Existence and multiplicity of positive solutions for fractional Laplacian systems with nonlinear coupling

TL;DR

This paper proves the existence and multiplicity of positive solutions for coupled fractional Schr"odinger systems with small parameters, using variational methods and critical point theory, extending known results from single equations.

Contribution

It introduces new multiplicity results for coupled fractional Schr"odinger systems with shared potential minima, employing energy estimates, Nehari manifold, and Lusternik-Schnirelmann theory.

Findings

Two positive solutions exist for the coupled system.

Solutions concentrate at the common minimum point of the potentials.

Conditions for existence and nonexistence of least energy solutions are identified.

Abstract

It is well known that a single nonlinear fractional Schr\"odinger equation with a potential $V(x)$ and a small parameter $\varepsilon $ may have a positive solution that is concentrated at the nondegenerate minimum point of $V(x)$. In this paper, we can find two different positive solutions for two weakly coupled fractional Schr\"odinger systems with a small parameter $\varepsilon $ and two potentials $V_{1}(x)$ and $V_{2}(x)$ having the same minimum point are concentrated at the same point minimum point of $V_{1}(x)$ and $V_{2}\left(x\right) $. In fact that by using the energy estimates, Nehari manifold technique and the Lusternik-Schnirelmann theory of critical points, we obtain the multiplicity results for a class of fractional Laplacian system. Furthermore, the existence and nonexistence of least energy positive solutions are also explored.

Equations

$$\left\{\begin{array}[]{ll}\varepsilon^{2\alpha}(-\Delta)^{\alpha}u+V_{1}\left(x\right)u=\left(\mu_{1}|u|^{2p}+\beta\left(x\right)|u|^{q-1}|v|^{q+1}\right)u,&\text{in }\mathbb{R}^{N},\\ \varepsilon^{2\alpha}(-\Delta)^{\alpha}v+V_{2}\left(x\right)v=\left(\mu_{2}|v|^{2p}+\beta\left(x\right)|v|^{q-1}|u|^{q+1}\right)v,&\text{in }\mathbb{R}^{N}\,,\\ 0\leq u,v\in H^{\alpha}(\mathbb{R}^{N}),&\text{in }\mathbb{R}^{N},\end{array}\right.$$

$$V_1\left(z_{1,i}\right)={\lambda }_1\equiv \min \left\{V_1\left(x\right):x\in {\mathbb{R}}^N\right\}>0\text{ for all }i=1,2\dots ,k$$

$$V_2\left(z_{2,j}\right)={\lambda }_2\equiv \min \left\{V_2\left(x\right):x\in {\mathbb{R}}^N\right\}>0\text{ for all }j=1,2\dots ,\ell ;$$

$${\epsilon }^{2\alpha }(-△{)}^{\alpha }u+V(x)u=f(x,u),\text{in}{\mathbb{R}}^N.$$

$$i\epsilon \frac{\partial u}{\partial t}={\epsilon }^{2\alpha }(-△{)}^{\alpha }u+V(x)u-f(x,u)\text{in}{\mathbb{R}}^N,$$

$$\left\{\begin{array}[]{ll}(-\triangle)^{\alpha}u+\lambda_{l}u=\mu_{l}|u|^{2p}u,&\text{in }\mathbb{R}^{N},\\ u\in H^{\alpha}(\mathbb{R}^{N}),\quad u>0,&\text{in }\mathbb{R}^{N}\,,\end{array}\right.$$

$$2_{\alpha}^{\ast}=\left\{\begin{array}[]{ll}\frac{2N}{N-2\alpha},&\text{{if }}N>2\alpha,\\ +\infty,&\text{{if }}N\leq 2\alpha.\end{array}\right.$$

$$\frac{C_1}{1+|x{|}^{N+2\alpha }}\le U(x)\le \frac{C_2}{1+|x{|}^{N+2\alpha }},\forall x\in {\mathbb{R}}^N,$$

$$ker{L}_0=span\{{\partial }_{x_1}U,\cdots ,{\partial }_{x_N}U\}.$$

$$\left\{\begin{array}[]{ll}-\varepsilon^{2}\Delta u+u=\left(|u|^{2p}+\beta|u|^{p-1}|v|^{p+1}\right)u,&\text{in }\mathbb{R}^{N},\\ -\varepsilon^{2}\Delta v+\omega^{2}v=\left(|v|^{2p}+\beta|v|^{p-1}|u|^{p+1}\right)v,&\text{in }\mathbb{R}^{N},\end{array}\right.$$

$$\left\{\begin{array}[]{ll}-\varepsilon^{2}\Delta u+V_{1}\left(x\right)u=\mu_{1}u^{3}+\beta uv^{2},&\text{in }\mathbb{R}^{N},\\ -\varepsilon^{2}\Delta v+V_{2}\left(x\right)v=\mu_{2}v^{3}+\beta vu^{2},&\text{in }\mathbb{R}^{N},\end{array}\right.$$

$${\epsilon }^{-N}{E}_{\epsilon }(u_{\epsilon ,1},u_{\epsilon ,2})\to {\alpha }_{{\lambda }_1,{\mu }_1}+{\alpha }_{{\lambda }_2,{\mu }_2},$$

$${\alpha }_{{\lambda }_l,{\mu }_l}=\inf \left\{I_{{\lambda }_l,{\mu }_l}\left(u\right)|u\in {\mathbf{M}}_{{\lambda }_l,{\mu }_l}\right\},$$

$$I_{{\lambda }_l,{\mu }_l}\left(u\right)$$

$${\mathbf{M}}_{{\lambda }_l,{\mu }_l}$$

$$V_l(z_l^{\epsilon })$$

$$U_{ϵ,l}(y)$$

$$\frac{|z_1^{\epsilon }-z_2^{\epsilon }|}{\epsilon }$$

$$-\Delta u+{\lambda }_{l}u={\mu }_l{u}^3\text{ in }{\mathbb{R}}^N,u>0\text{ in }{\mathbb{R}}^N.$$

$${\epsilon }^{-N}{\int }_{B^N\left(z_l^{\epsilon };\epsilon R\right)}{u}_{\epsilon ,l}^4\ge d>0,$$

$$\left\{\begin{array}[]{l}\varepsilon^{2\alpha}(-\Delta)^{\alpha}u+P_{1}(x)u=\left(|u|^{2p}+\beta|u|^{p-1}|v|^{p+1}\right)u,\quad\text{in}~{}\mathbb{R}^{N},\\ \varepsilon^{2\alpha}(-\Delta)^{\alpha}v+P_{2}(x)v=\left(|v|^{2p}+\beta|v|^{p-1}|u|^{p+1}\right)v,\quad~{}\text{in}~{}\mathbb{R}^{N},\end{array}\right.$$

$$J_{\epsilon }\left(u,v\right)$$

$${\Vert \left(u,v\right)\Vert }_H^2$$

$$H^{\alpha }\left({\mathbb{R}}^N\right)=\left\{u\in L^2\left({\mathbb{R}}^N\right):{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{|u(x)-u(y){|}^2}{|x-y{|}^{N+2\alpha }}\mathrm{d}x\mathrm{d}y<+\infty \right\}.$$

$${\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{|u(x)-u(y){|}^2}{|x-y{|}^{N+2\alpha }}\mathrm{d}x\mathrm{d}y=2C(N,\alpha ){\int }_{{\mathbb{R}}^N}|(-△{)}^{\frac{\alpha }{2}}u{|}^2\mathrm{d}x,$$

$$C(N,\alpha )=\left({\int }_{{\mathbb{R}}^N}\frac{1-cos{\zeta }_1}{|\zeta {|}^{N+2\alpha }}\mathrm{d}\zeta \right),\zeta =\left({\zeta }_1,{\zeta }_2,\cdots ,{\zeta }_N\right).$$

$$\mathbf{N}_{\varepsilon}=\left\{(u,v)\in\mathbf{T}:\begin{array}[]{l}\left\|u\right\|_{V_{1}}^{2}=\mu_{1}\int_{\mathbb{R}^{N}}\left|u^{+}\right|^{2p+2}\mathrm{d}x+\int_{\mathbb{R}^{N}}\beta(x)\left|u^{+}\right|^{q+1}\left|v^{+}\right|^{q+1}\mathrm{d}x,\\ \left\|v\right\|_{V_{2}}^{2}=\mu_{2}\int_{\mathbb{R}^{N}}\left|v^{+}\right|^{2p+2}\mathrm{d}x+\int_{\mathbb{R}^{N}}\beta(x)\left|u^{+}\right|^{q+1}\left|v^{+}\right|^{q+1}\mathrm{d}x\end{array}\right\},$$

$$\mathbf{T}=\left\{\left(u,v\right)\in H:u\not\equiv 0\text{ and }v\not\equiv 0\right\}.$$

$$c_{\epsilon }=\inf \left\{J_{\epsilon }\left(u,v\right):\left(u,v\right)\in {\mathbf{N}}_{\epsilon }\right\},$$

$$\left\{\begin{array}[]{ll}(-\Delta)^{\alpha}u+V_{1}\left(\varepsilon x\right)u=\left(\mu_{1}|u|^{2p}+\beta\left(\varepsilon x\right)|u|^{q-1}|v|^{q+1}\right)u,&\text{in }\mathbb{R}^{N},\\ (-\Delta)^{\alpha}v+V_{2}\left(\varepsilon x\right)v=\left(\mu_{2}|v|^{2p}+\beta\left(\varepsilon x\right)|v|^{q-1}|u|^{q+1}\right)v,&\text{in }\mathbb{R}^{N}\,,\\ u,v\in H^{\alpha}(\mathbb{R}^{N}),\quad u,v>0,&\text{in }\mathbb{R}^{N}.\end{array}\right.$$

$${\hat{I}}_l\left(u\right)=\frac{1}{2}{\int }_{{\mathbb{R}}^N}\left(|(-△{)}^{\frac{\alpha }{2}}u{|}^2+{\lambda }_l{u}^2\right)\mathrm{d}x-\frac{{\mu }_l}{2p+2}{\int }_{{\mathbb{R}}^N}{\left|u^{+}\right|}^{2p+2}\mathrm{d}x,l=1,2.$$

$$⟨{\hat{I}}_l^{\prime }(u),\phi ⟩={\int }_{{\mathbb{R}}^N}\left((-△{)}^{\frac{\alpha }{2}}u(-△{)}^{\frac{\alpha }{2}}v+{\lambda }_{l}u\phi \right)\mathrm{d}x-{\mu }_l{\int }_{{\mathbb{R}}^N}{\left|u^{+}\right|}^{2p}{u}^{+}\phi \mathrm{d}x,l=1,2,$$

Full text

††footnotetext: G. Che was supported by the Fundamental Research Funds for the Central Universities of Central South University (Grant No. 2017zzts058). H. Chen was supported by the National Natural Science Foundation of China (Grant No. 11671403). T.-F. Wu was supported in part by the Ministry of Science and Technology, Taiwan (Grant No. 106-2115-M-390-001-MY2) and the National Center for Theoretical Sciences, Taiwan. ∗ Corresponding author.  E-mail addresses: tfwu@nuk.edu.tw (T.-F. Wu).   E-mail addresses: cheguofeng222@163.com (G. Che), math_chb@163.com (H. Chen).

Keywords: Fractional Laplacian systems; Lusternik–Schnirelmann theory; Positive solutions; Variational methods

2010 Mathematics Subject Classification. Primary: 35J50, 35A01; Secondary:

35B38

1 Introduction

In this paper, we are concerned with the following nonlinear systems of two weakly coupled fractional Schrödinger equations with nonconstant potentials:

[TABLE]

where $\alpha\in(0,1)$, $0<q\leq p\leq 1$ if $N\leq 4\alpha$ and $0<q\leq p<\frac{2\alpha}{N-2\alpha}$ if $N>4\alpha$, $0<\varepsilon\ll 1$ is a small parameter and $\mu_{l},~{}l=1,2,$ are positive constants. Throughout this paper, we assume that the potentials $V_{1}(x),V_{2}(x)$ and the coupling function $\beta(x)$ satisfy the following hypotheses:

• $\left(D_{1}\right)$

$V_{1}\left(x\right)$ and $V_{2}\left(x\right)$ are bounded and uniformly continuous functions on $\mathbb{R}^{N};$

• $\left(D_{2}\right)$

the potentials $V_{1}\left(x\right)$ and $V_{2}\left(x\right)$ have isolated gobal minimum points at $z_{1,1},z_{1,2},\ldots,z_{1,k}$ and $z_{2,1},z_{2,2},\ldots,z_{2,\ell}$, respectively. Here, an isolated gobal minimum point $z_{1,i}$ of $V_{1}\left(x\right)$ (similarly to $z_{2,j}$ of $V_{2}(x)$) means that $z_{1,i}$ is the unique minimum point of $V_{1}\left(x\right)$ in $B_{r_{0}}(z_{1,i})$ (a neighborhood of $z_{1,i})$, where $r_{0}$ is a positive constant. Moreover,

[TABLE]

and

[TABLE]

• $\left(D_{3}\right)$

$\beta\left(x\right)$ is a nonpositive, bounded and uniformly continuous function on $\mathbb{R}^{N};$

• $\left(D_{4}\right)$

there exist a positive integer $1\leq m\leq\min\left\{k,\ell\right\}$ and a constant $c_{0}>0$ such that $z_{1,i}=z_{2,j}$ and $\beta\left(x\right)\leq-c_{0}$ for all $x\in B_{r_{0}}(z_{1,i})$ for all $1\leq i,j\leq m,$ where $r_{0}>0$ as in condition $\left(D_{2}\right).$

In recent years great interest has been devoted to the study of elliptic equations involving the fractional Laplacian operator $(-\triangle)^{\alpha},~{}\alpha\in(0,1)$. This type of operators appears in a quite natural way in many different applications, such as, fractional quantum mechanics, geophysical fluid dynamics, population dynamics, phase transitions, flames propagation, anomalous diffusion, crystal dislocation, materials science, water waves and soft thin films. For more details about the applications, we refer the readers to [18, 29, 30] and the references therein.

Much attention has been paid to the single fractional Laplacian equation:

[TABLE]

Eq. $(\ref{1.1})$ is related to the stationary analogue of the fractional Schrödinger equation:

[TABLE]

which was introduced by [29, 30] through expanding the Feynman path integral from the Brownian–like to the Lévy–like quantum mechanical paths. In the past several decades, with the aid of variational methods, the existence, nonexistence, multiplicity, uniqueness, regularity and the asymptotic decay properties of solutions for Eq. $(\ref{1.1})$ have been obtained under various hypotheses on the potential $V(x)$ and the nonlinearity $f(x,u)$, see [11, 14, 17, 20, 22, 40] and the references therein. For instance, Frank, Lenzmann and Silvestre [20] studied the following single fractional Schrödinger equation:

[TABLE]

where $l=1,2$, and obtained the following properties of the ground states for Eq. $(\widehat{E}_{l})$.

Proposition 1.1

Let $N\geq 1,~{}\alpha\in(0,1)$ and $0<p<\frac{2_{\alpha}^{\ast}-2}{2}$, where

[TABLE]

Then the following hold:

$(i)$ (Uniqueness) The ground state solution $U\in H^{\alpha}(\mathbb{R}^{N})$ for Eq. $(\widehat{E}_{l})$ is unique.

$(ii)$ (Symmetry, regularity and decay) $U(x)$ is radial, positive and strictly decreasing in $|x|$. Moreover, the function $U(x)$ belongs to $H^{2\alpha+1}(\mathbb{R}^{N})\cap C^{\infty}(\mathbb{R}^{N})$ and satisfies

[TABLE]

with some constants $C_{2}\geq C_{1}>0$.

$(iii)$ (Non-degeneracy) The linearized operator $L_{0}=(-\triangle)^{\alpha}+1-(2p+1)U^{2p}$ is non-degenerate, i.e., its kernel is given by

[TABLE]

Very recently, many researchers have paid attention to the following Schrödinger system with constant potentials and $\beta$ is a real constant, i.e.

[TABLE]

which arises as a model for propagation of polarized laser beams in briefringent Kerr medium in nonlinear optics (see [4, 19, 28]). With the help of the critical point theory and the variational methods, there have been lots of results about the existence and multiplicity of nontrivial solutions for system (1.2), see [10, 21, 38] and the references therein. In the case of $p=1,N=2,3,\beta\in\mathbb{R}$ and the potentials may be not constants, we may get the following system:

[TABLE]

which appears in the Hartree–Fock theory for a double condensate, i.e., a binary mixture of Bose–Einstein condensates in two different hyperfine states $|1\rangle$ and $|2\rangle$ (cf. [16, 25, 42]). Physically, $u$ and $v$ are the corresponding condensate amplitudes, $\varepsilon^{2}=\frac{h^{2}}{2m}$ and $\mu_{j}=-(N_{j}-1)U_{jj}$, where $h$ is Planck constant, $m$ is the atom mass, $N_{j}$ is a fixed number of atoms in the hyperfine state $|j\rangle$. Recently, various results on the existence and concentration of solutions for system (1.3) have been obtained, we refer the readers to [26, 31, 32, 33, 39] and the references therein. Peng and Li [39] obtained the existence of multi–spike vector solutions for system system (1.3) with $\beta\neq 0$. Moreover, the attractive phenomenon for $\beta<0$ and the repulsive phenomenon for $\beta>0$ are also explored by the authors. Lin and Wu [33], use energy estimates and category theory to prove the nonuniqueness theorem, provided $\beta<0$. Lin and Wei [31] proved that there exists $\beta_{0}\in(0,\sqrt{\mu_{1}\mu_{2}})$ such that system (1.3) possesses a least energy solution whenever $\beta\in(-\infty,\beta_{0})$. When the potentials $V_{1}(x)$ and $V_{2}(x)$ satisfy:

$(\overline{V})$ $0<\inf\limits_{x\in\mathbb{R}^{N}}V_{l}(x)<\lim\limits_{|x|\rightarrow\infty}V_{l}(x)\leq\infty\text{ for }l=1,2.$

Lin and Wei [32] studied the minimization of the functional $E_{\varepsilon}$ for system (1.3) on $\mathbb{R}^{N},~{}N=2,3$. When $\beta<0$ and $\varepsilon>0$ sufficiently small, problem (1.3) has a least energy solution $(u_{\varepsilon,1},u_{\varepsilon,2}),$ such that

[TABLE]

as $\varepsilon$ goes to zero (up to a subsequence), where $\alpha_{\lambda_{l},\mu_{l}}$ is defined by

[TABLE]

[TABLE]

for $l=1,2$. Furthermore, if $V_{1}(x)$ and $V_{2}(x)$ are of $C^{2}$ functions with nondegenerate minimum points at $z_{1}$ and $z_{2}$, respectively, then $u_{\varepsilon,l}$ has only one maximum point $z_{l}^{\varepsilon}$ that satisfies

[TABLE]

as $\varepsilon\rightarrow 0$ (up to a subsequence), where $\omega_{\lambda_{l},\mu_{l}}\left(x\right)=\omega_{\lambda_{l},\mu_{l}}\left(\left|x\right|\right)$ is the energy minimizer of the minimum $\alpha_{\lambda_{l},\mu_{l}}>0$ (cf. [43]) and is the unique solution (cf. [27]) of

[TABLE]

Consequently, the $u_{\varepsilon,l}$’s satisfy

[TABLE]

as $\varepsilon\rightarrow 0$ (up to a subsequence), where $d$ and $R$ are positive constants independent of $\varepsilon$, $B^{N}\left(z_{l}^{\varepsilon};\varepsilon R\right)$ is an $N$ dimensional ball with a radius $\varepsilon R$ and a center at $z_{l}^{\varepsilon}$. Hereafter, the point $z_{l}^{\varepsilon}$ is defined as a concentration point of $u_{\varepsilon,l}$ if and only if (1.4) holds.

In the nonlocal case, that is, when $\alpha\in(0,1)$, even in the power type nonlinearities case, there are very few results for the fractional Laplacian systems. In [23], Guo and He studied the following fractional Schrödinger system with nonconstant potentials:

[TABLE]

where $\alpha\in(0,1)$, $0<p<\frac{2\alpha}{N-2\alpha}$, $\varepsilon>0$ is a small parameter and coupling function $\beta\equiv b>0$. Under some appropriate hypotheses on the potentials $P_{1}(x)$ and $P_{2}(x)$, the authors obtained the existence of nontrivial nonnegative solutions for system $(\ref{1.10})$ which concentrate around local minima of the potentials. When $\varepsilon=1$, $P_{1}(x)=1$ and $P_{2}(x)=\omega^{2\alpha},~{}\omega>0$, Guo and He [24] obtained the existence of a least energy solution for system $(\ref{1.10})$ on the Nehari manifold. Furthermore, the existence of least energy positive solution with both nontrivial components was also established. In [37], Lü and Peng studied system $(\ref{1.10})$ with $P_{1}(x)=P_{2}(x)=\varepsilon=1,~{}\left(|u|^{2p}+\beta|u|^{p-1}|v|^{p+1}\right)u$ and $\left(|v|^{2p}+\beta|v|^{p-1}|u|^{p+1}\right)v$ being replaced by $f(u)+\beta v$ and $g(v)+\beta u$, respectively. Under very weak assumptions on the nonlinear terms $f$ and $g$, they obtained the existence of positive vector solutions and vector ground state solutions for system $(\ref{1.10})$. Moreover, the asymptotic behavior of the solutions as $\beta\rightarrow 0$ was also analyzed by them. For the other related results about the fractional Laplacian system, we refer the readers to [8, 9, 12] and the references therein.

Motivated by [23, 31, 32, 33, 37], it is very natural for us to pose some questions, in particular, such as:

$(I)$ As pointed out in [23, 37], when the coupled nonlinear terms are replaced by $\beta\left(x\right)|u|^{q-1}u|v|^{q+1}$ and $\beta\left(x\right)|u|^{q+1}|v|^{q-1}v$ for the coupling function $\beta<0$ in $\mathbb{R}^{N}$ and $0<q\leq p,$ is the existence of positive solutions for system $(\ref{1.10})$ which concentrate around local minima of the potentials still true?

$\left(II\right)$ Can the relationship of the minimum points of the potentials $V_{1}(x)$ and $V_{2}(x)$ affects the number of positive solutions for system $(P_{\varepsilon})?$

$(III)$ Under our assumptions $(D_{1})$ and $(D_{2})$, can one prove that the positive solution of problem $(P_{\varepsilon})$ is a least energy solution? If not, can one give some appropriate conditions on the potentials to assure that the positive solution is a least energy solution of problem $(P_{\varepsilon})?$

In the present paper, by using the Nehari manifold technique, the energy estimates and the Lusternik–Schnirelmann theory of critical points, we study how the relationship of the minimum points of the potentials $V_{1}(x)$ and $V_{2}(x)$ affects the number of positive solutions for system $(P_{\varepsilon})$, provided $\beta$ is a nonpositive, bounded and uniformly continuous function on $\mathbb{R}^{N}$, and will give answers to Questions $(I)-(III)$. Moreover, whether the positive solution for problem $\left(P_{\varepsilon}\right)$ is a least energy solution depends on the relationship between $\liminf\limits_{|x|\rightarrow\infty}V_{l}(x)$,$\lim\limits_{|x|\rightarrow\infty}V_{l}(x)$ and $\lambda_{l},~{}l=1,2.$

Before we describe the main results, we need some known techniques. The energy functional $J_{\varepsilon}$ we consider that corresponds to problem $\left(P_{\varepsilon}\right)$ is given by, for each $\left(u,v\right)\in H=H^{\alpha}\left(\mathbb{R}^{N}\right)\times H^{\alpha}\left(\mathbb{R}^{N}\right),$

[TABLE]

where with $u^{+}=\max\{u,0\}$, $v^{+}=\max\{v,0\}$ and the norm $\|\cdot\|_{H},$ given by

[TABLE]

Note that the fractional Sobolev space $H^{\alpha}\left(\mathbb{R}^{N}\right)$ is given by:

[TABLE]

From Proposition 3.4, 3.6 of [13], we know that

[TABLE]

where

[TABLE]

It is well known that the energy functional $J_{\varepsilon}$ is of class $C^{1}$ in $H$ and the nonnegative solutions of problem $\left(P_{\varepsilon}\right)$ are the critical points of the energy functional $J_{\varepsilon}.$ As the energy functional $J_{\varepsilon}$ is not bounded below on $H$ and to prove the existence of nontrivial critical points of $J_{\varepsilon}$, it is useful to consider the functional on the Nehari manifold

[TABLE]

where

[TABLE]

Furthermore, we consider the minimization problem:

[TABLE]

we call the nontrivial critical point $(u,v)\in H$ of $J_{\varepsilon}$ is a least energy solution of problem $(P_{\varepsilon})$ if $J_{\varepsilon}(u,v)=c_{\varepsilon}$ and $(u,v)\in\mathbf{N}_{\varepsilon}$. Note that if there exists a nontrivial solution $(u,v)\in\mathbf{N}_{\varepsilon}$ of problem $(P_{\varepsilon})$ such that $J_{\varepsilon}(u,v)>c_{\varepsilon}$, then we call the solution $(u,v)$ is a higher energy solution of problem $(P_{\varepsilon})$.

Now we state our main results.

Theorem 1.2

$\left(i\right)$* Assume that the conditions $\left(D_{1}\right)-\left(D_{4}\right)$ hold. Then there exists $\varepsilon_{0}>0$ such that for every $\varepsilon\in\left(0,\varepsilon_{0}\right),$ problem $(P_{\varepsilon})$ has at least $k\times\ell+m$ positive solutions.

$\left(ii\right)$ Let $(\widehat{u}_{\varepsilon,i},\widehat{v}_{\varepsilon,j})$ be a positive solution of problem $(P_{\varepsilon})$ as in part $\left(i\right).$ Then there exist $z_{1,i}$ and $z_{2,j}$ which are isolated gobal minimum points of potentials $V_{1}\left(x\right)$ and $V_{2}\left(x\right),$ respectively such that $(\widehat{u}_{\varepsilon,i},\widehat{v}_{\varepsilon,j})$ concentrating at $(z_{1,i},z_{2,j})$ as $\varepsilon\rightarrow 0.$*

Theorem 1.3

$\left(i\right)$* If $\liminf\limits_{|x|\rightarrow\infty}V_{l}(x)\equiv V_{l,\infty}>\lambda_{l}$ for all $l=1,2,$ then there exists $0<\varepsilon_{\ast\ast}\leq\varepsilon_{0}$ such that for every $\varepsilon<\varepsilon_{\ast\ast},$ we can find at least one least energy solution in the these solutions of Theorem 1.2 $\left(i\right)$.

$\left(ii\right)$ If $\lim\limits_{|x|\rightarrow\infty}V_{l}(x)=\lambda_{l}$ for all $l=1,2,$ then all of the solutions of Theorem 1.2 $\left(i\right)$ are higher energy.*

Remark 1.1

Compared with the local operator $-\Delta$, the operator $(-\Delta)^{\alpha}$ with $\alpha\in(0,1)$ on $\mathbb{R}^{N}$ is nonlocal, which can be expressed as follows: the quantity $(-\Delta)^{\alpha}u(x)$ depend on not only the values of $u$ in a neighborhood of $x$ (as is the case for the Laplacian), but also the values of $u$ at any point $y\in\mathbb{R}^{N}$, and it is expected that the standard techniques for $-\Delta$ cannot be used directly.

Remark 1.2

The main difficulty when dealing with problem $(P_{\varepsilon})$ lies in the lack of compactness of the embedding from $H^{\alpha}(\mathbb{R}^{N})$ into $L^{r}(\mathbb{R}^{N}),~{}2\leq r<2_{\alpha}^{\ast}$, which prevents us from using the variational methods in a standard way. We solve this difficulty by using the Nehari manifold technique and the energy estimates, see Section 3 for details.

Notation 1.1

Throughout this paper, we shall denote $|\cdot|_{r}$ the $L^{r}$-norm for $1\leq r\leq+\infty$ and $C$ various positive generic constants, which may vary from line to line. $B^{N}(x,r)$ denotes a ball centered at $x$ with radius $r$ in $\mathbb{R}^{N}$. Also if we take a subsequence of a sequence $\{(u_{n},v_{n})\}$ we shall denote it again by $\{(u_{n},v_{n})\}$. We use $o(1)$ to denote any quantity which tends to zero as $n\rightarrow\infty$ and $o_{\varepsilon}(1)$ to denote any quantity which tends to zero as $\varepsilon\rightarrow 0$.

The remainder of this paper is as follows. In Section 2, we give some preliminaries. In Section 3, we construct the Palais–Smale (PS) sequences. In Section 4, we prove Theorem 1.2. In Section 5, we prove Theorem 1.3.

2 Preliminaries

First of all, it is easy to see that if we make the change of variables $x=\varepsilon z$, then we can rewrite Eq. $(P_{\varepsilon})$ as the following equivalent equation:

[TABLE]

Now we present some related results about Eq. $\left(\widehat{E}_{l}\right)$. It is obvious that Eq. $\left(\widehat{E}_{l}\right)$ is variational, and its solutions are the critical points of the functional $\widehat{I}_{l}\left(u\right)$ defined in $H^{\alpha}(\mathbb{R}^{N})$ as

[TABLE]

Furthermore, one can see that $\widehat{I}_{l}$ is a $C^{1}$ functional with the derivative given by

[TABLE]

for all $\varphi\in H^{\alpha}(\mathbb{R}^{N})$, where $\widehat{I}_{l}^{\prime}$ denotes the Fréchet derivative of $\widehat{I}_{l}.$

Define the Nehari manifold

[TABLE]

Then, by Frank, Lenzmann and Silvestre [20], we may assume that Eq. $\left(\widehat{E}_{l}\right)$ has a unique least energy positive solution $\widehat{\omega}_{l}$ (which up to translation) such that

[TABLE]

Furthermore, $\widehat{\omega}_{l}$ is radial, i.e., $\widehat{\omega}_{l}(x)=\widehat{\omega}_{l}(|x|)$.

To prove the main results, we will show some technical lemmas, whose proofs follow with the same type of arguments found in [31]. However for the readers’ convenience we will write their proofs. we need the following lemmas.

Lemma 2.1

Assume that $\varepsilon>0$ and $(u,v)\in\mathbf{N}_{\varepsilon}.$ Then

[TABLE]

and

[TABLE]

Furthermore, if $\int_{\mathbb{R}^{N}}\beta(x)\left|u^{+}\right|^{q+1}\left|v^{+}\right|^{q+1}\mathrm{d}x<0,$ then all the inequalities of $\left(\ref{2.1}\right)$ and $\left(\ref{2.2}\right)$ become strict, i.e.

[TABLE]

and

[TABLE]

Proof. Let $\left(u,v\right)\in\mathbf{\ N}_{\varepsilon}$. Set $u_{\varepsilon}\left(x\right)=u\left(\varepsilon x\right)$ and $v_{\varepsilon}\left(x\right)=v\left(\varepsilon x\right)$ for $x\in\mathbb{R}^{N}$. Due to $\beta(x)\leq 0$ in $\mathbb{R}^{N}$ and $\left(u,v\right)\in\mathbf{N}_{\varepsilon}$, it is obvious that

[TABLE]

and

[TABLE]

Let

[TABLE]

Then by (2) and (2), it is easy to verify that $0<s_{\varepsilon},t_{\varepsilon}\leq 1$, $s_{\varepsilon}u_{\varepsilon}\in\widehat{\mathbf{M}}_{1}$ and $t_{\varepsilon}v_{\varepsilon}\in\widehat{\mathbf{M}}_{2}$. Hence,

[TABLE]

and

[TABLE]

Thus by $\left(\ref{2.3}\right)-\left(\ref{2.6}\right)$, we obtain  (2.1) and (2.2). Similarly, one may follow the above argument to show that all the inequalities of $\left(\ref{2.1}\right)$ and (2.2) become strict if $\int_{\mathbb{R}^{N}}\beta(x)\left|u^{+}\right|^{q+1}\left|v^{+}\right|^{q+1}\mathrm{d}x<0$. Therefore, we may complete the proof. $\square$

The integral $\int_{\mathbb{R}^{N}}\beta(x)\left|u^{+}\right|^{q+1}\left|v^{+}\right|^{q+1}\mathrm{d}x$ may play an important role in the quantity of the energy functional $J_{\varepsilon}$. Here we state the crucial energy estimates, given by

Lemma 2.2

Let $\varepsilon,\sigma>0$ and $\left(u,v\right)\in\mathbf{N}_{\varepsilon}.$ If $\int_{\mathbb{R}^{N}}\beta(x)\left|u^{+}\right|^{q+1}\left|v^{+}\right|^{q+1}\mathrm{d}x\leq-\sigma\varepsilon^{N},$ then

[TABLE]

where

[TABLE]

and

[TABLE]

Furthermore, if $0<\sigma<(2p+2)\sqrt{\widehat{\alpha}_{1}\widehat{\alpha}_{2}},$ then

[TABLE]

for all $\left(u,v\right)\in\mathbf{N}_{\varepsilon}$ and $\varepsilon^{-N}J_{\varepsilon}\left(u,v\right)\leq\widehat{\alpha}_{1}+\widehat{\alpha}_{2}+\overline{\delta}.$

Proof. Assume $\left(u,v\right)\in\mathbf{N}_{\varepsilon}$ and $\sigma>0$ such that

[TABLE]

Let $\widetilde{u}\left(x\right)=u\left(\varepsilon x\right)$ and $\widetilde{v}\left(x\right)=v\left(\varepsilon x\right).$ Then

[TABLE]

where

[TABLE]

Hence, by (2.11), (2) and $\left(u,v\right)\in\mathbf{N}_{\varepsilon}$, we have

[TABLE]

[TABLE]

and

[TABLE]

Moreover, it follows from (2), (2.14) and (2.15) that

[TABLE]

Set

[TABLE]

Then, by $\left(\ref{2.13}\right)-\left(\ref{2.15}\right)$, we have

[TABLE]

and

[TABLE]

Besides, $\widetilde{s}\widetilde{u}\in\widehat{\mathbf{M}}_{1}$ and $\widetilde{t}\widetilde{v}\in\widehat{\mathbf{M}}_{2}$ for all $\varepsilon>0$, $0<q\leq p\leq 1$ if $N\leq 4\alpha$ and $0<q\leq p\leq\frac{2\alpha}{N-2\alpha}$ if $N>4\alpha$. It follows from (2), (2.17) and (2.18) that

[TABLE]

[TABLE]

Now we wish to claim (2.7). Suppose both

[TABLE]

and

[TABLE]

Then by (2.9), we have

[TABLE]

and

[TABLE]

Hence,

[TABLE]

and

[TABLE]

Combining (2), (2.20) and (2.21), we obtain

[TABLE]

thus (2.7) holds. On the other hand, suppose either

[TABLE]

or

[TABLE]

Then by (2.19), we obtain

[TABLE]

i.e.

[TABLE]

where $\overline{\delta}$ is defined in (2.8). Therefore, by (2.22), we may complete the proof of (2.7). Since (2.7) holds for all $\left(u,v\right)\in\mathbf{N}_{\varepsilon}$ satisfying (2.13), then we may conclude that

[TABLE]

for all $\left(u,v\right)\in\mathbf{N}_{\varepsilon}$ with $\varepsilon^{-N}J_{\varepsilon}\left(u,v\right)\leq\widehat{\alpha}_{1}+\widehat{\alpha}_{2}+\overline{\delta}.$ Hence, by (2.23) and Lemma 2.1, we may obtain (2.10) for all $\left(u,v\right)\in\mathbf{N}_{\varepsilon}$ with $\varepsilon^{-N}J_{\varepsilon}\left(u,v\right)\leq\widehat{\alpha}_{1}+\widehat{\alpha}_{2}+\overline{\delta}$, provided $0<\sigma<(2p+2)\sqrt{\widehat{\alpha}_{1}\widehat{\alpha}_{2}}$. This completes the proof. $\square$

To prove Theorems 1.2 and 1.3, we need another lemma as follows:

Lemma 2.3

Let $c_{\varepsilon}$ be as in (1.7). Then $\varepsilon^{-N}c_{\varepsilon}\geq\widehat{\alpha}_{1}+\widehat{\alpha}_{2}$ for $\varepsilon>0$ that is sufficiently small. Furthermore, if $\varepsilon^{-N}c_{\varepsilon}=\widehat{\alpha}_{1}+\widehat{\alpha}_{2},$ then problem $(P_{\varepsilon})$ does not have a least energy solution.

Proof. By $q\leq p$ and Lemma 2.1, we have

[TABLE]

Hence, $\varepsilon^{-N}c_{\varepsilon}\geq\widehat{\alpha}_{1}+\widehat{\alpha}_{2}$ for all $\varepsilon>0$. Now we want to prove that problem $\left(P_{\varepsilon}\right)$ does not have a least energy solution if $\varepsilon^{-N}c_{\varepsilon}=\widehat{\alpha}_{1}+\widehat{\alpha}_{2}$. Argue by contradiction that there exists $\left(u_{0},v_{0}\right)\in\mathbf{N}_{\varepsilon}$ that is a least energy solution of problem $\left(P_{\varepsilon}\right)$ such that

[TABLE]

Hence by the maximum principle for the fractional Laplacian [41], $u_{0},v_{0}>0$ in $\mathbb{R}^{N}$. So

[TABLE]

and then by Lemma 2.1 and $q\leq p$, we obtain

[TABLE]

which would contradict (2.24). Therefore, we may complete the proof. $\square$

Now, we need to introduce a generalized barycenter map. By this we mean a continuous map $\Phi:L^{2}\left(\mathbb{R}^{N}\right)\backslash\left\{0\right\}\rightarrow\mathbb{R}^{N}$, which is equivariant with respect to the action of the group of euclidian motions in $\mathbb{R}^{N}$, that is, for every $x\in\mathbb{R}^{N}$, every orthogonal $N\times N$ matrix $A$ and every $u\in L^{2}\left(\mathbb{R}^{N}\right)\backslash\left\{0\right\}$, one has

[TABLE]

and

[TABLE]

where $(\xi\ast u)(x)=u(x-\xi)$ and $(u\circ\varepsilon)(x)=u\left(\varepsilon x\right)$. This property is easily built into the construction. Indeed, if $\Phi_{1}$ satisfies (2.25), then $\Phi$ defined by $|u|_{2}^{\frac{2}{N}}\Phi_{1}(u\circ|u|_{2}^{\frac{2}{N}})$ satisfies (2.25) and (2.26). Note that the map

[TABLE]

has the invariance properties (2.25) and (2.26), but it is neither well defined on $L^{2}(\mathbb{R}^{N})\backslash\{0\}$ nor on $H^{\alpha}(\mathbb{R}^{N})\backslash\{0\}$.

The conditions $\left(D_{1}\right)$ and $\left(D_{2}\right)$ may imply $z_{1,i}^{\varepsilon}\rightarrow P_{1,i}$ and $z_{2,j}^{\varepsilon}\rightarrow P_{2,j}$ as $\varepsilon\rightarrow 0$ (up to a subsequence), where $P_{1,i}$ and $P_{2,j}$ are global minimum points of $V_{1}\left(x\right)$ and $V_{2}\left(x\right)$, respectively. Generically, $(P_{1,i},P_{2,j})$ may not be equal to $(z_{1,i},z_{2,j}),$ because $V_{1}\left(x\right)$ and $V_{2}\left(x\right)$ may have multiple minimum points. To find the positive solutions $(\widehat{u}_{\varepsilon,i},\widehat{v}_{\varepsilon,j})$ concentrating at $(z_{1,i},z_{2,j})$, we may consider the minimization problem of $J_{\varepsilon}$ over the subset $N_{i,j}(\varepsilon)$ of $\mathbf{N}_{\varepsilon}$, where $N_{i,j_{2}}\left(\varepsilon\right)$ is defined by

[TABLE]

where $C_{s}\left(x\right)$ is a cube defined by $C_{s}\left(x\right)=\underset{n=1}{\overset{N}{\Pi}}\left(x_{n}-s,x_{n}+s\right)$ with the boundary $\partial C_{s}\left(x\right)$ for $0<s<r_{0}$ ($r_{0}$ is given in $(D_{2})$ and $\left(D_{4}\right)$), and $x=\left(x_{1},\ldots,x_{N}\right)\in\mathbb{R}^{N}$ such that $\overline{C_{s}\left(z_{1,i}\right)}\subset B_{r_{0}}(z_{1,i}),\overline{C_{s}\left(z_{2,j}\right)}\subset B_{r_{0}}(z_{2,j})$ and

[TABLE]

Moreover, by condition $\left(D_{4}\right),$

[TABLE]

Next, we consider the boundary of $N_{i,j}\left(\varepsilon\right)$ as follows:

[TABLE]

Now we consider the minimization of the functional $J_{\varepsilon}$ over $N_{i,j}\left(\varepsilon\right)$ and $O_{i,j}\left(\varepsilon\right)$, respectively, and denote the corresponding minima as

[TABLE]

The upper bound of $\gamma_{i,j}\left(\varepsilon\right)$ is given by:

Lemma 2.4

For each $\delta>0$, there exists $\varepsilon_{\delta}>0$ such that for $\varepsilon\in\left(0,\varepsilon_{\delta}\right)$, there holds

[TABLE]

Proof. First, we define test functions $u_{\varepsilon,i}$ and $v_{\varepsilon,j}$ by

[TABLE]

where $x^{\varepsilon}=\frac{\sqrt{\varepsilon}}{2}e$, $e\in S^{N-1}=\left\{x\in\mathbb{R}^{N}:\left|x\right|=1\right\}$ and $\widehat{\omega}_{l}$ is the unique positive radial solution of Eq. $\left(\widehat{E}_{l}\right)$ for $l=1,2.$ Notice that $\widehat{I}_{l}\left(\omega_{l}\right)=\widehat{\alpha}_{l}$ for $l=1,2.$ Moreover, for $0<\varepsilon<1$, the function $\psi_{\varepsilon}\in C^{1}\left(\mathbb{R}^{N},\left[0,1\right]\right)$ with compact support satisfies

[TABLE]

and $\left|\nabla\psi_{\varepsilon}\right|\leq 2$ in $\mathbb{R}^{N}$. Obviously,

[TABLE]

Then by (2), (2.30) and Lemma 3.2 of [32], it is easy to find two positive numbers $t_{\varepsilon,i},s_{\varepsilon,j}$ such that $\left(t_{\varepsilon,i}u_{\varepsilon,i},s_{\varepsilon,j}v_{\varepsilon,j}\right)\in\mathbf{N}_{\varepsilon}$ when $\varepsilon$ is sufficiently small and $\left(t_{\varepsilon,i},s_{\varepsilon,j}\right)\rightarrow\left(1,1\right)$ as $\varepsilon\rightarrow 0^{+},$ uniformly for $e\in S^{N-1}$. Moreover, by (2.25), (2.26) and $\left(t_{\varepsilon,i},s_{\varepsilon,j}\right)\rightarrow(1,1),$ as $\varepsilon\rightarrow 0^{+}$, we have

[TABLE]

Similarly, $\Phi\left(s_{\varepsilon,j}v_{\varepsilon,j}\right)=z_{2,j}+o_{\varepsilon}\left(1\right).$ Thus, $\Phi\left(t_{\varepsilon,i}u_{\varepsilon,i}\right)\in C_{s}\left(z_{1,i}\right)$ and $\Phi\left(s_{\varepsilon,j}v_{\varepsilon,j}\right)\in C_{s}\left(z_{2,j}\right)$ for $\varepsilon$ sufficiently small, and $\left(t_{\varepsilon,i}u_{\varepsilon,i},s_{\varepsilon,j}v_{\varepsilon,j}\right)\in N_{i,j}\left(\varepsilon\right)$. On the other hand, by (2), (2.30) and $\left(t_{\varepsilon,i},s_{\varepsilon,j}\right)\rightarrow(1,1)$ as $\varepsilon\rightarrow 0^{+}$, it is easy to check that

[TABLE]

Here we have used the facts that $V_{1}(z_{1,i})=\lambda_{1}$ and $V_{2}(z_{2,j})=\lambda_{2}$. Therefore, by (2.28) and (2.31), we may complete the proof of Lemma 2.4. $\square$

On the other hand, we may describe the lower bound of $\widetilde{\gamma}\left(\varepsilon\right)$ as follows:

Lemma 2.5

There exist positive numbers $\widetilde{\delta}$ and $\varepsilon_{\widetilde{\delta}}$ such that for every $\varepsilon\in\left(0,\varepsilon_{\widetilde{\delta}}\right)$

[TABLE]

Proof. We shall prove Lemma 2.5 by contradiction. Suppose there exist $1\leq i\leq k,1\leq j\leq\ell$ and a sequence $\left\{\varepsilon_{n}\right\}_{n=1}^{\infty}\subset\mathbb{R}^{+}$ such that $\varepsilon_{n}\rightarrow 0$ and

[TABLE]

as $n\rightarrow\infty$. Then there exists $\left\{\left(u_{n},v_{n}\right)\right\}_{n=1}^{\infty}$ such that $\left(u_{n},v_{n}\right)\in O_{i,j}\left(\varepsilon_{n}\right)$ for $n\in\mathbb{N}$, and

[TABLE]

Let $\widetilde{u}_{n}\left(x\right)=u_{n}\left(\varepsilon_{n}x\right)$ and $\widetilde{v}_{n}\left(x\right)=v_{n}\left(\varepsilon_{n}x\right)$. From (2.32), there holds

[TABLE]

It follows from (2.33) and Lemma 2.2 that

[TABLE]

Since $\left(u_{n},v_{n}\right)\in O_{i,j}\left(\varepsilon_{n}\right)$ for $n\in\mathbb{N}$, then from (2.34), we obtain

[TABLE]

and

[TABLE]

Therefore, by (2.35), (2.36) and a similar argument to the proof of Lemma 3.3 in [36] (or see [7, lemma 3.1]), we may identify a contradiction. This completes the proof. $\square$

3 Palais–Smale Sequences

Fix $0<\sigma<(2p+2)\sqrt{\widehat{\alpha}_{1}\widehat{\alpha}_{2}}$ arbitrarily. Then by Lemma 2.2,

[TABLE]

and

[TABLE]

for all $\left(u,v\right)\in N_{i,j}\left(\varepsilon\right)$ with $\varepsilon^{-N}J_{\varepsilon}\left(u,v\right)\leq\widehat{\alpha}_{1}+\widehat{\alpha}_{2}+\delta_{0}$, where $\delta_{0}=\delta_{0}(\sigma)>0$. Moreover, by Lemmas 2.4 and 2.5, there exist $\delta_{0}>0$ and $\varepsilon_{0}=\varepsilon_{0}\left(\delta_{0}\right)>0$ such that

[TABLE]

Here, we may set $0<\delta_{0}\leq\min\left\{\overline{\delta},\widetilde{\delta},\widehat{\alpha}_{1},\widehat{\alpha}_{2}\right\}/2$, then we have the following results.

Lemma 3.1

For each $\left(u,v\right)\in N_{i,j}\left(\varepsilon\right)$ with

[TABLE]

there exist $b>0$ and a differentiable function

[TABLE]

such that $\left(s\left(0,0\right),t\left(0,0\right)\right)=\left(1,1\right)$, the function $\left(s\left(\widetilde{u},\widetilde{v}\right)\left(u-\widetilde{u}\right),t\left(\widetilde{u},\widetilde{v}\right)\left(v-\widetilde{v}\right)\right)\in N_{i,j}\left(\varepsilon\right)$ and

[TABLE]

for all $\left(\widetilde{u},\widetilde{v}\right)\in B\left(0;b\right).$ Moreover,

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Proof. Let $L_{\beta}\left(u,v\right)=\int_{\mathbb{R}^{N}}\beta(x)\left|u^{+}\right|^{q+1}\left|v^{+}\right|^{q+1}\mathrm{d}x.$ Define a function $F:H\times\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ given by

[TABLE]

where

[TABLE]

and

[TABLE]

Because $\left(u,v\right)\in N_{i,j}\left(\varepsilon\right)$ and $J_{\varepsilon}\left(u,v\right)<\min\left\{\varepsilon^{N}\left(\widehat{\alpha}_{1}+\widehat{\alpha}_{2}+\delta_{0}\right),\widetilde{\gamma}_{i,j}\left(\varepsilon\right)\right\},$ we have $F\left(0,0,1,1\right)=\left(0,0\right)$ and $\|(u,v)\|_{H}\leq C$ for some $C>0$. By a direct computation, we obtain

[TABLE]

[TABLE]

and

[TABLE]

Hence, by (2.10), we obtain

[TABLE]

since $\beta\left(x\right)\leq 0$ in $\mathbb{R}^{N}$, $0<q\leq p\leq 1$ if $N\leq 4\alpha$ and $0<q\leq p\leq\frac{2\alpha}{N-2\alpha}$ if $N>4\alpha$. Thus, in view of the Implicit Function Theorem, there exists a differentiable function

[TABLE]

such that $\left(s\left(0\right),t\left(0\right)\right)=\left(1,1\right)$ and

[TABLE]

This function is equivalent to $\left(s\left(\widetilde{u},\widetilde{v}\right)\left(u-\widetilde{u}\right),t\left(\widetilde{u},\widetilde{v}\right)\left(v-\widetilde{v}\right)\right)\in\mathbf{N}_{\varepsilon}$. Moreover, by the continuity of $J_{\varepsilon}$ and $\left(s,t\right),$ we have

[TABLE]

and

[TABLE]

if $b$ is sufficiently small. $\square$

Now we may use Lemma 3.1 to find a $(PS)_{\gamma_{i,j}\left(\varepsilon\right)}$ sequence as follows:

Proposition 3.2

For each $1\leq i\leq k,1\leq j\leq\ell$ and $\varepsilon\in\left(0,\varepsilon_{0}\right)$, there exists a sequence $\left\{\left(u_{n},v_{n}\right)\right\}\subset N_{i,j}\left(\varepsilon\right)$ such that

[TABLE]

Proof. By the Ekeland variational principle [15], there exists a minimizing sequence $\left\{\left(u_{n},v_{n}\right)\right\}\subset N_{i,j}\left(\varepsilon\right)$ such that

[TABLE]

and

[TABLE]

for $\left(w_{1},w_{2}\right)\in\mathbf{N}_{\varepsilon}$ and $n\geq n_{0}$, where $n_{0}$ is a sufficiently large constant independent of $\left(w_{1},w_{2}\right)$. Applying Lemma 3.1 for each $\left(u_{n},v_{n}\right),$ we obtain the function

[TABLE]

such that $\left(s_{n}\left(\widetilde{u},\widetilde{v}\right)\left(u_{n}-\widetilde{u}\right),t_{n}\left(\widetilde{u},\widetilde{v}\right)\left(v_{n}-\widetilde{v}\right)\right)\in N_{i,j}\left(\varepsilon\right),$ for $\left(\widetilde{u},\widetilde{v}\right)\in B\left(0;b_{n}\right),$ where $b_{n}>0$. Fix $\left(u,v\right)\in H\setminus\{(0,0)\}$ arbitrarily. For $0<\rho<b_{n},$ let $\left(\phi_{\rho},\psi_{\rho}\right)=\left(\rho u,\rho v\right)\,\left\|\left(u,v\right)\right\|_{H}^{-1}$, $y_{\rho}=s_{n,\rho}u_{n,\rho}$ and $z_{\rho}=t_{n,\rho}v_{n,\rho}.$ Hereafter, $s_{n,\rho}=s_{n}\left(\phi_{\rho},\psi_{\rho}\right)$, $t_{n,\rho}=t_{n}\left(\phi_{\rho},\psi_{\rho}\right)$, $u_{n,\rho}=u_{n}-\phi_{\rho}$ and $v_{n,\rho}=v_{n}-\psi_{\rho}$. Then $\left(y_{\rho},z_{\rho}\right)\in N_{i,j}\left(\varepsilon\right)$. By $\left(\ref{3.5}\right)$, there holds

[TABLE]

Because $(y_{\rho},z_{\rho})\rightarrow(u_{n},v_{n})$ as $\rho\rightarrow 0^{+},$ it follows from (3.6) that

[TABLE]

On the other hand, due to $\left(y_{\rho},z_{\rho}\right)=\left(s_{n,\rho}u_{n,\rho}\,,t_{n,\rho}v_{n,\rho}\right)\in N_{i,j}\left(\varepsilon\right)$, we have

[TABLE]

and

[TABLE]

Then it is easy to verify that

[TABLE]

Combining $\left(\ref{3.7}\right)$ and (3), we obtain

[TABLE]

By $\left(\ref{3.3}\right)$ and (3.4), there exists a positive constant $C$ that is independent of $\rho$ and $n$ such that

[TABLE]

and

[TABLE]

Setting $\rho\rightarrow 0^{+}$ on (3), we obtain

[TABLE]

Here we have used (3.10), (3.11) and the fact that $\left(y_{\rho},z_{\rho}\right)\rightarrow\left(u_{n},v_{n}\right)$ as $\rho\rightarrow 0^{+}$. Therefore,

[TABLE]

Therefore, by (3.12), we may complete the proof of Proposition 3.2. $\square$

To prove Theorem 1.2, we need the (PS) condition of $J_{\varepsilon}$ as follows:

Proposition 3.3

*Assume that $\left\{\left(u_{n},v_{n}\right)\right\}$ is a sequence in $N_{i,j}\left(\varepsilon\right)$ satisfying

$\left(i\right)\ J_{\varepsilon}\left(u_{n},v_{n}\right)\rightarrow\varepsilon^{N}\theta_{\varepsilon}$ as $n\rightarrow\infty,$ where $\theta_{\varepsilon}<\widehat{\alpha}_{1}+\widehat{\alpha}_{2}+\delta_{0};$

$\left(ii\right)\ J_{\varepsilon}^{\prime}\left(u_{n},v_{n}\right)\rightarrow 0\;$strongly in $H^{\ast}$ as $n\rightarrow\infty.$

Then there exists a convergent subsequence such that as $n\rightarrow\infty$, $\left(u_{n},v_{n}\right)\rightarrow\left(u_{0},v_{0}\right)$ strongly in $H,$ where $\left(u_{0},v_{0}\right)\in N_{i,j}\left(\varepsilon\right).$*

Proof. It follows from (1.6) and $(i)$ that

[TABLE]

and $\left\{\left(u_{n},v_{n}\right)\right\}$ is bounded in $H.$ Thus, there exists a convergent subsequence of $\left\{\left(u_{n},v_{n}\right)\right\}$ (denoted as $\left\{\left(u_{n},v_{n}\right)\right\}$ for notation convenience) such that as $n\rightarrow\infty$,

[TABLE]

where $\left(u_{0},v_{0}\right)\in H$. By $0<q\leq p\leq 1$ if $N\leq 4\alpha$, $0<q\leq p<\frac{2\alpha}{N-2\alpha}$ if $N>4\alpha$, then in view of (3.13) and the Sobolev compact embedding, we have

[TABLE]

and

[TABLE]

Now we claim that there exist a subsequence $\left\{\left(u_{n},v_{n}\right)\right\}$ and a sequence $\{z_{n}\}\subset\mathbb{R}^{N}$ such that

[TABLE]

where $d_{0}$ and $R$ are positive constants that are independent of $n.$ Suppose on the contrary. Thus, for all $R>0$,

[TABLE]

Then, similarly to the argument of Lemma I.1 in [34] (see also [43]), we have

[TABLE]

this implies that

[TABLE]

Therefore, by $(ii)$, (3.13), (3.14) and (3.17), the functions $u_{0}$ and $v_{0}$ satisfy

[TABLE]

and

[TABLE]

On the other hand, by hypethesis $\left(ii\right)$ and (3.15), we have

[TABLE]

Furthermore, it follows from (3.15) and (3.17) that

[TABLE]

Thus, by (3.18)–(3.20) and the strong maximum principle for the fractional Laplacian [41], we have $u_{0}\equiv 0$ or $v_{0}\equiv 0$. Without loss of generality, we may assume that $u_{0}\equiv 0$. By Lemma 2.1 and the concentration–compactness principle (cf. [34, 35]), there are positive constants $R,b$ and a sequence $\left\{x_{n}\right\}_{n=1}^{\infty}\subset\mathbb{R}^{N}$ such that

[TABLE]

By $\left(\ref{3.14}\right)$, $\left(\ref{3.21}\right)$ and $u_{0}\equiv 0$, it is easy to verify that $\left\{x_{n}\right\}$ is an unbounded sequence in $\mathbb{R}^{N}.$ Let $\widehat{u}_{n}\left(x\right)=(-x_{n}\ast u_{n})(x)$. Then sequence $\left\{\widehat{u}_{n}\right\}$ is bounded in $H^{\alpha}\left(\mathbb{R}^{N}\right),$ so we may assume that there exists $\widehat{u}_{0}\in H^{\alpha}\left(\mathbb{R}^{N}\right)$ such that

[TABLE]

and

[TABLE]

It follows from $\left(\ref{3.21}\right)$ and $\left(\ref{3.23}\right)$ that $\widehat{u}_{0}\not\equiv 0$ in $\mathbb{R}^{N}.$ Set $w_{n}=\widehat{u}_{n}-\widehat{u}_{0}.$ Then we may divide the proof into the following cases:

Case $I:$ $\int_{\mathbb{R}^{N}}\left(|(-\triangle)^{\frac{\alpha}{2}}w_{n}|^{2}+\lambda_{1}w_{n}^{2}\right)\mathrm{d}x\rightarrow 0$ as $n\rightarrow\infty;$

Case $II:$ $\int_{\mathbb{R}^{N}}\left(|(-\triangle)^{\frac{\alpha}{2}}w_{n}|^{2}+\lambda_{1}w_{n}^{2}\right)\mathrm{d}x\geq b$ for large $n$ and for some constant $b>0.$

Suppose Case $I$ holds. Then

[TABLE]

which implies $\left|\Phi\left(u_{n}\right)\right|\rightarrow\infty$ as $n\rightarrow\infty,$ this contradicts $\Phi\left(u_{n}\right)\in C_{s}\left(z_{1,i}\right).$ Here, we have used the fact that $\left\{x_{n}\right\}$ is a unbounded sequence in $\mathbb{R}^{N}.$

Suppose Case $II$ holds. Since $\ J_{\varepsilon}^{\prime}(u_{n},v_{n})\rightarrow 0\;$strongly in $H^{\ast},$ then by (3.17), we have

[TABLE]

and

[TABLE]

By $\left(\ref{3.24}\right)$ and Brézis–Lieb lemma [5], we obtain

[TABLE]

Because $\int_{\mathbb{R}^{N}}\left(|(-\triangle)^{\frac{\alpha}{2}}\omega_{n}|^{2}+\lambda_{1}w_{n}^{2}\right)dx\geq b$ for large $n,$ it is easy to find a sequence $\left\{s_{n}\right\}\subset\mathbb{R}^{+}$ with $s_{n}\rightarrow 1$ as $n\rightarrow\infty$ such that

[TABLE]

and so

[TABLE]

Analogously,

[TABLE]

Furthermore, $\left\langle J_{\varepsilon}^{\prime}(u_{n},v_{n}),\left(x_{n}\ast\widehat{u}_{0},0\right)\right\rangle\rightarrow 0.$ By $\left(\ref{3.22}\right)$ and $\left(\ref{3.23}\right),$ we have

[TABLE]

which implies that

[TABLE]

It follows from $\left(\ref{1.14}\right),$ $\left(\ref{3.17}\right)$ and Brézis–Lieb lemma [5] that

[TABLE]

which implies that

[TABLE]

However, $\theta_{\varepsilon}<\widehat{\alpha}_{1}+\widehat{\alpha}_{2}+\delta_{0}$ and $0<\delta_{0}\leq\frac{1}{2}\min\{\widehat{\alpha}_{1},\widehat{\alpha}_{2}\}.$ Hence, we find a contradiction and complete the proof of (3.16). Let $\left(\widetilde{u}_{n},\widetilde{v}_{n}\right)=\left(-z_{n}\ast u_{n},-z_{n}\ast v_{n}\right)$ for $n\in\mathbb{N}$, where the operation $\ast$ is defined in (2.25). Then as for (3.13)–(3.14), we have

[TABLE]

where $\left(\widetilde{u}_{0},\widetilde{v}_{0}\right)\in H$. Besides, it follows from $\left(\ref{3.16}\right)$ and $\left(\ref{3.28}\right)$ that

[TABLE]

which implies $\widetilde{u}_{0}^{+}\not\equiv 0$ and $\widetilde{v}_{0}^{+}\not\equiv 0$ in $\mathbb{R}^{N}.$ Because $\beta\left(x\right)$ and $V_{l}(x)$ are uniformly continuous for each $l\in\{1,2\}$, the sequences $\left\{-z_{n}\ast\beta\right\}$ and $\left\{-z_{n}\ast V_{l}\right\}$ are equicontinuous, the Arzelà–Ascoli theorem allows us to assume that $-z_{n}\ast\beta\rightarrow\beta_{\infty}\in C\left(\mathbb{R}^{N}\right)$ and $-z_{n}\ast V_{l}\rightarrow V_{l}^{0}\in C\left(\mathbb{R}^{N}\right)$ uniformly on bounded subsets of $\mathbb{R}^{N}.$ Then

[TABLE]

and

[TABLE]

for all $\varphi_{1},\varphi_{2}\in C_{0}^{\infty}\left(\mathbb{R}^{N}\right)$. Therefore, by (3.25), (3.26) and the hypothesis $(ii)$, we obtain

[TABLE]

i.e., $\left(\widetilde{u}_{0},\widetilde{v}_{0}\right)$ is a nonnegative solution of the following problem

[TABLE]

Hence

[TABLE]

and

[TABLE]

Then in view of Lemma 2.3, we obtain

[TABLE]

where

[TABLE]

for $(u,v)\in H.$ Let $\left(\widehat{u}_{n},\widehat{v}_{n}\right)=\left(u_{n}-z_{n}\ast\widetilde{u}_{0},v_{n}-z_{n}\ast\widetilde{v}_{0}\right),$ i.e.

[TABLE]

By (3.25), it is evident that

[TABLE]

and

[TABLE]

Furthermore, since $-z_{n}\ast\beta\rightarrow\beta_{\infty}(x)$ and $-z_{n}\ast V_{l}\rightarrow V_{l}^{0}(x)$ uniformly on bounded subsets for $l=1,2$, we may conclude that

[TABLE]

and

[TABLE]

On the other hand, it follows from Brézis–Lieb Lemma (cf.[5]) that

[TABLE]

and

[TABLE]

By (3.28), (3.29) and $\left(\ref{3.33}\right)-\left(\ref{3.39}\right)$, we have

[TABLE]

[TABLE]

and

[TABLE]

Now we want to show the strong convergence of $\left\|\left(\widehat{u}_{n},\widehat{v}_{n}\right)\right\|_{H}\rightarrow 0$ as $n\rightarrow\infty.$ Suppose not, then $\left\|\left(\widehat{u}_{n},\widehat{v}_{n}\right)\right\|_{H}\geq c_{0}$ for large $n$, where $c_{0}$ is a positive constant independent of $n$. Hence, it follows from (3.38), (3.39), Lemma 2.1 and Lemma 2.3 that

[TABLE]

for $n$ sufficiently large. Thus, in view of (3.30), (3.40) and (3.41), we obtain

[TABLE]

which is a contradiction with $\theta_{\varepsilon}<\widehat{\alpha}_{1}+\widehat{\alpha}_{2}+\delta_{0}$. Hence, the strong convergence $\left\|\left(\widehat{u}_{n},\widehat{v}_{n}\right)\right\|_{H}\rightarrow 0$ as $n\rightarrow\infty$ holds, i.e.

[TABLE]

It follows from (2.25) and (3.42) that

[TABLE]

i.e.

[TABLE]

Since $\Phi\left(u_{n}\right)\in C_{s}\left(z_{1,i}\right)$ for $n\in\mathbb{N}$, then (3.43) implies that $\{z_{n}\}$ is a bounded sequence in $\mathbb{R}^{N}$. As a result, we may assume $z_{n}\rightarrow z_{0}\in\mathbb{R}^{N}$ as $n\rightarrow\infty$. Thus, (3.42) becomes

[TABLE]

Since $\left(u_{n},v_{n}\right)\in N_{i,j}\left(\varepsilon\right)$ for $n\in\mathbb{N}$, then by (3.44), we have $\left(z_{0}\ast\widetilde{u}_{0},z_{0}\ast\widetilde{v}_{0}\right)\in N_{i,j}\left(\varepsilon\right)\cup O_{i,j}\left(\varepsilon\right).$ In view of the assumption $\left(i\right)$, (3.1) and (3.44), we obtain

[TABLE]

so $\left(z_{0}\ast\widetilde{u}_{0},z_{0}\ast\widetilde{v}_{0}\right)\notin O_{i,j}\left(\varepsilon\right),$ i.e., $\left(z_{0}\ast\widetilde{u}_{0},z_{0}\ast\widetilde{v}_{0}\right)\in N_{i,j}\left(\varepsilon\right)$. Therefore, we may complete the proof of Proposition 3.3. $\square$

Theorem 3.4

For each $0<\varepsilon<\varepsilon_{0},1\leq i\leq k$ and $1\leq j\leq\ell,$ problem $\left(P_{\varepsilon}\right)$ has a positive solution $\left(\widehat{u}_{\varepsilon,i},\widehat{v}_{\varepsilon,j}\right)\in N_{i,j}\left(\varepsilon\right).$

Proof. From Propositions 3.2 and 3.3, we can prove Theorem 3.4. $\square$

4 Proof of Theorem 1.2

Throughout this section, we define

[TABLE]

The main goal of this section is to show

[TABLE]

where $0<\delta_{\varepsilon}\rightarrow 0$ as $\varepsilon\rightarrow 0^{+}$. Here $cat(\cdot)$ is the standard Lusternik–Schnirelman category (cf. [6]).

From Lemma 4.1 in [33] and Lemma 2.2 in [31], we obtain

Lemma 4.1

Let $\left(u_{\varepsilon},v_{\varepsilon}\right)$ be a constrained critical point of $J_{\varepsilon}$ on $\mathbf{N}_{\varepsilon}$ with

[TABLE]

Then $\nabla J_{\varepsilon}\left(u_{\varepsilon},v_{\varepsilon}\right)=0$ on $H^{\ast}.$

Therefore, from Theorem 2.3 in [2], Proposition 3.3 and Lemma 4.1, we have

Proposition 4.2

Suppose $cat\left(M_{i,j}(\varepsilon,\delta_{0})\right)\geq k,$ where $k\in\mathbb{N}$ and

[TABLE]

Then the functional $J_{\varepsilon}$ has at least $k$ critical points in $M_{i,j}(\varepsilon,\delta_{0})$.

Now we define a function by

[TABLE]

This leads to the following results.

Lemma 4.3

Suppose that $z_{1,i}=z_{2,j}.$ Then there exist $\widetilde{\varepsilon}_{0}>0$ and $0<\delta_{1}\leq\delta_{0}$ such that for every $0<\varepsilon<\widetilde{\varepsilon}_{0},$ there holds

[TABLE]

Proof. We may prove this by contradiction. Suppose that there exists $\varepsilon_{n}>0$, for $n\in\mathbb{N},\varepsilon_{n}\rightarrow 0$ as $n\rightarrow\infty$ and $\left(u_{n},v_{n}\right)\in N_{i,j}\left(\varepsilon_{n}\right)$ such that

[TABLE]

and

[TABLE]

To obtain a contradiction with (4.2), it is sufficient to show that

[TABLE]

Then $\Phi\left(u_{n}\right)=\Phi\left(v_{n}\right)$ and

[TABLE]

Let $\widetilde{u}_{n}\left(x\right)=u_{n}\left(\varepsilon_{n}x\right)$ and $\widetilde{v}_{n}\left(x\right)=v_{n}\left(\varepsilon_{n}x\right).$ Clearly, $\Phi\left(\widetilde{u}_{n}\right)=\Phi\left(\widetilde{v}_{n}\right)$ and $\Phi\left(\widetilde{u}_{n}\right),\Phi\left(\widetilde{v}_{n}\right)\in C_{s/\varepsilon_{n}}\left(z_{1,i}\right).$ By a similar argument to the proof of Lemma 2.5 and the proof of Lemma 3.3 in [36], we can obtain

[TABLE]

Moreover, we can conclude that $\widehat{I}_{1}\left(\widetilde{u}_{n}\right)=\widehat{\alpha}_{1}+o\left(1\right),\widehat{I}_{2}\left(\widetilde{v}_{n}\right)=\widehat{\alpha}_{2}+o\left(1\right)$ and

[TABLE]

Thus, there exist $t_{n},s_{n}>0$ with $t_{n},s_{n}\rightarrow 1$ as $n\rightarrow\infty$ such that $\overline{u}_{n}=t_{n}\widetilde{u}_{n}\in\widehat{\mathbf{M}}_{1}$ and $\overline{v}_{n}=s_{n}\widetilde{v}_{n}\in\widehat{\mathbf{M}}_{2},$ which implies that $\widehat{I}_{1}\left(\overline{u}_{n}\right)=\widehat{\alpha}_{1}+o\left(1\right)$ and $\widehat{I}_{2}\left(\overline{v}_{n}\right)=\widehat{\alpha}_{2}+o\left(1\right)$. Furthermore, by the Ekeland varitional principle [15], we may assume that $\left\{\overline{u}_{n}\right\}$ is a (PS)${}_{\widehat{\alpha}_{1}}$–sequence of $\widehat{I}_{1}$ in $H^{\alpha}\left(\mathbb{R}^{N}\right)$ and that $\left\{\overline{v}_{n}\right\}$ is a (PS)${}_{\widehat{\alpha}_{2}}$–sequence of $\widehat{I}_{2}$ in $H^{\alpha}\left(\mathbb{R}^{N}\right)$, respectively. Applying the concentration-compactness principle of Lions [34, 35], there are positive constants $R,b_{0}$ and two sequences $\left\{x_{n}\right\},\left\{y_{n}\right\}\subset\mathbb{R}^{N}$ such that

[TABLE]

Let $\widehat{u}_{n}(x)=\overline{u}_{n}\left(x+x_{n}\right)$ for $x\in\mathbb{R}^{N}$ and $n\in\mathbb{N}$. Then, due to translation invariance, it is obvious that $\left\{\widehat{u}_{n}\right\}$ is also a (PS) ${}_{\widehat{\alpha}_{1}}$–sequence of $\widehat{I}_{1}$ in $H^{\alpha}\left(\mathbb{R}^{N}\right).$ Hence by $\left(\ref{4.5}\right)$, we may assume that there exists a subsequence of $\left\{\widehat{u}_{n}\right\}$ such that

[TABLE]

Now we set $w_{n}=\widehat{u}_{n}-u_{0}.$ Then it follows from $(\ref{4.6})$ and Brézis–Lieb Lemma [5] that

[TABLE]

and

[TABLE]

Combining $\left(\ref{4.7}\right),\left(\ref{4.8}\right)$ and $\left\{\widehat{u}_{n}\right\}$ is a (PS)${}_{\widehat{\alpha}_{1}}$–sequence of $\widehat{I}_{1}$ in $H^{\alpha}\left(\mathbb{R}^{N}\right),$ we obtain

[TABLE]

and

[TABLE]

Consequently, $\widehat{u}_{n}\rightarrow u_{0}$ strongly in $H^{\alpha}\left(\mathbb{R}^{N}\right)$ and $\widehat{I}_{1}\left(u_{0}\right)=\widehat{\alpha}_{1}.$ Similarly, $\widehat{v}_{n}\rightarrow v_{0}$ strongly in $H^{\alpha}\left(\mathbb{R}^{N}\right)$ and $\widehat{I}_{2}\left(v_{0}\right)=\widehat{\alpha}_{2},$ where $\widehat{v}_{n}(x)=\overline{v}_{n}\left(x+y_{n}\right)$ for $x\in\mathbb{R}^{N}$ and $n\in\mathbb{N}$. Therefore, we obtain

[TABLE]

as $n\rightarrow\infty$, where $u_{0}$ and $v_{0}$ are positive solutions of Eq. $\left(\widehat{E}_{1}\right)$ and Eq. $\left(\widehat{E}_{2}\right),$ respectively (cf. [3] and [20]). From $\Phi\left(\widetilde{u}_{n}\right),\Phi\left(\widetilde{v}_{n}\right)\in C_{s/\varepsilon_{n}}\left(0\right)$ $\widehat{u}_{n}(x)=\overline{u}_{n}\left(x+x_{n}\right),\widehat{v}_{n}(x)=\overline{v}_{n}\left(x+y_{n}\right)$ and $\left(\ref{4.9}\right),$ we have

[TABLE]

and

[TABLE]

and so

[TABLE]

Without loss of generality, we may assume $\varepsilon_{n}x_{n}\rightarrow x_{0}\in\overline{C_{s}\left(z_{1,i}\right)},\varepsilon_{n}y_{n}\rightarrow y_{0}\in\overline{C_{s}\left(z_{2,j}\right)}.$ By condition $\left(D_{2}\right),$ we must have that

[TABLE]

and

[TABLE]

this implies that $x_{0}=y_{0}=z_{1,i}=z_{2,j}.$ Moreover,

[TABLE]

and

[TABLE]

which implies that

[TABLE]

It follows from condition $\left(D_{4}\right),$ $\varepsilon_{n}x_{n}\rightarrow 0$ as $n\rightarrow\infty,\left(\ref{2.34}\right)$ and $(\ref{4.9})$ that

[TABLE]

for $n$ sufficiently large. However, by (4.3) and Lemma 2.2, we obtain

[TABLE]

which is a contradiction with (4.11). This completes the proof. $\square$

For $0<\varepsilon<\widetilde{\varepsilon}_{0}$, a map $F_{\varepsilon}^{\left(i,j\right)}:S^{N-1}\rightarrow H$ can be written as

[TABLE]

where $\widetilde{\varepsilon}_{0}$ is given by Lemma 4.3. Here $\left(t_{\varepsilon,i}u_{\varepsilon,i},s_{\varepsilon,j}v_{\varepsilon,j}\right)$ is as in the proof of Lemma 2.4. Note that $\left(t_{\varepsilon,i},s_{\varepsilon,j}\right)\rightarrow(1,1)$ as $\varepsilon\rightarrow 0^{+}$ and by $\left(\ref{2.28}\right)$

[TABLE]

Then, from (4.12) and Lemma 2.3, we obtain

[TABLE]

where

[TABLE]

Here we have used the facts that the map $F_{\varepsilon}$ is continuous and that the set $F_{\varepsilon}\left(S^{N-1}\right)$ is compact. As a result, (4.13) and (4.14) imply:

Lemma 4.4

Suppose that $z_{1,i}=z_{2,j}.$ Then there exists $0<\sigma_{\varepsilon}<\delta_{\varepsilon}$ such that

[TABLE]

By (4.14) and Lemma 4.3, we may define

[TABLE]

by

[TABLE]

Then we can prove the following results.

Lemma 4.5

Suppose that $z_{1,i}=z_{2,j}.$ Then there exists a positive number $\varepsilon_{\ast}\leq\widetilde{\varepsilon}_{0}$ such that for $0<\varepsilon<\varepsilon_{\ast},$ the map $G_{\varepsilon}^{\left(i,j\right)}\circ F_{\varepsilon}^{\left(i,j\right)}:S^{N-1}\rightarrow S^{N-1}$ is homotopic to the identity.

Proof. Let $\overline{G}_{\varepsilon}^{\left(i,j\right)}:\Theta\rightarrow S^{N-1}$ satisfy

[TABLE]

where $\Theta=\left\{\left(u,v\right)\in H\backslash\left\{0,0\right\}:\left|h(u,v)\right|>0\right\}.$ By Lemma 4.3, one may regard the map $\overline{G}_{\varepsilon}^{\left(i,j\right)}$ as an extension of $G_{\varepsilon}^{\left(i,j\right)}$. By $\left(t_{\varepsilon,i},s_{\varepsilon,j}\right)\rightarrow(1,1)$ as $\varepsilon\rightarrow 0^{+}$, for $\theta\in\left[0,1/2\right)$,

[TABLE]

Hence as for the proof of Lemma 4.3, there exists a positive number $\varepsilon_{\ast}\leq\widetilde{\varepsilon}_{0}$ such that for $0<\varepsilon<\varepsilon_{\ast},$ there holds,

[TABLE]

On the other hand, for $e\in S^{N-1}$ and $\theta\in\left[1/2,1\right)$, we may set

[TABLE]

and

[TABLE]

where $z_{1,i}$ and $z_{2,j}$ are minimum points of $V_{1}(x)$ and $V_{2}(x)$, respectively. Then it is easy to verify that

[TABLE]

which implies that

[TABLE]

for $e\in S^{N-1}$ and $\theta\in\left[1/2,1\right)$. Consequently, due to $z_{1,i}=z_{2,j}$,

[TABLE]

One may remark that to assure $|h(f_{\varepsilon,i},g_{\varepsilon,j})|>0$ for $e\in S^{N-1}$ and $\theta\in\left[1/2,1\right)$, it is necessary for $z_{1,i}$ to be a common minimum point of the $V_{1}(x)$ and $V_{2}(x)$. Otherwise, if $V_{1}(x)$ and $V_{2}(x)$ have two different minimum points at $z_{1,i}$ and $z_{2,j}$, one may use (4.16) to find some $e\sim\frac{z_{1,i}-z_{2,j}}{|z_{1,i}-z_{2,j}|}\in S^{N-1}$ and $\theta\sim 1-\frac{\sqrt{\varepsilon}}{2|z_{1,i}-z_{2,j}|}\in\left[1/2,1\right)$ such that $h(f_{\varepsilon,i},g_{\varepsilon,j})=0$ and $(f_{\varepsilon,i},g_{\varepsilon,j})\not\in\Theta$.

By (4.15) and (4.17), we may define

[TABLE]

by

[TABLE]

Then $\zeta_{\varepsilon}^{\left(i,j\right)}\left(0,e\right)=\overline{G}_{\varepsilon}^{\left(i,j\right)}\left(F_{\varepsilon}^{\left(i,j\right)}\left(e\right)\right)=G_{\varepsilon}^{\left(i,j\right)}\left(F_{\varepsilon}^{\left(i,j\right)}\left(e\right)\right)$ and $\zeta_{\varepsilon}^{\left(i,j\right)}\left(1,e\right)=e.$ It is easy to verify that

[TABLE]

On the other hand, by (4.16) and $z_{1,i}=z_{2,j}$, we have

[TABLE]

Consequently, by (4.18) and (4.19), we know that $\zeta_{\varepsilon}^{\left(i,j\right)}\in C\left(\left[0,1\right]\times S^{N-1},S^{N-1}\right)$ and

[TABLE]

[TABLE]

provided that $0<\varepsilon<\varepsilon_{\ast}.$ Therefore, we may complete the proof. $\square$

By Lemma 4.5 and Adachi–Tanaka’s Lemma 2.5 (cf. [1]), we obtain

[TABLE]

for $z_{1,i}=z_{2,j}$ and $0<\varepsilon<\varepsilon_{\ast}$. Thus, by Proposition 4.2, $J_{\varepsilon}$ has at least two critical points in $M_{i,j}(\varepsilon,\delta_{\varepsilon})$, where $M_{i,j}(\varepsilon,\delta_{\varepsilon})=\left[\varepsilon^{-N}\,J_{\varepsilon}\leq\widehat{\alpha}_{1}+\widehat{\alpha}_{2}+\delta_{\varepsilon}\right]_{i,j}\,.$ This implies that problem $\left(P_{\varepsilon}\right)$ has two positive solutions $\left(\widehat{u}_{\varepsilon,i},\widehat{v}_{\varepsilon,j}\right),\left(\widetilde{u}_{\varepsilon,i},\widetilde{v}_{\varepsilon,j}\right)\in N_{i,j}\left(\varepsilon\right)\,$ such that $\left|h\left(\widehat{u}_{\varepsilon,i},\widehat{v}_{\varepsilon,j}\right)\right|>0$ and $\left|h\left(\widetilde{u}_{\varepsilon,i},\widetilde{v}_{\varepsilon,j}\right)\right|>0.$ So we may conclude that:

Theorem 4.6

Suppose that $z_{1,i}=z_{2,j}.$ Then for each $0<\varepsilon<\varepsilon_{\ast},$ problem $\left(P_{\varepsilon}\right)$ has at least two positive solutions $\left(\widehat{u}_{\varepsilon,i},\widehat{v}_{\varepsilon,j}\right),\left(\widetilde{u}_{\varepsilon,i},\widetilde{v}_{\varepsilon,j}\right)\in M_{i,j}(\varepsilon,\delta_{\varepsilon})$ such that

[TABLE]

We are now ready to prove Theorem 1.2. By Theorem 3.4 and Theorem 4.6, we can conclude that problem $\left(P_{\varepsilon}\right)$ has at least $k\times\ell+m$ positive solutions. As for the proofs of $(\ref{4.9})$ and $(\ref{4.10}),$ we may use $(\ref{4.16})$ and a similar argument to that in the proof of [32, Theorem 2.4], we can conclude that part $\left(ii\right)$ is holds.

5 Proof of Theorem 1.3

By Lemmas 2.2 and 2.4, there exists a positive function $\delta_{\varepsilon}$ with $\delta_{\varepsilon}\rightarrow 0$ as $\varepsilon\rightarrow 0$ such that the sublevel set

[TABLE]

is nonempty. Then we have the following results.

Lemma 5.1

Assume that $\liminf\limits_{|x|\rightarrow\infty}\,V_{l}(x)\equiv V_{l,\infty}>\lambda_{l}$ for all $l=1,2.$ Then

[TABLE]

where $\mathbf{B}=\underset{1\leq i\leq k,1\leq j\leq\ell}{\cup}\left[C_{s/2}\left(z_{1,i}\right)\times C_{s/2}\left(z_{2,j}\right)\right].$

Proof. Let $\varepsilon_{n}\rightarrow 0$ as $n\rightarrow\infty;$ for any $n\in\mathbb{N}$, there exists $\left(u_{n},v_{n}\right)\in\widehat{M}(\varepsilon,\delta_{\varepsilon_{n}})$ such that

[TABLE]

To prove $\left(\ref{5.1}\right),$ it is sufficient to find points $\left(x_{n},y_{n}\right)\in\mathbf{B}$ such that

[TABLE]

possibly up to subsequence. For any $n\in\mathbb{N},$ let $\widetilde{u}_{n}\left(x\right)=u_{n}\left(\varepsilon_{n}x\right)$ and $\widetilde{v}_{n}\left(x\right)=v_{n}\left(\varepsilon_{n}x\right).$ Then similar to the argument in the proof of Lemma 2.5, we have

[TABLE]

and

[TABLE]

Furthermore, there exists $\left\{\left(\widetilde{x}_{n},\widetilde{y}_{n}\right)\right\}\subset\mathbb{R}^{N}\times\mathbb{R}^{N}$ such that

$(i)\ \widetilde{u}_{n}\left(\cdot+\widetilde{x}_{n}\right)$ converges strongly in $H^{\alpha}\left(\mathbb{R}^{N}\right)$ to $\widehat{u}_{0},$ a positive ground state solution of $\left(\widehat{E}_{1}\right);$

$(ii)\ \widetilde{v}_{n}\left(\cdot+\widetilde{y}_{n}\right)$ converges strongly in $H^{\alpha}\left(\mathbb{R}^{N}\right)$ to $\widehat{v}_{0},$ a positive ground state solution of $\left(\widehat{E}_{2}\right).$

Let us prove that $\left\{\left(\varepsilon_{n}\widetilde{x}_{n},\varepsilon_{n}\widetilde{y}_{n}\right)\right\}$ is a bounded sequence in $\mathbb{R}^{N}\times\mathbb{R}^{N}.$ Arguing by contradiction, we assume that $\left|\varepsilon_{n}\widetilde{x}_{n}\right|\rightarrow\infty$ as $n\rightarrow\infty.$ Then

[TABLE]

which is a contradiction. Thus, $\left\{\left(\varepsilon_{n}\widetilde{x}_{n},\varepsilon_{n}\widetilde{y}_{n}\right)\right\}$ is bounded and converges to some $\left(x_{0},y_{0}\right)$ (up to a subsequence). We are left to prove $\left(x_{0},y_{0}\right)=\left(z_{1,i},z_{2,j}\right)$ for some $1\leq i\leq k,1\leq j\leq\ell.$ Because

[TABLE]

and

[TABLE]

which implies that $V_{1,\infty}\left(x_{0}\right)=\lambda_{1}$ and $V_{2,\infty}\left(y_{0}\right)=\lambda_{2},$ that is, there exists $1\leq i\leq k,1\leq j\leq\ell$ such that $\left(x_{0},y_{0}\right)=\left(z_{1,i},z_{2,j}\right).$ Take $x_{n}=\varepsilon_{n}\widetilde{x}_{n}$ and $y_{n}=\varepsilon_{n}\widetilde{y}_{n}.$ Then $\left(\ref{5.2}\right)$ holds. $\square$

Lemma 5.2

Assume that $\liminf\limits_{|x|\rightarrow\infty}\,V_{l}(x)\equiv V_{l,\infty}>\lambda_{l}$ for all $l=1,2.$ Then there exists a positive number $\varepsilon_{\ast\ast}$ such that for every $\varepsilon<\varepsilon_{\ast\ast},$ we have $\widehat{M}(\varepsilon,\delta_{\varepsilon})\subset\underset{1\leq i\leq k,1\leq j\leq\ell}{\cup}N_{i,j}\left(\varepsilon\right).$

Proof. By Lemma 5.1, we can find $\varepsilon_{\ast\ast}>0$ such that for every $\varepsilon<\varepsilon_{\ast\ast},$ we have

[TABLE]

or

[TABLE]

where $\mathbf{B}=\underset{1\leq i\leq k,1\leq j\leq\ell}{\cup}\left[C_{s/2}\left(z_{1,i}\right)\times C_{s/2}\left(z_{2,j}\right)\right].$ This implies that

[TABLE]

Thus, $\widehat{M}(\varepsilon,\delta_{\varepsilon})\subset\underset{1\leq i\leq k,1\leq j\leq\ell}{\cup}N_{i,j}\left(\varepsilon\right).$ $\square$

Lemma 5.3

Assume that $\lim\limits_{|x|\rightarrow\infty}V_{l}(x)=\lambda_{l}$ for all $l=1,2.$ Then

[TABLE]

Proof. To prove Lemma 5.3, we may define the test functions $u_{R}$ and $v_{R}$ by

[TABLE]

where $R>1$, $e\in S^{N-1}=\left\{x\in\mathbb{R}^{N}:\left|x\right|=1\right\}$, $\widehat{\omega}_{l}$ is the unique positive radial solution of Eq. $\left(\widehat{E}_{l}\right)$ for $l=1,2$, and the function $\psi_{R}\in C^{1}\left(\mathbb{R}^{N},\left[0,1\right]\right)$ with compact support satisfies

[TABLE]

and $\left|\nabla\psi_{R}\right|\leq 2$ in $\mathbb{R}^{N}.$ Notice that $\widehat{I}_{l}\left(\widehat{\omega}_{l}\right)=\widehat{\alpha}_{1}$ for $l=1,2.$ Obviously,

[TABLE]

By $(\ref{5.3}),$ it is easy to find two positive numbers $t_{R}$ and $s_{R}$ such that

[TABLE]

and

[TABLE]

It follows from $(\ref{5.4})-(\ref{5.6})$ that $\left(t_{R}u_{R},s_{R}v_{R}\right)\in\mathbf{N}_{\varepsilon}$ and

[TABLE]

We want to show $\left(t_{R},s_{R}\right)\rightarrow\left(1,1\right)$ as $R\rightarrow\infty.$ Using $(\ref{5.3}),(\ref{5.5})$ and change of variables, it is easy to check that

[TABLE]

where $o_{R}\left(1\right)\rightarrow 0$ as $R\rightarrow\infty.$ Hence, $t_{R}\rightarrow 1$ as $R\rightarrow\infty$ because $\lim_{|x|\rightarrow\infty}V_{1}(x)=\lambda_{1}$ and

[TABLE]

Similarly, we may obtain $s_{R}\rightarrow 1$ as $R\rightarrow\infty$. Consequently, due to $\lim\limits_{|x|\rightarrow\infty}V_{l}(x)=\lambda_{l}$ for $l=1,2$, we have

[TABLE]

This implies

[TABLE]

Thus, by Lemma 2.3, we have $\varepsilon^{-N}c_{\varepsilon}=\widehat{\alpha}_{1}+\widehat{\alpha}_{2}$ for all $\varepsilon>0.$ $\square$

We are now ready to prove Theorem 1.3. Theorem 1.3 can be obtained directly from Lemma 2.3, Lemma 5.2 and Lemma 5.3.