Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent

TL;DR

This paper investigates the existence of positive ground state solutions for a fractional Laplacian system with mixed critical and subcritical nonlinearities, identifying parameter regimes for existence and non-existence.

Contribution

It introduces new conditions on parameters ensuring the existence of positive ground state solutions using the Nehari manifold approach.

Findings

Existence of solutions for small  when   

Existence of solutions for large  when   

Non-existence of solutions below a certain    threshold

Abstract

In this paper, we consider the following fractional Laplacian system with one critical exponent and one subcritical exponent \begin{equation*} \begin{cases} (-\Delta)^{s}u+\mu u=|u|^{p-1}u+\lambda v & x\in \ \mathbb{R}^{N}, (-\Delta)^{s}v+\nu v = |v|^{2^{\ast}-2}v+\lambda u& x\in \ \mathbb{R}^{N},\\ \end{cases} \end{equation*} where $(-\Delta)^{s}$ is the fractional Laplacian, $0<s<1,\ N>2s, \ \lambda <\sqrt{\mu\nu },\ 1<p<2^{\ast}-1~ and~\ 2^{\ast}=\frac{2N}{N-2s}$~ is the Sobolev critical exponent. By using the Nehari\ manifold, we show that there exists a $\mu_{0}\in(0,1)$, such that when $0<\mu\leq\mu_{0}$, the system has a positive ground state solution. When $\mu>\mu_{0}$, there exists a $\lambda_{\mu,\nu}\in[\sqrt{(\mu-\mu_{0})\nu},\sqrt{\mu\nu})$ such that if $\lambda>\lambda_{\mu,\nu}$, the system has a positive ground state solution, if $\lambda<\lambda_{\mu,\nu}$, the system has no ground state solution.

Equations

$$\{\begin{array}{ll}(-\Delta {)}^{s}u+\mu u=|u{|}^{p-1}u+\lambda v& x\in {\mathbb{R}}^N,\\ (-\Delta {)}^{s}v+\nu v=|v{|}^{2^{\ast }-2}v+\lambda u& x\in {\mathbb{R}}^N,\end{array}$$

$$(-\Delta {)}^{s}u+u=f(u)\text{in}{\mathbb{R}}^N$$

$$\{\begin{array}{ll}(-\Delta {)}^{s}u=F(u,v)& x\in {\mathbb{R}}^N,\\ (-\Delta {)}^{s}v=G(u,v)& x\in {\mathbb{R}}^N,\end{array}$$

$$\{\begin{array}{ll}(-\Delta {)}^{s}u+\mu u=|u{|}^{p-1}u+\lambda v& x\in {\mathbb{R}}^N,\\ (-\Delta {)}^{s}v+\nu v=|v{|}^{2^{\ast }-2}v+\lambda u& x\in {\mathbb{R}}^N,\end{array}$$

$$-(-\Delta {)}^{s}u(x)=\frac{C(N,s)}{2}{\int }_{{\mathbb{R}}^N}\frac{u(x+y)+u(x-y)-2u(x)}{|y{|}^{N+2s}}dy,x\in {\mathbb{R}}^N$$

$$C(N,s)={\left({\int }_{{\mathbb{R}}^N}\frac{1-\cos ({\varsigma }_1)}{|\varsigma {|}^{N+2s}}d\varsigma \right)}^{-1}=2^{2s}{\pi }^{-\frac{N}{2}}\frac{\Gamma (\frac{N+2s}{2})}{\Gamma (2-s)}s(1-s).$$

$$⟨u,v{⟩}_{D^s({\mathbb{R}}^N)}=\frac{C(N,s)}{2}{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{(u(x)-u(y))(v(x)-v(y))}{|y-x{|}^{N+2s}}dxdy,$$

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

$$⟨u,v⟩={\int }_{{\mathbb{R}}^N}\left((-\Delta {)}^{\frac{s}{2}}u\cdot (-\Delta {)}^{\frac{s}{2}}v+uv\right)dx,\Vert u{\Vert }_{H^s({\mathbb{R}}^N)}^2=⟨u,u⟩.$$

$$C_{p+1}=\underset{u\in H^s({\mathbb{R}}^N))\setminus \{0\}}{\inf }\frac{\Vert u{\Vert }_{H^s({\mathbb{R}}^N)}^2}{({\int }_{{\mathbb{R}}^N}|u{|}^{p+1}dx{)}^{\frac{2}{p+1}}},$$

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

$$\Vert (u,v){\Vert }_{\mathcal{H}}^2=\Vert u{\Vert }_{H^s({\mathbb{R}}^N)}^2+\Vert v{\Vert }_{H^s({\mathbb{R}}^N)}^2=\Vert (u,v){\Vert }_{\mathcal{D}}^2+\Vert u{\Vert }_{L^2({\mathbb{R}}^N)}^2+\Vert v{\Vert }_{L^2({\mathbb{R}}^N)}^2,$$

$$E_{\mu ,\nu ,\lambda }(u,v)=$$

$$-\frac{1}{p+1}{\int }_{{\mathbb{R}}^N}|u{|}^{p+1}dx-\frac{1}{2^{\ast }}{\int }_{{\mathbb{R}}^N}|v{|}^{2^{\ast }}dx-\lambda {\int }_{{\mathbb{R}}^N}uvdx.$$

$$\mathbb{M}:={\mathbb{M}}_{\lambda }:={\mathbb{M}}_{\mu ,\nu ,\lambda }$$

$$={\int }_{{\mathbb{R}}^N}(|u{|}^{p+1}+|v{|}^{2^{\ast }})dx+2\lambda {\int }_{{\mathbb{R}}^N}uvdx\},$$

$${\mu }_0={\left[\frac{2s(p+1)}{N(p-1)}{S}_s^{\frac{N}{2s}}{C}_{p+1}^{-\frac{p+1}{p-1}}\right]}^{(\frac{p+1}{p-1}-\frac{N}{2s}{)}^{-1}}.$$

$$H_r^s({\mathbb{R}}^N)=\{\phi \in H^s({\mathbb{R}}^N):\phi \text{is radial}\}$$

$$D_r^s({\mathbb{R}}^N)=\{\phi \in D^s({\mathbb{R}}^N):\phi \text{is radial}\}.$$

$${\mathcal{H}}_r=\{(u,v)\in \mathcal{H}:u,v\text{are radial}\},$$

$${\mathcal{D}}_r=\{(u,v)\in \mathcal{D}:u,v\text{are radial}\},$$

$${\mathbb{M}}_r=\{(u,v)\in \mathbb{M}:u,v\text{are radial}\}.$$

$$\Gamma =\{\gamma \in \mathcal{C}([0,1],{\mathcal{H}}_r):\gamma (0)=(0,0),\gamma (1)=(u_0,v_0)\}.$$

$${\hat{A}}_{\mu ,\nu ,\lambda }:=\underset{\gamma \in \Gamma }{\inf }\underset{t\in [0,1]}{\max }{E}_{\mu ,\nu ,\lambda }(\gamma (t)).$$

$$E_{\mu ,\nu ,\lambda }(u_n,v_n)\to {\hat{A}}_{\mu ,\nu ,\lambda }\text{and}{E}_{\mu ,\nu ,\lambda }^{\prime }(u_n,v_n)\to 0\text{as}n\to +\infty .$$

$$E_{\mu ,\nu ,\lambda }(u,v)$$

$$-\frac{1}{2^{\ast }}{\int }_{{\mathbb{R}}^N}|v{|}^{2^{\ast }}dx-\lambda {\int }_{{\mathbb{R}}^N}uvdx$$

$$\ge \frac{1}{2}||(u,v)|{|}_{{\mathcal{D}}_r}^2+\frac{\tau }{2}{\int }_{{\mathbb{R}}^N}(u^2+v^2)-\frac{1}{p+1}{\int }_{{\mathbb{R}}^N}|u{|}^{p+1}dx-\frac{1}{2^{\ast }}{\int }_{{\mathbb{R}}^N}|v{|}^{2^{\ast }}dx$$

$$\ge \min (\frac{1}{2},\frac{\tau }{2})||(u,v)|{|}_{{\mathcal{H}}_r}^2-C_1||(u,v)|{|}_{{\mathcal{H}}_r}^{p+1}-C_2||(u,v)|{|}_{{\mathcal{H}}_r}^{2^{\ast }}.$$

$$E_{\mu ,\nu ,\lambda }(u,v)\ge \min (\frac{1}{2},\frac{\tau }{2})||(u,v)|{|}_{{\mathcal{H}}_r}^2-C_1||(u,v)|{|}_{{\mathcal{H}}_r}^{p+1}-C_2||(u,v)|{|}_{{\mathcal{H}}_r}^{2^{\ast }}>\frac{1}{4}{C}_3{\rho }^2>0.$$

$$E_{\mu ,\nu ,\lambda }(u_n,v_n)\to {\hat{A}}_{\mu ,\nu ,\lambda }\text{and}{E}_{\mu ,\nu ,\lambda }^{\prime }(u_n,v_n)\to 0\text{as}n\to +\infty ,$$

$$\Gamma =\{\gamma \in \mathcal{C}([0,1],{\mathcal{H}}_r):\gamma (0)=(0,0),\gamma (1)=(u_0,v_0)\}.$$

$${\overline{A}}_{\mu ,\nu ,\lambda }=\underset{{\mathcal{H}}_r\setminus \{(0,0)\}}{\inf }\underset{t>0}{\max }{E}_{\mu ,\nu ,\lambda }(tu,tv),$$

Full text

[TABLE]

where $(-\Delta)^{s}$ is the fractional Laplacian, $0<s<1,\ N>2s,\ \lambda<\sqrt{\mu\nu},\ 1<p<2^{\ast}-1~{}and~{}\ 2^{\ast}=\frac{2N}{N-2s}$  is the Sobolev critical exponent. By using the Nehari manifold, we show that there exists a $\mu_{0}\in(0,1)$, such that when $0<\mu\leq\mu_{0}$, the system has a positive ground state solution. When $\mu>\mu_{0}$, there exists a $\lambda_{\mu,\nu}\in[\sqrt{(\mu-\mu_{0})\nu},\sqrt{\mu\nu})$ such that if $\lambda>\lambda_{\mu,\nu}$, the system has a positive ground state solution, if $\lambda<\lambda_{\mu,\nu}$, the system has no ground state solution.

Keywords: fractional Laplacian; critical exponent; subcritical exponent; ground state solution

000 ∗ Corresponding author.

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

The authors were supported by the NSFC grant 11571125.

1 Introduction

In the past decades, the Laplacian equation or system has been widely investigated and there are many results about ground state solutions, multiple positive solutions, sign-changing solutions, etc(see [30, 31, 11, 13, 14, 12, 15] and references therein).

Compared to the Laplacian problem, the fractional Laplacian problem is non-local and more challenging. Recently, a great attention has been focused on the study of fractional and non-local operators of elliptic type, both for the pure mathematical research and in view of concrete real-world applications(see [4, 16, 28, 8, 34, 7, 36, 21] 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[2, 18, 29, 37] and references therein).

For the case of fractional Laplacian equation, the existence and nonexistence of solutions has been studied by many researchers. For example

[TABLE]

has been studied by many authors under various hypotheses on the nonlinearity $f$. Such as, Wang and Zhou [38] obtained the existence of a radial sign-changing solution for equation (1.1) by using variational method and Brouwer degree theory. When the nonlinearity $f$ satisfies the general hypotheses introduced by Berestycki and Lions [6], Chang and Wang [10] proved the existence of a radially symmetric ground state solution with the help of the Pohozǎev identity for (1.1). However, in all these works, they only consider the existence and nonexistence solutions, but there are few results about the uniqueness of solution for fractional Laplacian equation. In the remarkable papers [25][26], for the subcritical case, when $f(u)=|u|^{p-2}u,\ p\in(2,2^{\ast})$, R.L. Frank and E. Lenzmann [25] showed the uniqueness of non-linear ground states solutions to the equation (1.1) for one dimension case and R.L. Frank, E. Lenzmann and L. Silvestre [26] showed the general unique ground state solution to the equation (1.1) for dimension great than one.

It is also nature to study the coupled system. For the following fractional Laplacian system,

[TABLE]

has been investigated by many authors under various hypotheses on the nonlinearity $F(u,v)$ and $G(u,v)$. For example, when $F(u,v)=f(u)+\lambda v-u,G(u,v)=g(u)+\lambda u-v$, D.F. Lü, S.J. Peng [24] showed that under suitable condition of $f,g$, it has a vector ground state solution for $\lambda\in(0,1)$. When $F(u,v)=\mu_{1}|u|^{2^{\ast}-2}u+\frac{\alpha\gamma}{2^{\ast}}|u|^{\alpha-2}u|v|^{\beta},G(u,v)=\mu_{2}|v|^{2^{\ast}-2}v+\frac{\beta\gamma}{2^{\ast}}|u|^{\alpha}|v|^{\beta-2}v$, M. D. Zhen, J. C. He and H. Y. Xu [40] showed that the existence and nonexistence of ground state solutions under suitable condition of $\alpha,\beta,\gamma,s,N$ and Z. Guo, S. Luo and W. Zou [21] showed that under suitable condition of $\alpha,\beta,s,N$ the system has a positive ground state solution for all $\gamma>0$. When $F(u,v)=(|u|^{2p}+b|u|^{p-1}|v|^{p+1})u-u,G(u,v)=(|v|^{2p}+b|v|^{p-1}|u|^{p+1})v-\omega^{2\alpha}v$, Q. Guo and X. He [20] proved the existence of a least energy solution via Nehari manifold method and showed that if $b$ is large enough, it has a positive least energy solution. Note that in all these works, they only consider subcritical case or critical case. As far as we know, there are few results for the fractional Laplacian system with one subcritical equation and one critical equation. In this paper, we consider the following fractional Laplacian system with one critical exponent and one subcritical exponent on ${\mathbb{R}}^{N}$. In the case of Laplacian system, the problem has been investigated by Z. Chen, W. Zou in [12].

The system we consider is the following

[TABLE]

where $(-\Delta)^{s}$ is the fractional Laplacian, $0<s<1,\ N>2s,\ \lambda<\sqrt{\mu\nu},\ 1<p<2^{\ast}-1,\ 2^{\ast}=\frac{2N}{N-2s}$ is the Sobolev critical exponent. The fractional Laplacian $(-\Delta)^{s}$ is defined by

[TABLE]

with

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Let $D^{s}(\mathbb{R}^{N})$ be the Hilbert space defined as the completion of $C^{\infty}_{0}(\mathbb{R}^{N})$ with the scalar product

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and norm

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Let ${H^{s}(\mathbb{R}^{N})}$ be the Hilbert space of function in $\mathbb{R}^{N}$ endowed with the standard scalar product and norm

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Let $C_{p+1}$ be the sharp constant of the Sobolev embedding ${H^{s}(\mathbb{R}^{N})}\hookrightarrow L^{p+1}(\mathbb{R}^{N})$

[TABLE]

and let $S_{s}$ be the sharp imbedding constant of ${D^{s}(\mathbb{R}^{N})}\hookrightarrow L^{2^{\ast}}(\mathbb{R}^{N})$

[TABLE]

From [17] we have $S_{s}$ is attained in $\mathbb{R}^{N}$ by  $\widetilde{u}_{\epsilon,y}=\kappa(\varepsilon^{2}+|x-y|^{2})^{-\frac{N-2s}{2}}$, where $\kappa\neq 0\in\mathbb{R},\ \varepsilon>0$ and $y\in\mathbb{R}^{N}$.

Denote $\mathcal{H}={H^{s}(\mathbb{R}^{N})}\times{H^{s}(\mathbb{R}^{N})}$ and $\mathcal{D}={D^{s}(\mathbb{R}^{N})}\times{D^{s}(\mathbb{R}^{N})}$, with the norm given by

[TABLE]

where $||(u,v)||^{2}_{\mathcal{D}}=\|u\|^{2}_{D^{s}(\mathbb{R}^{N})}+\|v\|^{2}_{D^{s}(\mathbb{R}^{N})}$.

The energy functional associated with (1.2) is given by

[TABLE]

Define the Nehari manifold

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We say that $(u,v)$ is a nontrivial solution of (1.2) if $u\neq 0,v\neq 0$ and $(u,v)$ solves (1.2). Any nontrivial solution of (1.2) is in $\mathbb{M}$. Due to the fact that if we take $\varphi,\psi\in\mathcal{C}^{\infty}_{0}(\mathbb{R}^{N})$ with $\varphi,\psi\not\equiv 0$ and $supp(\varphi)\bigcap supp(\psi)=\emptyset$, then there exist $t_{1},t_{2}>0$ such that $(t_{1}\varphi,t_{2}\psi)\in\mathbb{M}$, so $\mathbb{M}\neq\emptyset$.

Let

[TABLE]

Our main result is:

Theorem 1.1**.**

*Assume $N>2s,\ 1<p<2^{\ast}-1$ and $\mu,\nu>0,\ 0<\lambda<\sqrt{\mu\nu}$. Let $\mu_{0}$ be in (1.6).

$(1)$ If $0<\mu\leq\mu_{0},$ then the system (1.2) has a positive ground state solution.

$(2)$ If $\mu>\mu_{0},$ then there exists $\lambda_{\mu,\nu}\in[\sqrt{(\mu-\mu_{0})\nu},\sqrt{\mu\nu})$ such that,*

$(i)$* if $\lambda<\lambda_{\mu,\nu}$, the system (1.2) has no ground state solution.*

$(ii)$* if $\lambda>\lambda_{\mu,\nu}$, the system (1.2) has a positive ground state solution.*

We sketch our idea of the proof. It is well known that the Sobolev embedding ${H^{s}(\mathbb{R}^{N})}\hookrightarrow L^{p}(\mathbb{R}^{N})$ are not compact for $2\leq p\leq 2^{\ast}$. Hence, the associated functional of problem (1.2) does not satisfy the Palais-Smale condition. In order to overcome the lack of compactness, we first set our work space in $H_{r}^{s}(\mathbb{R}^{N})\times H_{r}^{s}(\mathbb{R}^{N})$, where

[TABLE]

and $H^{s}_{r}(\mathbb{R}^{N})$ is endowed with the $H^{s}(\mathbb{R}^{N})$ topology: $\|\varphi\|_{H^{s}_{r}(\mathbb{R}^{N})}=\|\varphi\|_{H^{s}(\mathbb{R}^{N})}$. Let

[TABLE]

By properties of symmetric radial decreasing rearrangement, we know $C_{p+1}$ is achieved by radial functions in ${H^{s}(\mathbb{R}^{N})}$ and $S_{s}$ is achieved by radial functions in ${D^{s}(\mathbb{R}^{N})}$. By principle of symmetric criticality (Theorem 1.28 in [39]), the solutions for (1.2) in function space ${H_{r}^{s}(\mathbb{R}^{N})}\times{H_{r}^{s}(\mathbb{R}^{N})}$ are also the solutions in function space ${H^{s}(\mathbb{R}^{N})}\times{H^{s}(\mathbb{R}^{N})}$.

Second, we show that if the critical value $c$ of the functional (1.5) is strictly less than $\frac{s}{N}S^{\frac{N}{2s}}_{s}$, then the corresponding critical sequence will satisfy $(PS)_{c}$ condition.

Finally, we prove that mountain pass value for (1.5) is less than $\frac{s}{N}S^{\frac{N}{2s}}_{s}$ under some proper conditions.

Remark 1.1**.**

The proof of Theorem 1.1 is totally variational. We do not need to use the regularity of the solutions to the system (1.2)(For the fractional Laplacian equation, there are no general results for regularity(of the solutions) higher than the one derived from the Sobolev imbedding). The method we use here is different from the one used in[12], where they use a limiting argument to deal with the problem in Laplacian case and the $C^{2}$ regularity of the solutions are needed.

The paper is organized as follows. In section 2, we introduce some preliminaries that will be used to prove Theorem 1.1. In section 3, we prove Theorem 1.1.

2 Some Preliminaries

As mentioned earlier, we will only work in the radial function space. Set $\mathcal{H}_{r}$ and $\mathcal{D}_{r}$ as the following

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[TABLE]

with norm deduced from $\mathcal{H}$ and $\mathcal{D}$ respectively. Define the Nehari Manifold in $\mathcal{H}_{r}$ as

[TABLE]

Theorem 1.1 is proved by Mountain Pass Theorem [1], we first show that $E_{\mu,\nu,\lambda}$ has a $(PS)_{c}$ sequence in $\mathcal{H}_{r}$. Choose $\varphi,\psi\in\mathcal{C}^{\infty}_{0}(\mathbb{R}^{N})\cap H_{r}(\mathbb{R}^{N})$ with $\varphi,\psi\not\equiv 0$ and $supp(\varphi)\bigcap supp(\psi)=\emptyset$, then there exists $t_{0}>0$ such that $E_{\mu,\nu,\lambda}(t_{0}\varphi,t_{0}\psi)<0$ for all $t\geq t_{0}$. Take $(u_{0},v_{0})=(t\varphi,t\psi)$ with $t\geq t_{0}$ large enough. Let

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Define

[TABLE]

Lemma 2.1**.**

Under the condition $0<\lambda<\sqrt{\mu\nu}$, there exists a Palais-Smale sequence $\{(u_{n},v_{n})\}\subset\mathcal{H}_{r}$ such that

[TABLE]

Proof.

We first claim that $E_{\mu,\nu,\lambda}$ possesses a mountain pass geometry around $(0,0)$;

(1) there exist $\alpha,\rho>0$, such that $E_{\mu,\nu,\lambda}(u,v)>\alpha$ for all $||(u,v)||_{\mathcal{H}_{r}}=\rho$;

(2) there exist $(u_{0},v_{0})\in\mathcal{H}_{r}$ such that $||(u_{0},v_{0})||_{\mathcal{H}_{r}}>\rho$ and $E_{\mu,\nu,\lambda}(u_{0},v_{0})<0$.

To claim (1), since $\lambda<\sqrt{\mu\nu}$, we can take a small $\tau>0$ such that $\lambda^{2}=(\mu-\tau)(\nu-\tau)$, then by Sobolev imbedding,

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Choose $\rho>0$ sufficiently small, if $||(u,v)||^{2}_{\mathcal{H}_{r}}=\rho$, then

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(2) is obvious when we choose $t$ large enough such that $\|(u_{0},v_{0})\|_{\mathcal{H}_{r}}=t\|(\varphi,\psi)\|_{\mathcal{H}_{r}}>\rho$.

By Mountain Pass Theorem [1], for the constant $0<\widehat{A}_{\mu,\nu,\lambda}:=\inf\limits_{\gamma\in\Gamma}\max\limits_{t\in[0,1]}E_{\mu,\nu,\lambda}(\gamma(t)),$ there exist a $(PS)_{\widehat{A}_{\mu,\nu,\lambda}}$ sequence $\{(u_{n},v_{n})\}\subset\mathcal{H}_{r}$, that is

[TABLE]

where

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∎

Define

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and

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Lemma 2.2**.**

Under the condition $0<\lambda<\sqrt{\mu\nu}$, $\widehat{A}_{\mu,\nu,\lambda}=\overline{A}_{\mu,\nu,\lambda}=A_{\mu,\nu,\lambda}$.

Proof.

We first claim $\overline{A}_{\mu,\nu,\lambda}=A_{\mu,\nu,\lambda}$. For any $(u,v)\in\mathcal{H}_{r}$ with $(u,v)\not\equiv(0,0)$, there exist a unique $t_{\lambda,u,v}>0$ such that

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where $t_{\lambda,u,v}>0$ satisfies $\varphi(\lambda,u,v,t_{\lambda,u,v})=0$ and

[TABLE]

Which implies that $(t_{\lambda,u,v}u,t_{\lambda,u,v}v)\in\mathbb{M}_{r}$. From (2.2), we get for any $(u,v)\in\mathcal{H}_{r}$, $(u,v)\not\equiv(0,0)$,

[TABLE]

therefore $\overline{A}_{\mu,\nu,\lambda}\geq A_{\mu,\nu,\lambda}$. Similarly, we can show that $\overline{A}_{\mu,\nu,\lambda}\leq A_{\mu,\nu,\lambda}$, thus $\overline{A}_{\mu,\nu,\lambda}=A_{\mu,\nu,\lambda}$.

Next, we claim $\widehat{A}_{\mu,\nu,\lambda}=\overline{A}_{\mu,\nu,\lambda}.$

For any $\epsilon>0$, we can take a $(u,v)\not\equiv(0,0)$ such that

[TABLE]

We take a two dimensional space $\mathcal{S}$ in $\mathcal{H}_{r}$ contain $(u,v)$ and $(u_{0},v_{0})$ and we choose a large $R>\max(\|(t_{\lambda,u,v}u,t_{\lambda,u,v}v)\|,\|(u_{0},v_{0})\|)>0$, such that $E_{\mu,\nu,\lambda}(u,v)<0$ for all $(u,v)\in\mathcal{S}$ with $\|(u,v)\|=R$. Now, we define a path $\Gamma$ connecting $(0,0)$ and $(u_{0},v_{0})$ as follows. If $t\in[0,\frac{1}{2}]$, let $(u_{t},v_{t})=(\frac{2tRu}{\|u\|},\frac{2tRv}{\|v\|})$ which is the segment connecting $(0,0)$ and $(\frac{Ru}{\|u\|},\frac{Rv}{\|v\|})$. If $t\in(\frac{1}{2},\frac{3}{4})$, let $(u_{t},v_{t})$ be the arc in $\mathcal{S}$, satisfying $(u_{\frac{1}{2}},v_{\frac{1}{2}})=(\frac{Ru}{\|u\|},\frac{Rv}{\|v\|}),\ \ (u_{\frac{3}{4}},v_{\frac{3}{4}})=(\frac{Ru_{0}}{\|u_{0}\|},\frac{Rv_{0}}{\|v_{0}\|})$ and $\|(u_{t},v_{t})\|=R.$ If $t\in(\frac{3}{4},1]$ define $(u_{t},v_{t})=(\frac{4(1-t)Ru_{0}}{\|u_{0}\|}+(4t-3)u_{0},\frac{4(1-t)Rv_{0}}{\|v_{0}\|}+(4t-3)v_{0})$ which is a segment connecting $(\frac{Ru_{0}}{\|u_{0}\|},\frac{Rv_{0}}{\|v_{0}\|})$ and $(u_{0},v_{0}).$ Let $\gamma(t)=(u_{t}(\cdot),v_{t}(\cdot)),t\in[0,1],$ then it is easy to check that

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which implies that $\overline{A}_{\mu,\nu,\lambda}\geq\widehat{A}_{\mu,\nu,\lambda}.$ Thus $\widehat{A}_{\mu,\nu,\lambda}\leq\overline{A}_{\mu,\nu,\lambda}=A_{\mu,\nu,\lambda}.$

Let $\{(u_{n},v_{n})\}$ is a $(PS)_{\widehat{A}_{\mu,\nu,\lambda}}$ sequence, then

[TABLE]

We claim $\{(u_{n},v_{n})\}$ is bounded in $\mathcal{H}_{r}$. For $n$ large enough, we have since $\lambda<\sqrt{\mu\nu}$, we can take a small $\tau>0$ such that $\lambda^{2}=(\mu-\tau)(\nu-\tau)$, then by Sobolev imbedding

[TABLE]

Consequently, $\{(u_{n},v_{n})\}$ is bounded in $\mathcal{H}_{r}$. Since $\widehat{A}_{\mu,\nu,\lambda}>0$, $\{(u_{n},v_{n})\}$ is not $(0,0)$.

Let $(t_{n}u_{n},t_{n}v_{n})\in\mathbb{M}_{r}$, from

[TABLE]

it is easy to see that $t_{n}\to 1$ as $n\to\infty$. So

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Consequently, $\widehat{A}_{\mu,\nu,\lambda}=\overline{A}_{\mu,\nu,\lambda}=A_{\mu,\nu,\lambda}$. This completes the proof of Lemma 2.2. ∎

Remark 2.1**.**

By properties of symmetric radial decreasing rearrangement, it is easy to show that the value $\widehat{A}_{\mu,\nu,\lambda}=\overline{A}_{\mu,\nu,\lambda}=A_{\mu,\nu,\lambda}$ defined in $\mathcal{H}_{r}$ is the same as the value defined in $\mathcal{H}$. So the ground state solution in $\mathcal{H}_{r}$ is also the ground state solution in $\mathcal{H}$.

In order to prove Theorem 1.1, we need the following lemma.

Lemma 2.3**.**

([5]Brezis-Lieb Lemma) Let $\{u_{n}\}\subset L^{p+1}(\mathbb{R}^{N})$, $0<p<\infty$. If $\{u_{n}\}$ is bounded in $L^{p+1}(\mathbb{R}^{N})$ and $u_{n}\rightarrow u\ \text{a.e on}\ \mathbb{R}^{N}$, then

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Next, we borrow some ideas form [12]. Define

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[TABLE]

and denote $f_{\beta}=f_{\beta,1}$. Then we have following Lemma.

Lemma 2.4**.**

For any $(u,v)\in\mathcal{H}_{r}$ with $u\neq 0$ and $v\neq 0$, there holds

[TABLE]

Proof.

Since $2\lambda uv\leq\frac{\lambda^{2}}{\nu}u^{2}+\nu v^{2}$, we have

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Moreover, there exist $t_{1},t_{2}>0$ such that

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Since $f_{\mu-\frac{\lambda^{2}}{\nu}}(0)=0,\ g(0)=0$ and $f_{\mu-\frac{\lambda^{2}}{\nu}}(tu)$ is increasing in $[0,t_{1}]$, decreasing in $[t_{1},+\infty)$, $g(tv)$ is increasing in $[0,t_{2}]$, decreasing in $[t_{2},+\infty)$. Thus, if $t_{1}<t_{2}$, then $g(t_{1}v)>0$ and $E_{\mu,\nu,\lambda}(t_{1}u,t_{1}v)>f_{\mu-\frac{\lambda^{2}}{\nu}}(t_{1}u)=\max\limits_{t>0}f_{\mu-\frac{\lambda^{2}}{\nu}}(tu).$ If $t_{1}\geq t_{2}$, then $f_{\mu-\frac{\lambda^{2}}{\nu}}(t_{2}u)>0$ and $E_{\mu,\nu,\lambda}(t_{2}u,t_{2}v)>g(t_{2}v)=\max\limits_{t>0}g(tv).$ This completes the proof. ∎

By [25, 26], there exists a unique positive radial ground state solution $w$ to the equation $(-\Delta)^{s}u+u=u^{p}$ in $H^{s}(\mathbb{R}^{N})$. By (1.3),

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where $f_{1}$ is defined in (2.5). Let $w_{\beta,\gamma}(x):=\beta^{\frac{1}{p-1}}\gamma^{-\frac{1}{p-1}}w(\beta^{\frac{1}{2s}}x)$, then $w_{\beta,\gamma}(x)$ is the unique positive radial solution of $(-\Delta)^{s}u+\beta u=\gamma u^{p},\ u\in H^{s}(\mathbb{R}^{N})$ with the energy

[TABLE]

For convenience, we denote $w_{\beta}=w_{\beta,1}$. Define $\alpha=N(\frac{1}{p+1}-\frac{1}{2^{\ast}})\in(0,s),$ then

[TABLE]

Let $\bar{\mu}_{0}=\frac{\alpha}{s}(\frac{s-\alpha}{s})^{\frac{N-2s}{2s}(\frac{p+1}{p-1}-\frac{N}{2s})^{-1}}$, we have the following Lemma:

Lemma 2.5**.**

There exists a constant $\mu_{0}>0$ with $0<\mu_{0}<\overline{\mu}_{0}$ such that

[TABLE]

Proof.

By (1.6) and (2.7), we have $f_{\mu_{0}}(w_{\mu_{0}})=\frac{s}{N}S^{\frac{N}{2s}}_{s}.$ Since $p<2^{\ast}-1$, we have

[TABLE]

From (2.7) it is easy to obtain (2.8). In order to prove $\overline{\mu}_{0}>\mu_{0}$, by (2.8) we need to show $f_{\overline{\mu}_{0}}(w_{\overline{\mu}_{0}})>\frac{s}{N}S^{\frac{N}{2s}}_{s}.$ By Hölder inequality and Young inequality, we have

[TABLE]

If we choose $C_{0}>0,\ \ \epsilon_{0}>0$ such that

[TABLE]

then, we have

[TABLE]

and

[TABLE]

This implies that $C_{p+1}>C_{0}$. Combining this with (2.7), we obtain

[TABLE]

If we choose

[TABLE]

we have $f_{\overline{\mu}_{0}}(w_{\overline{\mu}_{0}})>\frac{s}{N}S^{\frac{N}{2s}}_{s}$. Thus, $\overline{\mu}_{0}>\mu_{0}.$ This completes the proof. ∎

Remark 2.2**.**

From the proof of the above lemma, though the exact values of $C_{p+1}$ and $\mu_{0}$ are unknown, we have a lower bound estimate for $C_{p+1}$ and an upper bound estimate for $\mu_{0}$:

[TABLE]

[TABLE]

Since $\alpha\in(0,s)$, it is easy to deduce that $\overline{\mu}_{0}<1$.

For any $\mu>\mu_{0},\ \ \nu>0$, we define a $C^{1}$ function $h_{\mu,\nu}:(0,+\infty)\rightarrow\mathbb{R}$ by

[TABLE]

Then,

[TABLE]

Thus, $h_{\mu,\nu}(a)\rightarrow+\infty$ as $a\rightarrow 0_{+}$ and $h_{\mu,\nu}(a)$ is increasing in $[\sqrt{\frac{\mu}{\nu}},+\infty).$ Therefore, there exists $a_{\mu,\nu}\in(0,\sqrt{\frac{\mu}{\nu}})$ such that

[TABLE]

Since $h_{\mu,\nu}(\sqrt{\frac{\mu}{\nu}})<\sqrt{\mu\nu}$, we have

[TABLE]

Lemma 2.6**.**

*(1)If $0<\mu\leq\mu_{0},$ then $A_{\mu,\nu,\lambda}<\frac{s}{N}S^{\frac{N}{2s}}_{s}.$

(2)If $\mu>\mu_{0}$, then there exists a $\lambda_{\mu,\nu}\in[\sqrt{(\mu-\mu_{0})\nu},\widetilde{\lambda}_{\mu,\nu})$ such that*

$(i)$* if $0<\lambda\leq\lambda_{\mu,\nu},$ then $A_{\mu,\nu,\lambda}=\frac{s}{N}S^{\frac{N}{2s}}_{s}$;*

$(ii)$* if $\lambda_{\mu,\nu}<\lambda<\sqrt{\mu\nu},$ then $A_{\mu,\nu,\lambda}<\frac{s}{N}S^{\frac{N}{2s}}_{s},$

where $\widetilde{\lambda}_{\mu,\nu}$ is from (2.10).*

Proof.

(1) If $\mu\in(0,\mu_{0}),$ by Lemma 2.5, we have

[TABLE]

Thus $A_{\mu,\nu,\lambda}<\frac{s}{N}S^{\frac{N}{2s}}_{s}.$

When $\mu=\mu_{0},$ then $A_{\mu_{0},\nu,\lambda}\leq f_{\mu_{0}}(w_{\mu_{0}})=\frac{s}{N}S^{\frac{N}{2s}}_{s}.$ Assume by contradiction that $A_{\mu_{0},\nu,\lambda}=\frac{s}{N}S^{\frac{N}{2s}}_{s}$, then

[TABLE]

Thus, $(w_{\mu_{0}},0)$ is a ground state solution of (1.2). Since $\lambda>0$, if $(w_{\mu_{0}},0)$ is a ground state solution of (1.2), then $w_{\mu_{0}}\equiv 0.$ This contract with $w_{\mu_{0}}\neq 0,$ so $A_{\mu_{0},\nu,\lambda}<\frac{s}{N}S^{\frac{N}{2s}}_{s}.$

(2) In order to prove second part of Lemma 2.6, we divide into two steps.

Step 1 We first show that for any fixed $\mu>\mu_{0},\ \nu>0,$ if $0<\lambda\leq\sqrt{(\mu-\mu_{0})\nu}$, then

[TABLE]

First, we show that the single equation

[TABLE]

has no nontrivial solutions.

In deed, by the similar arguments as Lemma 2.1 and Lemma 2.2, we know that the energy functional $I(v)$ for equation (2.13) has mountain pass structure and bounded $(PS)_{c}$ sequence. It is well know that if the least energy for $I(v)$ strictly less than $\frac{s}{N}S^{\frac{N}{2s}}_{s}$, then (2.13) has nontrivial solutions.

Let us now consider the cut-off function $\eta(x)\in C^{\infty}_{0}(\mathbb{R}^{N},[0,1])$ such that $0\leq\eta\leq 1,\eta=1\ \text{on}\ B(0,r)$ and $\eta=1\text{on}\ \mathbb{R}^{N}\setminus B(0,2r).$ For every $\epsilon>0,$ we let $v_{\epsilon}=\eta(x)V_{\epsilon}(x)$, where $V_{\epsilon}(x)=\epsilon^{-\frac{N-2s}{2}}V(\frac{x}{\epsilon})$ , $S_{s}$ is attained by $V(x)$. Then, the following estimates holds true(Proposition 21 in [35])

[TABLE]

[TABLE]

[TABLE]

Then, using the above estimates and by the similar arguments as [35], we can show that the least energy for equation (2.13) can not strictly less than $\frac{s}{N}S^{\frac{N}{2s}}_{s}$. So, (2.13) has no nontrivial solutions.

Thus, it is easily seen that $S_{s}$ is also the sharp constant of

[TABLE]

which implies that

[TABLE]

On the other hand, assume $0<\lambda\leq\sqrt{(\mu-\mu_{0})\nu},$ then $\mu-\frac{\lambda^{2}}{\nu}\geq\mu_{0}$. By the same arguments as Lemma 2.2, we have

[TABLE]

By (1.4), we have

[TABLE]

For any $(u,v)\in\mathcal{H}_{r}\setminus\{(0,0)\},$ if $v=0,$ then $\max\limits_{t>0}E_{\mu,\nu,\lambda}(tu,0)=\max\limits_{t>0}f_{\mu}(tu)\geq\frac{s}{N}S^{\frac{N}{2s}}_{s}.$ If $u=0,$ then $\max\limits_{t>0}E_{\mu,\nu,\lambda}(0,tv)=\max\limits_{t>0}g(tv)\geq\frac{s}{N}S^{\frac{N}{2s}}_{s}.$ If $u\neq 0$ and $v\neq 0$, then by Lemma 2.4 and Lemma 2.5, we have

[TABLE]

Thus,

[TABLE]

Combining (2.14) with (2.15), we obtain $A_{\mu,\nu,\lambda}=\frac{s}{N}S^{\frac{N}{2s}}_{s}.$

Step 2 We prove $(i)-(ii)$ in $(2)$.

Let $0<\lambda<\sqrt{\mu\nu}$, we define

[TABLE]

where $a_{\mu,\nu}$ is from (2.10).

Then,

[TABLE]

By Lemma 2.5 we have that $A_{0}\leq\frac{s}{N}S^{\frac{N}{2s}}_{s}$ is equivalent to

[TABLE]

By (2.9) and (2.10), we can deduce that the above inequality is equivalent to $\lambda>\widetilde{\lambda}_{\mu,\nu}.$ Combining this with (2.11), for any $\lambda\in[\widetilde{\lambda}_{\mu,\nu},\sqrt{\mu\nu}),$ we have $A_{\mu,\nu,\lambda}<\frac{s}{N}S^{\frac{N}{2s}}_{s}.$ Define

[TABLE]

Then, by (2.12), we know $\lambda_{\mu,\nu}\in[\sqrt{(\mu-\mu_{0})\nu},\widetilde{\lambda}_{\mu,\nu}]$ and for any $\lambda\in(\lambda_{\mu,\nu},\sqrt{\mu\nu}),$ there holds $A_{\mu,\nu,\lambda}<\frac{s}{N}S^{\frac{N}{2s}}_{s}.$ This completes the proof of $(ii)$.

We show that $A_{\mu,\nu,\lambda_{\mu,\nu}}=\frac{s}{N}S^{\frac{N}{2s}}_{s},$ which implies $\lambda_{\mu,\nu}<\widetilde{\lambda}_{\mu,\nu}$ immediately.

By (2.14), we have $A_{\mu,\nu,\lambda_{\mu,\nu}}\leq\frac{s}{N}S^{\frac{N}{2s}}_{s}.$ By the definition of $\lambda_{\mu,\nu},$ there exists $\lambda_{n}<\lambda_{\mu,\nu},\ n\geq 1$ such that

[TABLE]

For any $(u,v)\in\mathcal{H}_{r}\setminus\{(0,0)\},$ there exist $t_{n}>0$ such that $\max\limits_{t>0}E_{\mu,\nu,\lambda_{n}}(tu,tv)=E_{\mu,\nu,\lambda_{n}}(t_{n}u,t_{n}v).$ Since $\lambda_{n}\rightarrow\lambda_{\mu,\nu},$ we have $t_{n}\rightarrow t_{0}$ as $n\rightarrow+\infty,$ where $t_{0}$ satisfies $\max\limits_{t>0}E_{\mu,\nu,\lambda_{n}}(tu,tv)=E_{\mu,\nu,\lambda_{n}}(t_{0}u,t_{0}v).$ Then,

[TABLE]

This implies

[TABLE]

Thus, $A_{\mu,\nu,\lambda_{\mu,\nu}}=\frac{s}{N}S^{\frac{N}{2s}}_{s}.$

Next, we claim $A_{\mu,\nu,\lambda}$ is non-increasing with respect to $\lambda>0.$

Since

[TABLE]

then we have

[TABLE]

Let $\lambda_{1}<\lambda_{2}.$ Then for any $(u,v)\in\mathcal{H}_{r}\setminus\{(0,0)\}$ and $t>0$ we have

[TABLE]

Thus, $A_{\mu,\nu,\lambda_{1}}\geq A_{\mu,\nu,\lambda_{2}}.$ Consequently, $A_{\mu,\nu,\lambda}$ is non-increasing with respect to $\lambda>0.$

Combining this fact with (2.12), we see that $(i)$ holds. This completes the proof. ∎

3 Proof of Theorem 1.1

Proof of Theorem 1.1.

By principle of symmetric criticality (Theorem 1.28 in [39]), the solutions for (1.2) in function space ${H_{r}^{s}(\mathbb{R}^{N})}\times{H_{r}^{s}(\mathbb{R}^{N})}$ are also the solutions in function space ${H^{s}(\mathbb{R}^{N})}\times{H^{s}(\mathbb{R}^{N})}$.

We prove Theorem 1.1 by two steps. First, we prove the existence of ground state solutions for system (1.2) in step 1, then we claim there exist a positive ground state solution.

Step 1. Prove the existence of ground state solutions for system (1.2).

By (2.1) and the proof of Lemma 2.2, there exists a bounded sequence $\{(u_{n},v_{n})\}\subset\mathcal{H}_{r}$, such that

[TABLE]

Thus, by Sobolev Imbedding Theorem, there exist $(u,v)\in\mathcal{H}_{r}$ such that

[TABLE]

If $(u,v)\not\equiv(0,0)$, then we have

[TABLE]

Let $w_{n}=u_{n}-u$ and $\sigma_{n}=v_{n}-v$. We claim

[TABLE]

[TABLE]

By Lemma 2.3, there holds

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Since

[TABLE]

and

[TABLE]

Combining (3.5)-(3.9) with (3.10) , we obtain identity (3.3).

Since, for any $(\phi,\varphi)\in\mathcal{H}_{r}$, we have

[TABLE]

By Vatali’s Theorem [32], we have

[TABLE]

Similarly, we have

[TABLE]

By (3.11), (3.12), (3.13), we obtain identity (3.4).

Thus, by (3.2),(3.3), (3.4) and Lemma 2.2, we deduce

[TABLE]

which gives that

[TABLE]

On the other hand, by (3.4), we have

[TABLE]

Since $\lambda<\sqrt{\mu\nu}$, we can take a small $\tau>0$ such that $\lambda^{2}=(\mu-\tau)(\nu-\tau)$, by Sobolev imbedding, (3.14) and (3.15), we find

[TABLE]

which implies that

[TABLE]

As a result, $(u,v)$ is a critical point of $E_{\mu,\nu,\lambda}$ and satisfies

[TABLE]

By Lemma 2.2, we obtain

[TABLE]

If $(u,v)\equiv(0,0)$, by (3.15) and

[TABLE]

We have

[TABLE]

Assuming by contradiction, we can assume without loss of generality that

[TABLE]

By (3.15) and Cauchy-Schwarz inequality, we have

[TABLE]

By Sobolev imbedding $D_{r}^{s}(\mathbb{R}^{N})\hookrightarrow L^{2^{\ast}}(\mathbb{R}^{N})$, we have

[TABLE]

Combining (3.17) with (3.18), we can deduce that

[TABLE]

Let $n\rightarrow+\infty$ in (3.16), we obtain

[TABLE]

This contradict with Lemma 2.6.

Consequently, $E_{\mu,\nu,\lambda}(u,v)=\widehat{A}_{\mu,\nu,\lambda}=A_{\mu,\nu,\lambda}$ and $E^{\prime}_{\mu,\nu,\lambda}(u,v)=0$. That is $(u,v)$ is a nontrivial solution of system (1.2).

Step 2. We claim that there exist a positive ground state solution.

Since

[TABLE]

hence,

[TABLE]

Then, for the minimizing sequence $(u_{n},v_{n})\in\mathbb{M}_{r}$, we have

[TABLE]

this implies that there exists $t_{n}\in(0,1]$ such that $(t_{n}|u_{n}|,t_{n}|v_{n}|)\in\mathbb{M}_{r}.$ Hence, we can choose a minimizing sequence $(\overline{u}_{n},\overline{v}_{n})=(t_{n}|u_{n}|,t_{n}|v_{n}|)$ and the weak limit $(\overline{u},\overline{v})$ is nonnegative. By Strong maximum principle for fractional Laplacian( see, Proposition 2.17 in [37]), we have $\overline{u}$ and $\overline{v}$ are both positive.

Next, we claim if $\mu>\mu_{0}$ and $0<\lambda<\lambda_{\mu,\nu}$, then system (1.2) has no ground state solution.

Assume by contradiction that there exist $\lambda\in(0,\lambda_{\mu,\nu})$ such that system (1.2) has a ground state solution $(u_{\lambda},v_{\lambda})\neq(0,0)$. Then $E_{\mu,\nu,\lambda}(u_{\lambda},v_{\lambda})=A_{\mu,\nu,\lambda}=\frac{s}{N}S^{\frac{N}{2s}}_{s}.$ By (2.16) and (2.17) we may assume that $u_{\lambda}\geq 0,v_{\lambda}\geq 0,$ by Strong maximum principle for fractional Laplacian( see Proposition 2.17 in [37]), we have $u_{\lambda}>0,v_{\lambda}>0.$ If we take $\lambda_{1}\in(\lambda,\lambda_{\mu,\nu})$, then by Lemma 2.6 and (2.2), we have

[TABLE]

a contradiction. This completes the proof. ∎