Vert ^{2}-\int _{\mathbb {R}^{N}}\varphi (t,x)\mathrm {d}x, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} \varphi (t,x)&=F_{u}(x,u,v)\left( \frac{t^{2}-1}{2}u+t\omega \right) +F_{v}(x,u,v)\left( \frac{t^{2}-1}{2}v+ts\right) \\&\quad +F(x,u,v)-F(x,tu+\omega ,tv+s). \end{aligned} \end{aligned}$$

Suppose that $(u(x),v(x))\ne (0,0)$. Similarly as in [20, 21], we show that $\varphi (t,x)\le 0$. In fact, it follows from $(F_{6})$ that $\varphi (0,x)\le 0$. By $(F_{4})$, we obtain $\varphi (t,x)\rightarrow -\infty$ as $t\rightarrow \infty$. Let $t_{0}>0$ be such that

$$\begin{aligned} \varphi (t_{0},x)=\max \limits _{t\ge 0}\varphi (t,x). \end{aligned}$$

We may assume that $t_{0}>0$ and $\partial _{t}\varphi (t_{0},x)=0$; therefore,

$$\begin{aligned} \begin{aligned} F_{u}&(x,u,v)\big (t_{0}u+\omega \big )+F_{v}(x,u,v)\big (t_{0}v+s\big )\\&=F_{u}(x,t_{0}u+\omega ,t_{0}v+s)u+F_{v}(x,t_{0}u+\omega ,t_{0}v+s)v. \end{aligned} \end{aligned}$$

If $F_{u}(x,u,v)(t_{0}u+\omega )\le 0$, $F_{v}(x,u,v)(t_{0}v+s)\le 0$, then by $(F_{6})$, we have

$$\begin{aligned} \begin{aligned} \varphi (t_{0},x)&\le \frac{-t_{0}^{2}-1}{2}F_{u}(x,u,v)u+\frac{-t_{0}^{2}-1}{2}F_{v}(x,u,v)v +F(x,u,v)\\&\quad -F(x,t_{0}u+\omega ,t_{0}v+s)\le 0. \end{aligned} \end{aligned}$$

Otherwise, $(F_{5})$ implies that

$$\begin{aligned} \varphi (t_{0},x)&\le F_{u}(x,u,v)\left( \frac{t_{0}^{2}-1}{2}u+t_{0}\omega \right) +F_{v}(x,u,v)\left( \frac{t_{0}^{2}-1}{2}v+t_{0}s\right) \\&\quad +\frac{\big (F_{u}(x,u,v)u\big )^{2}-\big (F_{u}(x,u,v)(t_{0}u+\omega )\big )^{2}}{2F_{u}(x,u,v)u}\\&\quad +\frac{\big (F_{v}(x,u,v)v\big )^{2}-\big (F_{v}(x,u,v)(t_{0}v+s)\big )^{2}}{2F_{v}(x,u,v)v}\\&=-\frac{\big (F_{u}(x,u,v)\omega \big )^{2}}{2F_{u}(x,u,v)u}-\frac{\big (F_{v}(x,u,v)s\big )^{2}}{2F_{v}(x,u,v)v}\\&\le 0. \end{aligned}$$

The proof is complete. $\square$

Lemma 3.4

J is coercive on $\mathcal {N}$, i.e., $J(u,v)\rightarrow \infty$ as $\Vert (u,v)\Vert \rightarrow \infty$, $(u,v)\in \mathcal {N}$.

Proof

Suppose that $\Vert (u_{n},v_{n})\Vert \rightarrow \infty$ as $n\rightarrow \infty$, $(u_{n},v_{n})\in \mathcal {N}$ and $J(u_{n},v_{n})\le c_{1}$ for some constant $c_{1}>0$. Let $(\omega _{n},s_{n})=\frac{(u_{n},v_{n})}{\Vert (u_{n},v_{n})\Vert }$, since $E_{2,\gamma }$ is reflective and compactly embedded in $L^{2}_{loc}(\mathbb {R}^{N})\times L^{2}_{loc}(\mathbb {R}^{N})$. Then, there exists $(\omega ,s)\in E_{2,\gamma }$ such that $(\omega _{n},s_{n})\rightharpoonup (\omega ,s)$ and $(\omega _{n}(x),s_{n}(x))\rightarrow (\omega (x),s(x))$ a.e. on $\mathbb {R}^{N}$. Furthermore, there exists a sequence $(y_{n})_{n\in \mathbb {N}}\subset \mathbb {R}^{N}$ such that

$$\begin{aligned} \liminf \limits _{n\rightarrow \infty }\int _{B(y_{n},1)}\left| \left( \omega _{n}^{+},s_{n}^{+}\right) \right| ^{2}\mathrm {d}x>0. \end{aligned}$$

(3.6)

Otherwise, it follows from Lions lemma (see Lemma 1.21 in [28]) that $(\omega _{n}^{+},s_{n}^{+})\rightarrow (0,0)$ in $L^{t}(\mathbb {R}^{N})\times L^{t}(\mathbb {R}^{N})$ for $t\in (2,2^{*})$. Then, $\int _{\mathbb {R}^{N}}F(x,t\omega _{n}^{+},ts_{n}^{+})\mathrm {d}x\rightarrow 0$ for any $t>0$. For any $t>0$, in view of Lemma 3.3 (c), we obtain

$$\begin{aligned} \begin{aligned} c_{1}&\ge \limsup \limits _{n\rightarrow \infty }J(u_{n},v_{n})\ge \limsup \limits _{n\rightarrow \infty }J\left( t\omega _{n}^{+},ts_{n}^{+}\right) =\frac{t^{2}}{2}\left\| (\omega _{n}^{+},s_{n}^{+})\right\| _{\mu _{n}}^{2}. \end{aligned} \end{aligned}$$

(3.7)

It follows from Lemma 2.4 (a) and (3.2) that

$$\begin{aligned} \begin{aligned} \frac{1}{2}&\left( \left\| (u^{+}_{n},v^{+}_{n}\right) \right\| ^{2}-\left\| (u^{-}_{n},v^{-}_{n})\right\| ^{2}) -c\min \left\{ \Vert (u_{n},v_{n})\Vert ^{2}_{2,\gamma },\Vert (u_{n},v_{n})\Vert ^{\gamma }_{2,\gamma }\right\} \\&\ge J(u_{n},v_{n})\ge c_{inf}=\inf \limits _{N}J>0. \end{aligned} \end{aligned}$$

If $\liminf \limits _{n\rightarrow \infty }\Vert (u_{n},v_{n})\Vert _{2,\gamma }=0$, then up to a subsequence, $\Vert (u_{n},v_{n})\Vert _{2,\gamma }\rightarrow 0,~n\rightarrow \infty$, and for sufficiently large n

$$\begin{aligned} \begin{aligned} 2\Vert (u^{+}_{n},v^{+}_{n})\Vert ^{2}&\ge \Vert (u^{+}_{n},v^{+}_{n})\Vert ^{2}+\Vert (u^{-}_{n},v^{-}_{n})\Vert ^{2}+ 2c_{inf}\\&\quad +2c\min \left\{ \Vert (u_{n},v_{n})\Vert ^{2}_{2,\gamma },\Vert (u_{n},v_{n})\Vert ^{\gamma }_{2,\gamma }\right\} \\&\ge \Vert (u^{+}_{n},v^{+}_{n})\Vert ^{2}+\Vert (u^{-}_{n},v^{-}_{n})\Vert ^{2}+\Vert (u_{n},v_{n})\Vert ^{\gamma }_{2,\gamma }\\&=\Vert (u_{n},v_{n})\Vert ^{2}. \end{aligned} \end{aligned}$$

If $\liminf \limits _{n\rightarrow \infty }\Vert (u_{n},v_{n})\Vert _{2,\gamma }>0$, then there exists $c_{2}\in (0,1)$ such that for sufficiently large n

$$\begin{aligned} \begin{aligned} 2\Vert (u^{+}_{n},v^{+}_{n})\Vert ^{2}&\ge \Vert (u^{+}_{n},v^{+}_{n})\Vert ^{2}+\Vert (u^{-}_{n},v^{-}_{n})\Vert ^{2}+ 2c_{inf}\\&\quad +2c\min \left\{ \Vert (u_{n},v_{n})\Vert ^{2}_{2,\gamma },\Vert (u_{n},v_{n})\Vert ^{\gamma }_{2,\gamma }\right\} \\&\ge c_{2}\left( \Vert (u^{+}_{n},v^{+}_{n})\Vert ^{2}+\Vert (u^{-}_{n},v^{-}_{n})\Vert ^{2}+\Vert (u_{n},v_{n})\Vert ^{\gamma }_{2,\gamma }\right) \\&=c_{2}\Vert (u_{n},v_{n})\Vert ^{2}. \end{aligned} \end{aligned}$$

Hence, passing to a subsequence if necessary, $c_{3}=\inf \limits _{n\in \mathbb {N}}\Vert (\omega ^{+}_{n},s^{+}_{n})\Vert ^{2}>0$ and by (3.7), we have

$$\begin{aligned} c_{1}\ge \frac{t^{2}}{2}c_{3} \end{aligned}$$

for any $t\ge 0$. Then, we get a contradiction. Therefore, we may assume that $(y_{n})_{n\in \mathbb {N}}\subset \mathbb {R}^{N}$ and

$$\begin{aligned} \liminf \limits _{n\rightarrow \infty }\int _{B(y_{n},1)}|(\omega _{n}^{+},s_{n}^{+})|^{2}\mathrm {d}x>0. \end{aligned}$$

Since J and $\mathcal {N}$ are invariant under translations of the form $(u,v)\mapsto (u(\cdot -k),v(\cdot -k))$, $k\in \mathbb {Z}^{N}$, then we may assume that $(\omega ^{+}_{n},s^{+}_{n})\rightarrow (\omega ^{+},s^{+})$ in $L^{2}_{loc}(\mathbb {R}^{N})\times L^{2}_{loc}(\mathbb {R}^{N})$ and $(\omega ^{+},s^{+})\ne (0,0)$. Note that $|(\omega ^{+},s^{+})|\ne 0$, then $(u_{n}(x),v_{n}(x))=(\omega _{n}(x),s_{n}(x))\Vert (u_{n},v_{n})\Vert \rightarrow \infty$, and by $(F_{4})$, we have

$$\begin{aligned} \frac{F(x,u_{n}(x),v_{n}(x))}{\Vert (u_{n},v_{n})\Vert ^{2}}=\frac{F(x,u_{n}(x),v_{n}(x))}{|(u_{n},v_{n})|^{2}}| (\omega _{n}(x),s_{n}(x))|^{2}\rightarrow \infty ,~n\rightarrow \infty . \end{aligned}$$

Hence, it follows from Fatou’s lemma that

$$\begin{aligned} \begin{aligned} \frac{J_{\mu _{n}}(u_{n},v_{n})}{\Vert (u_{n},v_{n})\Vert ^{2}}&=\limsup \limits _{n\rightarrow \infty }\bigg (\frac{1}{2}\left( \Vert (\omega _{n}^{+},s_{n}^{+})\Vert ^{2} -\Vert (\omega _{n}^{-},s_{n}^{-})\Vert ^{2}\right) -\int _{\mathbb {R}^{N}}\frac{F(x,u_{n} ,v_{n})}{\Vert (u_{n},v_{n})\Vert ^{2}}\mathrm {d}x\bigg )\\&\rightarrow -\infty ,~n\rightarrow \infty , \end{aligned} \end{aligned}$$

which is a contradiction. The proof is complete. $\square$

Proof of Theorem 1.1

It follows from Lemma 2.4 (a) that $c_{inf}=\inf \limits _{\mathcal {N}}J>0$ and there exists a $(PS)_{c_{inf}}$-sequence $(u_{n},v_{n})_{n\in \mathbb {N}}$. By Lemma  3.4. we get that $(u_{n},v_{n})_{n\in \mathbb {N}}$ is bounded and passing to a subsequence $(u_{n},v_{n})\rightharpoonup (u,v)$ in $E_{2,\gamma }$. Then, there exists a subsequence $(y_{n})_{n\in \mathbb {N}}$ such that

$$\begin{aligned} \liminf \limits _{n\rightarrow \infty }\int _{B(y_{n},1)}|(u_{n}^{+},v_{n}^{+})|^{2}\mathrm {d}x>0. \end{aligned}$$

(3.8)

Otherwise, it follows from Lions lemma (see Lemma 1.21 in [28]) that $(u_{n}^{+},v_{n}^{+})\rightarrow (0,0)$ in $L^{t}(\mathbb {R}^{N})\times L^{t}(\mathbb {R}^{N})$ for $t\in (2,2^{*})$. From $(F_{2})$, we have

$$\begin{aligned} \begin{aligned} \Vert (u_{n}^{+},v_{n}^{+})\Vert ^{2}=&\langle J'(u_{n},v_{n}),(u^{+}_{n},v^{+}_{n}) \rangle +\int _{\mathbb {R}^{N}} \big (F_{u}(x,u_{n},v_{n})u^{+}_{n}+F_{v}(x,u_{n},v_{n})v^{+}_{n}\big )\mathrm {d}x\\&\rightarrow 0, \text{ as }~ n\rightarrow \infty . \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned} 0<c_{\inf }=\lim \limits _{n\rightarrow \infty }J(u_{n},v_{n})\le \frac{1}{2}\lim \limits _{n\rightarrow \infty } \Vert (u^{+}_{n},v^{+}_{n})\Vert ^{2}=0, \end{aligned}$$

which is a contradiction. Therefore, (3.8) holds and we may assume that there is a subsequence $(y_{n})_{n\in \mathbb {N}}$ such that

$$\begin{aligned} \liminf \limits _{n\rightarrow \infty }\int _{B(y_{n},r)}|(u_{n}^{+},v_{n}^{+})|^{2}\mathrm {d}x>0, \end{aligned}$$

(3.9)

for some $r>1$. Since $\Vert (u_{n}(\cdot +y_{n}),v_{n}(\cdot +y_{n}))\Vert =\Vert (u_{n},v_{n})\Vert$, then there exists $(u,v)\in E_{2,\gamma }$ such that $(u_{n}(\cdot +y_{n}),v_{n}(\cdot +y_{n}))\rightharpoonup (u,v)$ in $E_{2,\gamma }$ and $(u^{+}_{n}(\cdot +y_{n}),v^{+}_{n}(\cdot +y_{n}))\rightharpoonup (u^{+},v^{+})$ in $L^{2}_{loc}(\mathbb {R}^{N})\times L^{2}_{loc}(\mathbb {R}^{N})$. It follows from (3.9) that $(u^{+},v^{+})\ne (0,0)$ and then $(u,v)\ne (0,0)$. Since J and $\mathcal {N}$ are invariant under translations of the form $(u,v)\mapsto (u(\cdot +y),v(\cdot +y))$ for $y\in \mathbb {Z}^{N}$, then $J'(u,v)=0$. Hence, $(u,v)\in \mathcal {N}$ and $J(u,v)=c_{\inf }$. Since $(u,v)\in E_{2,\gamma }$ is a solution to problem (1.1), then by Corollary 2.3, we get $(u(x),v(x))\rightarrow (0,0)$ as $|x|\rightarrow \infty$. The proof is complete.

$\square$

4 Multiple Solutions

Observe that if $(u,v)\in E_{2,\gamma }$ is a critical point of J, then the orbit under the action of $\mathbb {Z}^{N}$, $O(u,v)=\big \{(u(\cdot -k),v(\cdot -k))|k\in \mathbb {Z}^{N}\big \}$ consists of critical points. Two critical points $(u_{1},v_{1}),~(u_{2},v_{2})\in E_{2,\gamma }$ are said to be geometrically distinct if $O(u_{1},v_{1})\cap O(u_{2},v_{2})=\emptyset$. It follows from Lemma 2.4(b) that $\psi = J\circ m: S^{+}\rightarrow \mathbb {R}$ is a $C^{1}$ map. In order to prove Theorem 1.2, we just need to prove that $\psi$ has infinitely many geometrically distinct critical points (Lemma 2.4(b)). The following lemma is crucial in the consideration of the multiplicity of critical points (see Lemma 2.14 in [18]).

Lemma 4.1

Let $d\ge c_{\inf }$, if $(u^{1}_{n},v^{1}_{n}),~(u^{2}_{n},v^{2}_{n})\subset \psi ^{d}=\{(u,v)\in S^{+}|\psi (u,v)\le d\}$ are two Palais–Smale sequences for $\psi$, then either $\Vert (u^{1}_{n}-u^{2}_{n},v^{1}_{n}-v^{2}_{n})\Vert \rightarrow 0,~n\rightarrow \infty$, or

$$\begin{aligned} \begin{aligned} \limsup \limits _{n\rightarrow \infty }\Vert (u^{1}_{n}-u^{2}_{n},v^{1}_{n}-v^{2}_{n})\Vert ^{2}&\ge \rho (d) \inf \bigg \{\Vert (u_{1}-u_{2},v_{1}-v_{2})\Vert \big |\psi '(u^{1},v^{1})\\&=\psi '(u^{2},v^{2})=0, ~(u^{1},v^{1})\ne (u^{2},v^{2})\in S^{+}\bigg \}, \end{aligned}\nonumber \\ \end{aligned}$$

(4.1)

where $\rho (d)$ depends on d but not on the particular choice of Palais–Smale sequences.

Proof

Let $(u^{1}_{n},v^{1}_{n}),~(u^{2}_{n},v^{2}_{n})\subset \psi ^{d}=\big \{(u,v)\in S^{+}|\psi (u,v)\le d\big \}$ be two Palais–Smale sequences for $\psi$. Let $(\omega ^{i}_{n},s^{i}_{n})=(m(u^{i}_{n},v^{i}_{n})),~i=1,2$, and we consider two cases:

Case 1:$\Vert ((\omega ^{1}_{n})^{+}-(\omega ^{2}_{n})^{+},(s^{1}_{n})^{+}-(s^{2}_{n})^{+})\Vert _{\gamma }\rightarrow 0$, and by $(F_{2})$, we have

$$\begin{aligned}&\left\| \left( \left( \omega ^{1}_{n}\right) ^{+}-\left( \omega ^{2}_{n}\right) ^{+},\left( s^{1}_{n}\right) ^{+}-\left( s^{2}_{n}\right) ^{+}\right) \right\| ^{2}\\&= \left\langle J'\left( \omega ^{1}_{n},s^{1}_{n}\right) , \left( \left( \omega ^{1}_{n}\right) ^{+}-\left( \omega ^{2}_{n}\right) ^{+},\left( s^{1}_{n}\right) ^{+}-\left( s^{2}_{n}\right) ^{+}\right) \right\rangle \\&\quad -\left\langle J'\left( \omega ^{2}_{n},s^{2}_{n}\right) , \left( \left( \omega ^{1}_{n}\right) ^{+}-\left( \omega ^{2}_{n}\right) ^{+},\left( s^{1}_{n}\right) ^{+}-\left( s^{2}_{n}\right) ^{+}\right) \right\rangle \\&\quad +\int _{\mathbb {R}^{N}}\left( F_{u}\left( x,\omega ^{1}_{n},s^{1}_{n}\right) -F_{u}\left( x,\omega ^{2}_{n},s^{2}_{n}\right) \right) \left( \left( \omega ^{1}_{n}\right) ^{+}-\left( \omega ^{2}_{n}\right) ^{+}\right) \mathrm {d}x\\&\quad +\int _{\mathbb {R}^{N}}\left( F_{v}\left( x,\omega ^{1}_{n},s^{1}_{n}\right) -F_{v}\left( x,\omega ^{2}_{n},s^{2}_{n}\right) \right) \left( \left( s^{1}_{n}\right) ^{+}-\left( s^{2}_{n}\right) ^{+}\right) \mathrm {d}x\\&\le \left\langle J'\left( \omega ^{1}_{n},s^{1}_{n}\right) , \left( \left( \omega ^{1}_{n}\right) ^{+}-\left( \omega ^{2}_{n}\right) ^{+},\left( s^{1}_{n}\right) ^{+}-\left( s^{2}_{n}\right) ^{+}\right) \right\rangle \\&\quad -\left\langle J'\left( \omega ^{2}_{n},s^{2}_{n}\right) , \left( \left( \omega ^{1}_{n}\right) ^{+}-\left( \omega ^{2}_{n}\right) ^{+},\left( s^{1}_{n}\right) ^{+}-\left( s^{2}_{n}\right) ^{+}\right) \right\rangle \\&\quad + C\left( \left\| \left( \omega ^{1}_{n},s^{1}_{n}\right) \right\| _{q}^{p-1}+\left\| \left( \omega ^{1}_{n},s^{1}_{n}\right) \right\| _{\gamma }^{\gamma -1}+ \left\| \left( \omega ^{2}_{n},s^{2}_{n}\right) \right\| _{q}^{p-1}\right. \\&\left. \quad +\left\| \left( \omega ^{2}_{n},s^{2}_{n}\right) \right\| _{\gamma }^{\gamma -1}\right) \left\| \left( \left( \omega ^{1}_{n}\right) ^{+}- \left( \omega ^{2}_{n}\right) ^{+},\left( s^{1}_{n}\right) ^{+}-\left( s^{2}_{n}\right) ^{+}\right) \right\| _{\gamma }, \end{aligned}$$

where $q=\frac{(p-1)\gamma }{\gamma -1}$ and $\gamma \le q\le p$. It follows from Lemma 2.4(b) that $(\omega ^{1}_{n},s^{1}_{n})_{n\in \mathbb {N}}$ and $(\omega ^{2}_{n},s^{2}_{n})_{n\in \mathbb {N}}$ are Palais–Smale sequences for J and by Lemma 3.4, we know that they are bounded in $E_{2,\gamma }$. Since the embeddings from $E_{2,\gamma }$ into $L^{\gamma }(\mathbb {R}^{N})\times L^{\gamma }(\mathbb {R}^{N})$ and into $L^{q}(\mathbb {R}^{N})\times L^{q}(\mathbb {R}^{N})$ are continuous, then we derive

$$\begin{aligned} \left\| \left( \left( \omega ^{1}_{n}\right) ^{+}-\left( \omega ^{2}_{n}\right) ^{+},\left( s^{1}_{n}\right) ^{+}-\left( s^{2}_{n}\right) ^{+}\right) \right\| \rightarrow 0,~n\rightarrow \infty . \end{aligned}$$

Note that if $(u,v)=(u^{+}+u^{-},v^{+}+v^{-})\in \mathcal {N}$, then $J(u,v)\ge c_{inf}$ and

$$\begin{aligned} \left\| \left( u^{+},v^{+}\right) \right\| \ge \left\{ \sqrt{2c_{\inf }}\Vert (u^{-},v^{-})\Vert \right\} . \end{aligned}$$

(4.2)

Similarly as in Lemma 2.13 in [18], we obtain

$$\begin{aligned} \begin{aligned} \left\| \left( u^{1}_{n}-u^{2}_{n},v^{1}_{n}-v^{2}_{n}\right) \right\|&=\left\| \frac{\left( \left( \omega ^{1}_{n}\right) ^{+},\left( s^{1}_{n}\right) ^{+}\right) }{\left\| \left( \left( \omega ^{1}_{n}\right) ^{+},\left( s^{1}_{n}\right) ^{+}\right) \right\| }-\frac{\left( \left( \omega ^{2}_{n}\right) ^{+},\left( s^{2}_{n}\right) ^{+}\right) }{\left\| \left( \left( \omega ^{2}_{n}\right) ^{+},\left( s^{2}_{n}\right) ^{+}\right) \right\| } \right\| \\&\le \sqrt{\frac{2}{c_{\inf }}}\left\| \left( \left( \omega ^{1}_{n}\right) ^{+}-\left( \omega ^{2}_{n}\right) ^{+}, \left( s^{1}_{n}\right) ^{+}-\left( s^{2}_{n}\right) ^{+}\right) \right\| . \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned} \left\| \left( u^{1}_{n}-u^{2}_{n},v^{1}_{n}-v^{2}_{n}\right) \right\| \rightarrow 0~n\rightarrow \infty . \end{aligned}$$

Case 2:$\Vert ((\omega ^{1}_{n})^{+}-(\omega ^{2}_{n})^{+},(s^{1}_{n})^{+}-(s^{2}_{n})^{+})\Vert _{\gamma }\not \rightarrow 0$. It follows from Lemma 1.21 in [18] that there exists a subsequence $(y_{n})\in \mathbb {Z}^{N}$ and $r>1$ such that

$$\begin{aligned} \liminf \limits _{n\rightarrow \infty }\int _{B\left( y_{n},1\right) }\bigg | \left( \left( \omega ^{1}_{n}\right) ^{+}-\left( \omega ^{2}_{n}\right) ^{+},\left( s^{1}_{n}\right) ^{+}-\left( s^{2}_{n}\right) ^{+}\right) \bigg |^{2}\mathrm {d}x>0. \end{aligned}$$

(4.3)

Then, we may assume that up to a subsequence,

$$\begin{aligned}&\left( \omega ^{1}_{n}\left( \cdot +y_{n}\right) ,s^{1}_{n}\left( \cdot +y_{n}\right) \right) \rightharpoonup \left( \omega ^{1},s^{1}\right) ,~\text{ in } ~E_{2,\gamma },\\&\left( \omega ^{2}_{n}\left( \cdot +y_{n}\right) ,s^{2}_{n}\left( \cdot +y_{n}\right) \right) \rightharpoonup \left( \omega ^{2},s^{2}\right) ,~\text{ in } ~E_{2,\gamma },\\&\left( \left( \omega ^{1}_{n}\right) ^{+}\left( \cdot +y_{n}\right) ,\left( s^{1}_{n}\right) ^{+}\left( \cdot +y_{n}\right) \right) \rightharpoonup \left( \left( \omega ^{1}\right) ^{+},\left( s^{1}\right) ^{+}\right) ,~~\text{ in }~ L^{2}_{loc}\left( \mathbb {R}^{N}\right) \times L^{2}_{loc}\left( \mathbb {R}^{N}\right) ,\\&\left( \left( \omega ^{2}_{n}\right) ^{+}\left( \cdot +y_{n}\right) ,\left( s^{2}_{n}\right) ^{+}\left( \cdot +y_{n}\right) \right) \rightharpoonup \left( \left( \omega ^{2}\right) ^{+},\left( s^{2}\right) ^{+}\right) ,~~\text{ in }~ L^{2}_{loc}\left( \mathbb {R}^{N}\right) \times L^{2}_{loc}\left( \mathbb {R}^{N}\right) , \end{aligned}$$

and

$$\begin{aligned}&\left\| \left( \left( \omega ^{1}_{n}\right) ^{+}\left( \cdot +y_{n}\right) ,\left( s^{1}_{n}\right) ^{+}\left( \cdot +y_{n}\right) \right) \right\| \rightarrow \alpha _{1},\\&\left\| \left( \left( \omega ^{2}_{n}\right) ^{+}\left( \cdot +y_{n}\right) ,\left( s^{2}_{n}\right) ^{+}\left( \cdot +y_{n}\right) \right) \right\| \rightarrow \alpha _{2}, \end{aligned}$$

for $\alpha _{1},~\alpha _{2}\ge \sqrt{2c_{inf}}$. It follows from (4.3) that $((\omega ^{1})^{+},(s^{1})^{+})\ne ((\omega ^{2})^{+},(s^{2})^{+})$ and then $(\omega ^{1},s^{1})\ne (\omega ^{2},s^{2})$. Since $m,~m^{-},~J^{-},~(J\circ m)^{-}$ are equivariant with respect to $\mathbb {Z}^{N}$-action, then $J^{-}(\omega ^{1},s^{1})=J^{-}(\omega ^{2},s^{2})=0$. Observe that if $(\omega ^{1},s^{1})\ne (0,0)$ and $(\omega ^{2},s^{2})\ne (0,0)$, then $(\omega ^{1},s^{1}),~(\omega ^{2},s^{2})\in \mathcal {N}$ and

$$\begin{aligned} \liminf \limits _{n\rightarrow \infty }&\left\| \left( u^{1}_{n}-u^{2}_{n},v^{1}_{n}-v^{2}_{n}\right) \right\| \\&= \liminf \limits _{n\rightarrow \infty }\left\| \left( \left( u^{1}_{n}-u^{2}_{n}\right) \left( \cdot +y_{n}\right) ,\left( v^{1}_{n}-v^{2}_{n}\right) \left( \cdot +y_{n}\right) \right) \right\| \\&=\liminf \limits _{n\rightarrow \infty }\left\| \frac{\left( \left( \omega ^{1}_{n}\right) ^{+},\left( s^{1}_{n}\right) ^{+}\right) \left( \cdot +y_{n}\right) }{\left\| \left( \left( \omega ^{1}_{n}\right) ^{+},\left( s^{1}_{n}\right) ^{+}\right) \left( \cdot +y_{n}\right) \right\| }-\frac{\left( \left( \omega ^{2}_{n}\right) ^{+},\left( s^{2}_{n}\right) ^{+}\right) \left( \cdot +y_{n}\right) }{\left\| \left( \left( \omega ^{2}_{n}\right) ^{+},\left( s^{2}_{n}\right) ^{+}\right) \left( \cdot +y_{n}\right) \right\| } \right\| \\&\ge \left\| \frac{\left( \left( \omega ^{1}_{n}\right) ^{+},\left( s^{1}_{n}\right) ^{+}\right) }{\alpha _{1}}-\frac{\left( \left( \omega ^{2}_{n}\right) ^{+},\left( s^{2}_{n}\right) ^{+}\right) }{\alpha _{2}}\right\| \\&\ge \min \left\{ \frac{\left\| \left( \left( \omega ^{1}_{n}\right) ^{+},\left( s^{1}_{n}\right) ^{+}\right) \right\| }{\alpha _{1}}, \frac{\left\| \left( \left( \omega ^{2}_{n}\right) ^{+},\left( s^{2}_{n}\right) ^{+}\right) \right\| }{\alpha _{2}} \right\} \left\| \frac{\left( \left( \omega ^{1}_{n}\right) ^{+},\left( s^{1}_{n}\right) ^{+}\right) }{\left\| \left( \left( \omega ^{1}_{n}\right) ^{+},\left( s^{1}_{n}\right) ^{+}\right) \right\| }\right. \\&\left. \quad -\frac{\left( \left( \omega ^{2}_{n}\right) ^{+},\left( s^{2}_{n}\right) ^{+}\right) }{\left\| \left( \left( \omega ^{2}_{n}\right) ^{+},\left( s^{2}_{n}\right) ^{+}\right) \right\| }\right\| \\&\ge \frac{\sqrt{2c_{\inf }}}{s\left( d\right) }\left\| m^{-}\left( \omega ^{1},s^{1}\right) -m^{-}\left( \omega ^{2},s^{2}\right) \right\| , \end{aligned}$$

where $s(d)=\text{ sup }\big \{\Vert (u^{+},v^{+})\Vert \big |(u,v)\in \mathcal {N},~J(u,v)\le d\big \}$. It follows from Lemma 2.4 that $m-(\omega ^{1}, s^{1}), m-(\omega ^{2}, s^{2})$ are critical points and we have (4.1). Note that if $(\omega ^{1},s^{1})=\{0,0\}$ or $(\omega ^{2},s^{2})=\{0,0\}$, then similarly as above, we prove that

$$\begin{aligned} \liminf \limits _{n\rightarrow \infty }\left\| \left( u^{1}_{n}-u^{2}_{n},v^{1}_{n}-v^{2}_{n}\right) \right\| \ge \frac{\sqrt{2c_{\inf }}}{s(d)} \end{aligned}$$

and again (4.1) holds. The proof is complete. $\square$

Proof of Theorem 1.2

It follows from Lemma 2.4(b) that (u, v) is equivariant with respect to the $\mathbb {Z}^{N}$-action given by $(u,v)\mapsto (u(\cdot -k),v(\cdot -k))$ for $k\in \mathbb {Z}^{N}$. Furthermore, J is even and (u, v) is odd. Thus, $\psi$ is even and invariant with respect to the $\mathbb {Z}^{N}$-action. Let $\mathcal {F}$ be the set of geometrically distinct critical points of $\psi$ and assume that $\mathcal {F}$ is finite. Then, similar to Lemma 2.13 in [18], we prove that

$$\begin{aligned} \begin{aligned} \inf&\bigg \{\Vert (u_{1}-u_{2},v_{1}-v_{2})\Vert \big |\psi '(u_{1},v_{1})= \psi '(u_{2},v_{2})=0,~(u_{1},v_{1})\ne (u_{2},v_{2})\in S^{+}\bigg \}\\&>0. \end{aligned} \end{aligned}$$

The discreteness of the Palais–Smale sequences in Lemma 4.1 allows us to repeat the following arguments: Lemma 2.15, Lemma 2.16 and proof of Theorem 1.2 in [18]. In fact, we show that for any $k\in \mathbb {N}$, there exists $(u,v)\in S^{+}$ such that $\psi '(u,v)=0$ and $\psi (u,v)=c_{k}$, where

$$\begin{aligned} c_{k}=\inf \big \{d\in \mathbb {R}|\gamma (\psi ^{d})\ge k \big \} \end{aligned}$$

and $\gamma$ denotes the usual Krasnoselskii genus (see [29]). Furthermore, $c_{k}<c_{k+1}$ for $k\in \mathbb {Z}^{N}$ and thus we get a contradiction (see [18] for detailed arguments). It follows from Lemma 2.4(b) that we get the existence of infinitely many geometrically distinct solutions to problem (1.1). The proof is complete. $\square$