Critical system involving fractional Laplacian

TL;DR

This paper investigates a critical fractional Laplacian system, establishing conditions for the existence or nonexistence of positive least energy solutions using the Nehari manifold approach.

Contribution

It introduces new existence and nonexistence results for positive solutions of a fractional Laplacian system with critical nonlinearity, under specific conditions.

Findings

Established existence of positive least energy solutions under certain conditions.

Proved nonexistence of solutions in some parameter regimes.

Applied Nehari manifold method to analyze the system.

Abstract

In this paper, we study the following critical system with fractional Laplacian: \begin{equation*} \begin{cases} (-\Delta)^{s}u= \mu_{1}|u|^{2^{\ast}-2}u+\frac{\alpha\gamma}{2^{\ast}}|u|^{\alpha-2}u|v|^{\beta} \ \ \ \text{in} \ \ \mathbb{R}^{n}, (-\Delta)^{s}v= \mu_{2}|v|^{2^{\ast}-2}v+\frac{\beta\gamma}{2^{\ast}}|u|^{\alpha}|v|^{\beta-2}v\ \ \ \ \text{in} \ \ \mathbb{R}^{n}, u,v\in D_{s}(\mathbb{R}^{n}). \end{cases} \end{equation*} By using the Nehari\ manifold,\ under proper conditions, we establish the existence and nonexistence of positive least energy solution of the system.

Equations

$$\{\begin{array}{l}(-\Delta {)}^{s}u={\mu }_1|u{|}^{2^{\ast }-2}u+\frac{\alpha \gamma }{2^{\ast }}|u{|}^{\alpha -2}u|v{|}^{\beta }\text{in}{\mathbb{R}}^n,\\ (-\Delta {)}^{s}v={\mu }_2|v{|}^{2^{\ast }-2}v+\frac{\beta \gamma }{2^{\ast }}|u{|}^{\alpha }|v{|}^{\beta -2}v\text{in}{\mathbb{R}}^n,\\ u,v\in D_s({\mathbb{R}}^n).\end{array}$$

$$\begin{cases}(-\Delta)^{s}u=\lambda u^{q}+u^{\frac{n+s}{n-s}}$$

&\text{in}\ \ \Omega,\\ u=0&\text{on}\ \partial\Omega,\\ \end{cases}$$

$$\begin{cases}(-\Delta)^{s}u=|u|^{2^{\ast}-2}u+f(x)$$

&\text{in}\ \ \Omega,\\ u=0&\text{on}\ \partial\Omega,\\ \end{cases}$$

$$\begin{cases}(-\Delta)^{s}u=\lambda|u|^{q-2}u+{\frac{2\alpha}{\alpha+\beta}}|u|^{\alpha-2}u|v|^{\beta}$$

&\text{in}\ \ \Omega,\\ (-\Delta)^{s}v=\mu|v|^{q-2}v+{\frac{2\beta}{\alpha+\beta}}|u|^{\alpha}|v|^{\beta-2}v&\text{in}\ \ \Omega,\\ u=0,v=0&\text{on}\ \partial\Omega,\end{cases}$$

$$(-\Delta {)}^{s}u(x)=C(n,s)P.V.{\int }_{R^n}\frac{u(x)-u(y)}{|y-x{|}^{n+2s}}dy,$$

$$C(n,s)={\left({\int }_{R^n}\frac{1-\cos (\varsigma )}{|\varsigma {|}^{n+2s}}d\varsigma \right)}^{-1}=2^{2s}{\pi }^{-\frac{n}{2}}\frac{\Gamma (\frac{n+2s}{2})}{\Gamma (2-s)}s(1-s),0<s<1.$$

$$\begin{cases}(-\Delta)^{s}u=\mu_{1}|u|^{2^{\ast}-2}u+\frac{\alpha\gamma}{2^{\ast}}|u|^{\alpha-2}u|v|^{\beta}$$

&\text{in}\ \ \mathbb{R}^{n},\\ (-\Delta)^{s}v=\mu_{2}|v|^{2^{\ast}-2}v+\frac{\beta\gamma}{2^{\ast}}|u|^{\alpha}|v|^{\beta-2}v&\text{in}\ \ \mathbb{R}^{n},\\ u,v\in D_{s}(\mathbb{R}^{n}),\end{cases}$$

$${\mu }_1,{\mu }_2>0,2^{\ast }=\frac{2n}{n-2s},n>2s,\alpha ,\beta >1,\text{mbox}and\alpha +\beta =2^{\ast }.$$

$$\Vert u{\Vert }_{D_s({\mathbb{R}}^n)}^2=\frac{C(n,s)}{2}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}\frac{|u(x)-u(y){|}^2}{|y-x{|}^{n+2s}}dxdy.$$

$$S_s=\underset{u\in D_s({\mathbb{R}}^n)\setminus 0}{\inf }\frac{\Vert u{\Vert }_{D_s({\mathbb{R}}^n)}^2}{({\int }_{R^n}|u{|}^{2^{\ast }}dx{)}^{\frac{2}{2^{\ast }}}}$$

$$S_s=\frac{\Vert {\tilde{u}}_{ϵ,y}|{|}_{D_s({\mathbb{R}}^n)}^2}{({\int }_{{\mathbb{R}}^n}|{\tilde{u}}_{ϵ,y}{|}^{2^{\ast }}dx{)}^{\frac{2}{2^{\ast }}}}.$$

$${\overline{u}}_{ϵ,y}(x)=\frac{{\tilde{u}}_{ϵ,y}(x)}{||{\tilde{u}}_{ϵ,y}|{|}_{2^{\ast }}}.$$

$$(-\Delta {)}^{s}u=|u{|}^{2^{\ast }-2}u\text{in}{\mathbb{R}}^n$$

$$||U_{\epsilon ,y}|{|}_{D_s({\mathbb{R}}^n)}^2={\int }_{{\mathbb{R}}^n}|U_{\epsilon ,y}{|}^{2^{\ast }}dx=(S_s{)}^{\frac{n}{2s}}.$$

$$E(u,v)=\frac{1}{2}(||u|{|}_{D_s({\mathbb{R}}^n)}^2+||v|{|}_{D_s({\mathbb{R}}^n)}^2)-\frac{1}{2^{\ast }}{\int }_{{\mathbb{R}}^n}({\mu }_1|u{|}^{2^{\ast }}+{\mu }_2|v{|}^{2^{\ast }}+\gamma |u{|}^{\alpha }|v{|}^{\beta })dx.$$

$$\mathbb{N}=\{(u,v)$$

$$||u|{|}_{D_s({\mathbb{R}}^n)}^2={\int }_{{\mathbb{R}}^n}({\mu }_1|u{|}^{2^{\ast }}+\frac{\alpha \gamma }{2^{\ast }}|u{|}^{\alpha }|v{|}^{\beta })dx,$$

$$||v|{|}_{D_s({\mathbb{R}}^n)}^2={\int }_{{\mathbb{R}}^n}({\mu }_2|v{|}^{2^{\ast }}+\frac{\beta \gamma }{2^{\ast }}|u{|}^{\alpha }|v{|}^{\beta })dx\}.$$

$$A:=\underset{(u,v)\in \mathbb{N}}{\inf }E(u,v)$$

$$=\underset{(u,v)\in \mathbb{N}}{\inf }\frac{s}{n}{\int }_{{\mathbb{R}}^n}({\mu }_1|u{|}^{2^{\ast }}+{\mu }_2|v{|}^{2^{\ast }}+\gamma |u{|}^{\alpha }|v{|}^{\beta })dx.$$

$$\{\begin{array}{l}{\mu }_1{k}^{\frac{2^{\ast }-2}{2}}+\frac{\alpha \gamma }{2^{\ast }}{k}^{\frac{\alpha -2}{2}}{l}^{\frac{\beta }{2}}=1,\\ {\mu }_2{l}^{\frac{2^{\ast }-2}{2}}+\frac{\beta \gamma }{2^{\ast }}{k}^{\frac{\alpha }{2}}{l}^{\frac{\beta -2}{2}}=1,\\ k,l>0.\end{array}$$

$$0<\gamma \le \frac{4ns}{(n-2s{)}^2}min\{\frac{{\mu }_1}{\alpha }(\frac{\alpha -2}{\beta -2}{)}^{\frac{\beta -2}{2}},\frac{{\mu }_2}{\beta }(\frac{\alpha -2}{\beta -2}{)}^{\frac{\alpha -2}{2}}\},$$

$$\gamma \ge \frac{4ns}{(n-2s{)}^2}max\{\frac{{\mu }_1}{\alpha }(\frac{2-\beta }{2-\alpha }{)}^{\frac{2-\beta }{2}},\frac{{\mu }_2}{\beta }(\frac{2-\alpha }{2-\beta }{)}^{\frac{2-\alpha }{2}}\},$$

$${\gamma }_1\in (0,\frac{4ns}{(n-2s{)}^2}max\{\frac{{\mu }_1}{\alpha }(\frac{2-\beta }{2-\alpha }{)}^{\frac{2-\beta }{2}},\frac{{\mu }_2}{\beta }(\frac{2-\alpha }{2-\beta }{)}^{\frac{2-\alpha }{2}}],$$

$$\gamma >={\gamma }_0:\triangleq \frac{4ns}{(n-2s{)}^2}max\{\frac{{\mu }_1}{\alpha }(\frac{2-\beta }{2-\alpha }{)}^{\frac{2-\beta }{2}},\frac{{\mu }_2}{\beta }(\frac{2-\alpha }{2-\beta }{)}^{\frac{2-\alpha }{2}}\},$$

$${\mathbb{N}}_1=\{$$

$$G_1(u,v)=||u|{|}_{D_s({\mathbb{R}}^n)}^2-{\int }_{{\mathbb{R}}^n}({\mu }_1|u{|}^{2^{\ast }}+\frac{\alpha \gamma }{2^{\ast }}|u{|}^{\alpha }|v{|}^{\beta })dx=0\},$$

$${\mathbb{N}}_2=\{$$

$$G_2(u,v)=||v|{|}_{D_s({\mathbb{R}}^n)}^2-{\int }_{{\mathbb{R}}^n}({\mu }_2|v{|}^{2^{\ast }}+\frac{\beta \gamma }{2^{\ast }}|u{|}^{\alpha }|v{|}^{\beta })dx=0\}.$$

$$E^{\prime }(u,v)({\phi }_1,{\phi }_2)$$

$$-{\int }_{{\mathbb{R}}^n}({\mu }_1|u{|}^{2^{\ast }-2}u{\phi }_1+{\mu }_2|v{|}^{2^{\ast }-2}v{\phi }_2)dx$$

$$-\frac{\gamma }{2^{\ast }}{\int }_{{\mathbb{R}}^n}(\alpha |u{|}^{\alpha -2}u{\phi }_1|v{|}^{\beta }+\beta |u{|}^{\alpha }|v{|}^{\beta -2}v{\phi }_2)dx,$$

$$G_1^{\prime }(u,v)({\phi }_1,{\phi }_2)$$

$$-\frac{\alpha \gamma }{2^{\ast }}{\int }_{{\mathbb{R}}^n}(\alpha |u{|}^{\alpha -2}u{\phi }_1|v{|}^{\beta }+\beta |u{|}^{\alpha }|v{|}^{\beta -2}v{\phi }_2)dx,$$

$$G_2^{\prime }(u,v)({\phi }_1,{\phi }_2)$$

Full text

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By using the Nehari manifold, under proper conditions, we establish the existence and nonexistence of positive least energy solution of the system.

Keywords: fractional Laplacian system; Nehari manifold; least energy solution

000 ∗ Corresponding author.

AMS Subject Classification: 35J50, 35B33, 35R11

The authors were supported by the NSFC grant 11571125.

E-mails: d201677010@hust.edu.cn; taoismnature@hust.edu.cn; hyxu@hust.edu.cn

1 Introduction

Recently, a great attention has been focused on the study of equations or systems involving the fractional Laplacian with nonlinear terms, both for their interesting theoretical structure and their concrete applications(see [3, 12, 19, 5, 6, 28, 24, 4, 29, 25, 16] and references therein). This type of operator arises in a quite natural way in many different contexts, such as, the thin obstacle problem, finance, phase transitions, anomalous diffusion, flame propagation and many others(see [1, 15, 20, 26] and references therein).

Compared to the Laplacian problem, the fractional Laplacian problem is nonlocal and more challenging. In 2007, L. Caffarelli and L. Silvestre [5] studied an extension problem related to the fractional Laplacian in ${\mathbb{R}}^{n}$, which can transform the nonlocal problem into a local problem in ${\mathbb{R}}_{+}^{n+1}$. This method can be extended to bounded regions and is extensively used in recent articles. For example, B. Barrios, E. Colorado, A. de Pablo and U. Sánchez [3] studied the following nonhomogeneous equation involving fractional Laplacian,

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and proved the existence and multiplicity of solutions under suitable conditions of $s$ and $q$. In the above, the fractional Laplacian operator $(-\Delta)^{s}$ is defined through the spectral decomposition using the powers of the eigenvalues of the positive Laplace operator $(-\Delta)$ with zero Dirichlet boundary data.

Furthermore, E. Colorado, A. de Pablo and U. Sánchez [12] studied the following fractional equation with critical Sobolev exponent

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where the existence and the multiplicity of solutions were proved under appropriate conditions on the size of $f$. For more recent advances on this topic, see [14, 17, 21] and references therein.

It is also natural to study the coupled system of equations. X. He, M. Squassina and W. Zou [18] considered the following fractional Laplacian system with critical nonlinearities on a bounded domain in ${\mathbb{R}}^{n}$

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using variational methods and a Nehari manifold decomposition, they prove that the system admits at least two positive solutions when the pair of parameters $(\lambda,\mu)$ belongs to certain subset of $\mathbb{R}^{2}$.

When $\Omega={\mathbb{R}}^{n}$, the Dirichlet-Neumann map used in [3, 12, 18] provides a formula for the fractional Laplacian in the whole space, which is equivalent to that obtained from Fourier Transform [5], where the operator has explicit expression,

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with

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In [16], Z. Guo, S. Luo and W. Zou studied the following critical system involving fractional Laplacian

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they showed the existence of positive least energy solution, which is radially symmetric with respect to some point in $\mathbb{R}^{n}$ and decays at infinity with certain rate. Q. Wang [28] studied a special case where $\alpha=\beta=\frac{2^{\ast}}{2}$ and $\frac{\alpha\gamma}{2^{\ast}}=\frac{\beta\gamma}{2^{\ast}}=\beta$, the author also showed the existence of positive least energy solution under suitable conditions.

In this paper, we study the existence of the least energy solutions for the system (1.1) with critical exponent. We assume

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Let ${D_{s}(\mathbb{R}^{n})}$ be Hilbert space as the completion of $C^{\infty}_{c}(\mathbb{R}^{n})$ equipped with the norm

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Let

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be the sharp imbedding constant of ${D_{s}(\mathbb{R}^{n})}\hookrightarrow L^{2^{\ast}}(\mathbb{R}^{n})$  and $S_{s}$ is attained (see [13]) in $\mathbb{R}^{n}$ by  $\widetilde{u}_{\epsilon,y}=\kappa(\varepsilon^{2}+|x-y|)^{-\frac{n-2s}{2}}$, where $\kappa\neq 0\in\mathbb{R},\ \varepsilon>0$ and $y\in\mathbb{R}^{n}$. That is

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We normalize $\widetilde{u}_{\epsilon,y}$ as follow, let

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By Lemma 2.12 in [16], $U_{\varepsilon,y}(x)=(S_{s})^{\frac{1}{2^{\ast}-2}}\overline{u}_{\epsilon,y}(x)$ is a positive ground state solution of

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and

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

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Define the Nehari manifold

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Finally, we say that $(u,v)$ is a nontrivial solution of (1.1) if $u\neq 0,v\neq 0$ and $(u,v)$ solves (1.1). Any nontrivial solution of (1.1) is in $\mathbb{N}$. It is easy to see that when the following algebra system (1.5) has a solution $(k,l)$, then $(\sqrt{k}U_{\varepsilon,y},\sqrt{l}U_{\varepsilon,y})$ is a nontrivial solution of (1.1).

In this paper, we get the existence and nonexistence of least energy solutions of (1.1) under certain conditions of $\gamma$, $n$ and $s$. Our existence results strongly depend on the following algebra system

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Our main results are:

Theorem 1.1**.**

If $\gamma<0,$  then $A=\frac{s}{n}(\mu_{1}^{-\frac{n-2s}{2s}}+\mu_{2}^{-\frac{n-2s}{2s}})S^{\frac{n}{2s}}_{s}$ and $A$ is not attained.

Theorem 1.2**.**

If $2s\leq n\leq 4s,\ \alpha>2,\ \beta>2$ and

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or $n>4s,\ 1<\alpha,\beta<2$ and

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then $A=\frac{s}{n}(k_{0}+l_{0})S_{s}^{\frac{n}{2s}}$ and $A$ is attained by $(\sqrt{k_{0}}U_{\varepsilon,y},\sqrt{l_{0}}U_{\varepsilon,y})$, where $(k_{0},l_{0})$ satisfies (1.5) and  $k_{0}=min\{k:(k,l)\ satisfies\ \eqref{int6}\}$.

Theorem 1.3**.**

Assume $n>4s$ and  $1<\alpha,\beta<2$ hold, there exists a

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such that for any  $\gamma\in(0,\gamma_{1})$, there exists a solution  $(k(\gamma),l(\gamma))$ of (1.5), satisfying $E(\sqrt{k(\gamma)}U_{\varepsilon,y},\sqrt{l(\gamma)}U_{\varepsilon,y})>A$ and $(\sqrt{k(\gamma)}U_{\varepsilon,y},\sqrt{l(\gamma)}U_{\varepsilon,y})$ is a positive solution of (1.1).

Remark 1.4**.**

Z. Guo, S. Luo and W. Zou [16] already showed the existence of positive least energy solutions for the system (1.1) with $n>4s$ and $1<\alpha,\ \beta<2$ for all $\gamma>0$. Theorem 1.2 and Theorem 1.3 tell that if

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the positive least energy solution of (1.1) has to have the form $(\sqrt{k(\gamma)}U_{\varepsilon,y},\sqrt{l(\gamma)}U_{\varepsilon,y})$; whereas, if $0<\gamma<\gamma_{1}$,  $(\sqrt{k(\gamma)}U_{\varepsilon,y},\sqrt{l(\gamma)}U_{\varepsilon,y})$ is not the least energy solution. However, it should be interesting to know whether $\gamma_{1}=\gamma_{0}$ or not. If $\gamma_{1}\neq\gamma_{0}$, what happens when $\gamma\in[\gamma_{1},\gamma_{0}]$.

The paper is organized as follows. In section 2, we introduce some preliminaries that will be used to prove theorems. In section 3, we prove Theorem 1.1. In section 4, we prove Theorem 1.2. The proof of Theorem 1.3 is given in section 5.

2 Some Preliminaries

For the case of $\gamma<0$, the following Lemma 2.1 shows that if the energy functional attains its minimum at some point $(u,v)\in\mathbb{N}$, then $(u,v)$ is a nontrivial solution of (1.1).

Lemma 2.1**.**

Assume $\gamma<0,$ if $A$ is attained by a couple $(u,v)\in\mathbb{N},$ then (u,v)  is a nontrivial solution of (1.1).

Proof.

Define

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Then $\mathbb{N}=\mathbb{N}_{1}\bigcap\mathbb{N}_{2}.$ By the Fréchet derivative, for any $\varphi_{1},\varphi_{2}\in D_{s}({\mathbb{R}^{n}})$, we have

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[TABLE]

where

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Then

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[TABLE]

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Suppose that $(u,v)\in\mathbb{N}$ is a minimizer for $E$ restricted to $\mathbb{N}$, then by the standard minimization theory, there exist two lagrange multipliers $L_{1},L_{2}\in\mathbb{R}$ such that,

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Then we have

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and

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Since

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hence

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Define the matrix

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then

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which means $L_{1}=L_{2}=0$, that is $E^{\prime}(u,v)=0.$

∎

Define functions

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then

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Remark 2.2**.**

If $(k,l)$ satisfies (1.5), then $(\sqrt{k}U_{\varepsilon,y},\sqrt{l}U_{\varepsilon,y})$  is a nontrivial solution of(1.1), where $U_{\varepsilon,y}$  satisfy (1.3) and (1.4). Hence the main work is to establish the existence of solutions to (1.5).

In order to prove the existence results for (1.5), we have the following Lemma 2.3.

Lemma 2.3**.**

Assume that $n>4s,1<\alpha,\beta<2,\gamma>0$, then

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has a solution $(k_{0},l_{0})$ such that

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where $(k_{0},l_{0})$ satisfies $k_{0}=min\{k:(k,l)\ satisfies\ \eqref{int6}\}$. Similarly, (2.2) has a solution $(k_{1},l_{1})$, such that

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Proof.

Solving $F_{1}(k,l)=0$ for $k,l>0,$ we have  $l=l(k)$ for all $k\in(0,\mu_{1}^{-\frac{2}{2^{\ast}-2}}).$ Then, substituting this into $F_{2}(k,l)=0,$ we have

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Let

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then (2.5) has a solution is equivalent to $f(k)=0$  has a solution in $(0,\mu_{1}^{-\frac{2}{2^{\ast}-2}})$. Since $1<\alpha,\beta<2$, we obtain

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then by the intermediate value theorem, there exists

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and

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Let $l_{0}=l(k_{0}),$ then $(k_{0},l_{0})$ is the solution of (2.2) and satisfies (2.3). Similarly, we can show (2.2) has a solution $(k_{1},l_{1})$, such that

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∎

Remark 2.4**.**

From the proof of Lemma 2.9 in the next a few pages, it is easy to see that system (1.5) has only one solution $(k,l)=(k_{0},l_{0})$ under the assumption that $2s<n<4s,\ \alpha>2,\ \beta>2$ and (1.6).

Remark 2.5**.**

Obviously, if $(\sqrt{k_{0}}U_{\varepsilon,y},\sqrt{l_{0}}U_{\varepsilon,y})\in\mathbb{N}$, then

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Next, in order to show $A\geq\frac{s}{n}(k_{0}+l_{0})S_{s}^{\frac{n}{2s}}$, we require the following lemmas.

Case 1. $n>4s,\ 1<\alpha,\beta<2$ and (1.7) hold.

Lemma 2.6**.**

Assume $n>4s,\ 1<\alpha,\beta<2$ and (1.7) hold, then

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[TABLE]

Proof.

Since

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hence

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and

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Next, we compute the second derivatives,

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we have

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By (1.7), we obtain

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Hence

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which means  $l(k)+k$  is strictly increasing in $[0,\mu_{1}^{-\frac{2}{2^{\ast}-2}}]$ and $k(l)+l$ is strictly increasing in $[0,\mu_{1}^{-\frac{2}{2^{\ast}-2}}].$ ∎

Lemma 2.7**.**

Assume $n>4s,\ 1<\alpha,\beta<2$ and (1.7) hold,  $(k_{0},l_{0})$ is obtained in Lemma 2.3. Then

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and

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Proof.

Since $k_{0}<\mu_{1}^{-\frac{2}{2^{\ast}-2}}.$ By Lemma 2.6, we have

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That is

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Similarly

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Hence

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To prove (2.7), by Lemma 2.3, we only need to show that $(k_{0},l_{0})=(k_{1},l_{1}).$ By (2.3), (2.4), we have $l_{0}\geq l_{1},\ k_{1}\geq k_{0}$. Suppose by contradiction that $k_{1}>k_{0}$, then

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hence

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Since $k(l)+l$ is strictly increasing for $l\in[0,\mu_{2}^{-\frac{2}{2^{\ast}-2}}],$ therefore $l_{1}>l_{0},$ which contradicts to $l_{0}\geq l$, then we have $k_{1}=k_{0}.$ Similarly, $l_{1}=l_{0}.$ ∎

Lemma 2.8**.**

Assume $n>4s,1<\alpha,\beta<2$  and (1.7) hold, then

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has an unique solution $(k,l)=(k_{0},l_{0}).$

Proof.

Obviously $(k_{0},l_{0})$ satisfies (2.8). Suppose $(\widetilde{k},\ \widetilde{l})$ is another solution of (2.8). Without loss of generality, we may assume that $\widetilde{k}>0,$ then $\widetilde{l}>0$. In fact, if $\widetilde{l}=0,$ then $\widetilde{k}\leq k_{0}+l_{0}$ and

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Therefore

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which contradicts to Lemma 2.7. In the following, we prove that $\widetilde{k}=k_{0}.$ Suppose by contradiction that $\widetilde{k}<k_{0},$ by the proof of Lemma 2.6, we have $k(l)$ is strictly increasing on $(0,(\frac{2-\beta}{\mu_{2}\alpha})^{\frac{2}{2^{\ast}-2}})$, and strictly decreasing on $((\frac{2-\beta}{\mu_{2}\alpha})^{\frac{2}{2^{\ast}-2}},\mu_{2}^{-\frac{2}{2^{\ast}-2}})$.

On the one hand, since $k(0)=k(\mu_{2}^{-\frac{2}{2^{\ast}-2}})=0$ and $0<\widetilde{k}<k_{0},$ therefore, there exist $l_{1},\ l_{2}$ satisfying $0<l_{1}<l_{2}<\mu_{2}^{-\frac{2}{2^{\ast}-2}},$ such that $k(l_{1})=k(l_{2})=\widetilde{k}$ and

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Furthermore, $F_{1}({\widetilde{k}},\ \widetilde{l})\geq 0,\ F_{2}({\widetilde{k}},\widetilde{l})\geq 0,$ we have $\widetilde{l}>l(\widetilde{k}),\ \widetilde{l}\leq l_{1}\ or\ \widetilde{l}\geq l_{2}.$

By (2.7), we see F_{2}({\widetilde{k}},l(\widetilde{k}))<0,\ by (2.9), we obtain  $l_{1}<l<l_{2}$, therefore $\widetilde{l}\geq l_{2}.$

On the other hand,  let $l_{3}=k_{0}+l_{0}-\widetilde{k},$ then $l_{3}>l_{0}$ and

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that is $k(l_{3})>\widetilde{k}.$ By (2.9), we have $l_{1}<l_{3}<l_{2}.$ Since $\widetilde{k}+\widetilde{l}\leq k_{0}+l_{0},$ we obtain that $\widetilde{l}\leq k_{0}+l_{0}-\widetilde{k}=l_{3}<l_{2}.$ This contradicts to $\widetilde{l}\geq l_{2},$ the proof completes.

∎

Case 2. $2s<n<4s,\ \alpha>2,\ \beta>2$ and (1.6) hold.

Lemma 2.9**.**

Assume $c,d\in\mathbb{R}$ satisfy

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If   $2s<n<4s,\ \alpha,\beta>2$ and (1.6) hold, then  $c+d\geq k+l,$ where $(k,l)\in{\mathbb{R}}^{2}$ is the unique solution of (1.5).

Proof.

Let $y=c+d,x=\frac{c}{d},y_{0}=k+l,x_{0}=\frac{k}{l}$, by (2.10) and (1.5) we have

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Then

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Let

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then

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Therefore, $g_{1}(x)$ increases in the interval $(0,(\frac{2\alpha\gamma}{2^{\ast}(2^{\ast}-2)\mu_{1}})^{\frac{2}{\beta-2}})$ and decreases in the interval $((\frac{2\alpha\gamma}{2^{\ast}(2^{\ast}-2)\mu_{1}})^{\frac{2}{\beta-2}},+\infty)$. $g_{2}(x)$ decreases in the interval $(0,\frac{\alpha-2}{\beta-2})$ and increases in the interval $(\frac{\alpha-2}{\beta-2},+\infty).$

Hence

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Then, by (1.6), we have

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Therefore, $f_{1}(x)$ is strictly decreasing in $(0,+\infty)$ and $f_{2}(x)$ is strictly increasing in $(0,+\infty).$ Due to the fact that

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there exists a unique  $x_{0}>0$, such that $f_{1}(x_{0})=f_{2}(x_{0})$ , which gives the uniqueness of $(k,l)$.

Since $f_{1}(x)\geq f_{1}(x_{0})$ for $x\leq x_{0}$ and $f_{2}(x)\geq f_{2}(x_{0})$ for $x\geq x_{0}$, we get

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which means $c+d\geq k+l.$ ∎

3 Proof of Theorem 1.1

Proof.

By Lemma 2.1, we obtain that when $-\infty<\gamma<0$ and if $A$ is attained by a couple $(u,v)\in\mathbb{N},$ then $(u,v)$   is a solution of (1.1). For any $(u,v)\in\mathbb{N},$

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Therefore

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Similarly

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Hence

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By lemma 2.12 in [16], we know $w_{\mu_{i}}=(\frac{S_{S}}{\mu_{i}})^{\frac{1}{2^{\ast}-2}}\overline{u}_{\epsilon,y}=(\frac{1}{\mu_{i}})^{\frac{1}{2^{\ast}-2}}U_{\epsilon,y}$ is the solution of the equation

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for $i=1,2.$

Let $e_{1}=(1,0,\cdots 0)\in\mathbb{R}^{n},\ (U(X),V_{R}(X))=(w_{\mu_{1}}(x),\ w_{\mu_{2}}(x+Re_{1}),$ where R is a positive constant. Since  $V_{R}(x)\in D_{s}(\mathbb{R}^{n})$ is a solution of

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we have, $V_{R}(X)\rightharpoonup 0\ in\ L^{2^{\ast}}(\mathbb{R}^{n})$ as $R\rightarrow+\infty,$ hence

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We will show that for $R>0$ sufficiently large, the system

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has a solution $(t_{R},s_{R})$ with

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which implies that

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Let us assume $(\star)$ first, then

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Let $R\rightarrow+\infty$, we get

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Therefore

[TABLE]

Suppose that $A$ is attained by some $(u,v)\in\mathbb{N}$, then $E(u,v)=A$. By Lemma 2.1, we know $(u,v)$ is a nontrivial solution of (1.1). By Strong maximum principle for fractional Laplacian( see, Proposition 2.17 in [26], Lemma 6 in [24]) and comparison principle in [22], we may assume that $u>0,\ v>0$ and

[TABLE]

then

[TABLE]

Hence

[TABLE]

Similarly, we have

[TABLE]

Then

[TABLE]

which contradicts to (3.2). Therefore, $A$ is not be obtained.

Now, we claim $(\star)$,

Proof of $(\star)$. Let

[TABLE]

we have $\theta\rightarrow 0$ as R sufficiently large. Then (3.1) has a solution is equivalent to that the following system has a solution at the neighbourhood of point $(1,1)$.

[TABLE]

By Taylor expansion at $(1,1)$, we have

[TABLE]

Therefore

[TABLE]

Let

[TABLE]

[TABLE]

we have that

[TABLE]

Let

[TABLE]

then

[TABLE]

Let

[TABLE]

then

[TABLE]

If we choose $x\in{\mathbb{R}^{2}}$ with $\|x\|\leq 2\|c\|\theta$, we get

[TABLE]

when $\theta$ is small enough. Since $T$ is continuous and $\theta\rightarrow 0,$ as $R\rightarrow\infty$, by Brouwer’s fixed point theorem, we get that the system (3.1) has a solution $(t_{R},s_{R})$ for all large $R$  with

[TABLE]

∎

4 Proof of Theorem 1.2

Proof.

By Remark 2.5, we have

[TABLE]

Let $(u_{i},\ v_{i})\in\mathbb{N}$ be a minimizing sequence for $A$, that is $E(u_{i},v_{i})\rightarrow A$ as $n\rightarrow\infty.$

Define

[TABLE]

then

[TABLE]

[TABLE]

Dividing both side of inequality by $S_{s}c_{i}$ and $S_{s}d_{i}$. Let

[TABLE]

we get

[TABLE]

that is

[TABLE]

Consequently, for the case  $2s<n<4s,\ \alpha>2,\ \beta>2$ and (1.6) hold, Lemma 2.9 ensures that

[TABLE]

For the case $n>4s,1<\alpha,\beta<2$ and (1.7) hold, Lemma 2.8 ensures

[TABLE]

Therefore

[TABLE]

Since $(u_{i},v_{i})\in\mathbb{N}$ is a minimizing sequence for $A$, we have

[TABLE]

By the definition of $S_{s}$ and (4.2), we have,

[TABLE]

Since

[TABLE]

we have

[TABLE]

that is

[TABLE]

thus

[TABLE]

Hence

[TABLE]

∎

5 Proof of Theorem 1.3

In order to proof Theorem 1.3, we use a result of Z. Guo, S. Luo and W. Zou in [16].

Theorem 5.1** (Theorem 1.1 of [16]).**

Assume $(H)$ holds, where

[TABLE]

Then (1.1) has a positive ground state solution $(U,\ V)$ for all $\gamma>0$, which is radially symmetric decreasing with the following decay condition

[TABLE]

That is $E(U,V)=A,$ where

[TABLE]

Lemma 5.2** (A result after Lemma 3.1 in [16]).**

Let $u_{\mu}=(\frac{S_{S}}{\mu})^{\frac{1}{2^{\ast}-2}}\overline{u}$ is a positive ground state solution of

[TABLE]

then

[TABLE]

Remark 5.3**.**

The Nahri manifold used in [16] is different from the Nahri manifold used in our paper. However, the positive solutions of (1.1) are certainly in both of the Nahri manifolds, therefore, Z. Guo, S. Luo and W. Zou’s result: Theorem 5.1 ensures that

[TABLE]

Proof of Theorem 1.3.

To obtain the existence of $(k(\gamma),\ l(\gamma)$ for $\gamma>0$, we define functions

[TABLE]

[TABLE]

let

[TABLE]

then

[TABLE]

Denote

[TABLE]

since $det(D)>0$, by implicit function theorem, we see that $k(\gamma),\ l(\gamma)$ are well defined and of class  $C^{1}$ in $(-\gamma_{2},\gamma_{2})$, for some $\gamma_{2}>0$ and

[TABLE]

Thus $(\sqrt{k_{\gamma}}U_{\varepsilon,y},\sqrt{l_{\gamma}}U_{\varepsilon,y})$ is a solution of (1.1).

Since

[TABLE]

there exists a $\gamma_{1}\in(0,\gamma_{2})$, such that

[TABLE]

By Lemma 5.2 and Remark 5.3, we get

[TABLE]

that is $(\sqrt{k_{\gamma}}U_{\varepsilon,y},\sqrt{l_{\gamma}}U_{\varepsilon,y})$ is another positive solution of (1.1). ∎