A Nonexistence Result for Choquard-Type Hamiltonian System

Zexi Wang

School of Mathematics and Statistics, Southwest University, Chongqing, China.

DOI: 10.4236/jamp.2023.113038                

Abstract

In this article, we establish a nonexistence result of nontrivial non-negative solutions for the following Choquard-type Hamiltonian system by the Pohožaev identity , when , , , , , and , where and denotes the convolution in .

Keywords

Nonexistence, Choquard-Type Hamiltonian System, Pohožaev Identity

1. Introduction and Statement of Main Result

Recently, a lot of attention has been focused on the study of the following Choquard-type Hamiltonian system

$\{\begin{array}{l}-\Delta u+u=\left(I_{{\mu }_1}\ast \frac{{\left|v\right|}^p}{{\left|x\right|}^{\alpha }}\right)\frac{{\left|v\right|}^{p-2}v}{{\left|x\right|}^{\alpha }}\mathrm{,}\text{in}{ℝ}^N\backslash \left\{0\right\}\mathrm{,}\\ -\Delta v+v=\left(I_{{\mu }_2}\ast \frac{{\left|u\right|}^q}{{\left|x\right|}^{\beta }}\right)\frac{{\left|u\right|}^{q-2}u}{{\left|x\right|}^{\beta }}\mathrm{,}\text{in}{ℝ}^N\backslash \left\{0\right\}\mathrm{,}\\ u\left(x\right)\mathrm{,}v\left(x\right)\to \mathrm{0,}\text{when}\left|x\right|\to \infty \mathrm{,}\end{array}$ (1.1)

where $N\ge 3$ , $0<{\mu }_1,{\mu }_2<N$ , $0\le \alpha \le \frac{{\mu }_1}{2}$ , $0\le \beta \le \frac{{\mu }_2}{2}$ , $p,q>1$ , $I_{{\mu }_i}=\frac{1}{{\left|x\right|}^{N-{\mu }_i}}$ for $i=1,2,$ $\ast$ is the convolution in ${ℝ}^N$ .

When $\alpha =\beta =0$ , ${\mu }_1={\mu }_2$ , $p=q$ , $u=v$ , (1.1) reduces to the following classic Choquard equation

$-\Delta u+u=\left(I_{{\mu }_1}\ast {\left|u\right|}^p\right){\left|u\right|}^{p-2}u\mathrm{,}\text{in}{ℝ}^N\mathrm{.}$ (1.2)

Equation (1.2) has a physical prototype, as pointed out by Lieb [1] , Choquard introduced the equation

$i{\psi }_t=-\Delta \psi -\left(I_2\ast {\left|\psi \right|}^2\right)\psi \mathrm{,}\left(x\mathrm{,}t\right)\in {ℝ}^3\times {ℝ}^{+}\mathrm{,}$

to describe an electron trapped in its own hole as an approximation to Hartree-Fock theory for a one component plasma. It also arises in multiple particle systems and Quantum Mechanics [2] . In a pioneering work, set $\psi \left(x\mathrm{,}t\right)={\text{e}}^{it}u\left(x\right)$ , then

$-\Delta u+u=\left(I_2\ast {\left|u\right|}^2\right)u\mathrm{,}\text{in}{ℝ}^3\mathrm{,}$ (1.3)

and Lieb [1] first obtained the existence and uniqueness of a ground state solution to (1.3) via variational methods. Lions [3] [4] considered the same problem and proved the existence and multiplicity of normalized solutions. The classification of positive solutions was first studied by Ma and Zhao [5] .

As for (1.2), Moroz and Van Schaftingen [6] studied the positivity, regularity, decay asymptotics and radial symmetry of ground state solutions for

$\frac{N-2}{N+{\mu }_1}<\frac{1}{p}<\frac{N}{N+{\mu }_1}$ . Meanwhile, they also proved that (1.2) has no nontrivial smooth $H^1$ solution for either $\frac{1}{p}\le \frac{N-2}{N+{\mu }_1}$ or $\frac{1}{p}\ge \frac{N}{N+{\mu }_1}$ by using the Pohožaev identity. The number $\frac{N+{\mu }_1}{N}$ and $\frac{N+{\mu }_1}{N-2}$ (if $N\ge 3$ ) are called the lower and upper critical exponents related to the Hardy-Littlewood-Sobolev inequality, respectively. Furthermore, if ${\mu }_1={\mu }_2$ , $0\le \alpha =\beta \le \frac{{\mu }_1}{2}$ and $p=q$ , $u=v$ in (1.1), Du et al. [7] also established the nonexistence result for

$p\ge \frac{N+{\mu }_1-2\alpha }{N-2}$ or $p\le \frac{N+{\mu }_1-2\alpha }{N}$ when $N\ge 3$ by the Pohožaev identity.

By using the method of moving planes in integral forms introduced by Chen et

al. [8] , Le [9] proved the following equation has no positive solution if $p<\frac{N+{\mu }_1}{N-2}$ ( $p\in ℝ$ when $N\le 2$ ), and every positive solution u has the form $u\left(x\right)=c{\left(\frac{\lambda }{{\lambda }^2+{\left|x-x_0\right|}^2}\right)}^{\frac{N-2}{2}}$ for some $c,\lambda >0$ and $x_0\in {ℝ}^N$ ,

$-\Delta u=\left(I_{{\mu }_1}\ast {\left|u\right|}^p\right){\left|u\right|}^{p-2}u\mathrm{,}\text{in}{ℝ}^N\mathrm{.}$

As for more investigations about Choquard equations, we refer to [10] .

For the Hamiltonian system, if $\alpha =\beta =0$ in (1.1), Maia and Miyagaki [11] studied the following Choquard-type Hamiltonian system

$\{\begin{array}{l}-\Delta u+u=\left(I_{{\mu }_1}\ast {\left|v\right|}^p\right){\left|v\right|}^{p-2}v\mathrm{,}\text{in}{ℝ}^N\mathrm{,}\\ -\Delta v+v=\left(I_{{\mu }_2}\ast {\left|u\right|}^q\right){\left|u\right|}^{q-2}u\mathrm{,}\text{in}{ℝ}^N\mathrm{,}\\ u\left(x\right)\mathrm{,}v\left(x\right)\to \mathrm{0,}\text{when}\left|x\right|\to \infty \mathrm{.}\end{array}$ (1.4)

However, different from (1.2), the structure of (1.4) makes it quite difficult to obtain the Pohožaev identity for (1.4). In the spirit of the method in [12] , Maia and Miyagaki used similar arguments to overcome this difficulty and obtained the Pohožaev identity, they proved that: if $N\ge 3$ , (1.4) has no nontrivial non-negative $C^2$ solution for $p\ge \frac{N+{\mu }_1}{N-2}$ and $q\ge \frac{N+{\mu }_2}{N-2}$ ; if $N=2$ , (1.4) has no nontrivial non-negative $C^2$ solution for $p\le \frac{2+{\mu }_1}{2}$ and $q\le \frac{2+{\mu }_2}{2}$ . The key

idea of [12] is that, for the Hamiltonian system of 2 equations, consider a pair of non-negative solutions $\left(u\mathrm{,}v\right)$ , define $U\mathrm{<mo>\; :\; </mo>}=x\cdot \nabla u\left(x\right)$ and $V\mathrm{<mo>\; :\; </mo>}=x\cdot \nabla v\left(x\right)$ . Through a straightforward calculation to obtain $v\Delta U$ and $u\Delta V$ , then by using the differential knowledge and $\left(u\mathrm{,}v\right)$ is solution, we can get a differential form of Pohožaev identity, which will produce the Pohožaev identity. This new idea also helps Kou and An [13] to generalize the well-known results of Mitidieri [14] and discuss the nonexistence result of positive solutions for the Hamiltonian system in a non-star shaped domain. By the method of moving planes in integral forms, Le [15] also showed that the following system has no positive classical solution

$\{\begin{array}{l}-\Delta u=\left(I_{{\mu }_1}\ast {\left|v\right|}^p\right){\left|v\right|}^{p-2}v\mathrm{,}\text{in}{ℝ}^N\mathrm{,}\\ -\Delta v=\left(I_{{\mu }_2}\ast {\left|u\right|}^q\right){\left|u\right|}^{q-2}u\mathrm{,}\text{in}{ℝ}^N\mathrm{,}\end{array}$

when $N\ge 3$ , $1<p\le \frac{N+{\mu }_1}{N-2}$ , $1<q\le \frac{N+{\mu }_2}{N-2}$ and $p+q<\frac{2N+{\mu }_1+{\mu }_2}{N-2}$ .

Other existence or nonexistence results of solutions for equations can be find, we refer readers to [16] [17] [18] [19] [20] and references therein.

Motivated by the aforementioned papers, in the present paper, we give a nonexistence result of nontrivial non-negative solutions for (1.1) with $\alpha \mathrm{,}\beta \ge 0$ by the Pohožaev identity.

The main result of this paper is the following:

Theorem 1.1. Assume that $N\ge 3$ , $0<{\mu }_1,{\mu }_2<N$ , $0\le \alpha \le \frac{{\mu }_1}{2}$ , $0\le \beta \le \frac{{\mu }_2}{2}$ , $p,q>1$ , and $\frac{N+{\mu }_1-2\alpha }{p}+\frac{N+{\mu }_2-2\beta }{q}\le 2\left(N-2\right)$ , if

$\begin{array}{l}\left(u\mathrm{,}v\right)\in \left(C^2\left({ℝ}^N\backslash \left\{0\right\}\right)\cap H^1\left({ℝ}^N\right)\cap L^{\frac{2Np}{N+{\mu }_1-2\alpha }}\left({ℝ}^N\right)\right)\\ \times \left(C^2\left({ℝ}^N\backslash \left\{0\right\}\right)\cap H^1\left({ℝ}^N\right)\cap L^{\frac{2Nq}{N+{\mu }_2-2\beta }}\left({ℝ}^N\right)\right)\end{array}$ is a pair of non-negative solutions of (1.1), then $\left(u,v\right)=\left(0,0\right)$ . In particular, (1.1) does not have a pair of nontrivial non-negative solutions for $p\ge \frac{N+{\mu }_1-2\alpha }{N-2}$ and $q\ge \frac{N+{\mu }_2-2\beta }{N-2}$ .

In Section 2, we will give the proof of Theorem 1.1. To facilitate reading, we use the notations:

• $C^2\left({ℝ}^N\backslash \left\{0\right\}\right)$ is the space of functions whose 2-th derivatives are continuous in ${ℝ}^N\backslash \left\{0\right\}$ .

• $C_0^{\infty }\left({ℝ}^N\right)$ is the space of functions infinitely differentiable with compact support in ${ℝ}^N$ .

• $L^p\left({ℝ}^N\right)$ , $p\in \left[\mathrm{1,}+\infty \right)$ is the usual Lebesgue space endowed with the norm ${\Vert u\Vert }_t={\left({\displaystyle {\int }_{{ℝ}^N}}{\left|u\right|}^t\text{d}x\right)}^{\frac{1}{t}}$ .

• $H^1\left({ℝ}^N\right)=\left\{u\in L^2\left({ℝ}^N\right)\mathrm{<mo>\; :\; </mo>}\nabla u\in L^2\left({ℝ}^N\right)\right\}$ is endowed with norm $\Vert u\Vert ={\left({\displaystyle {\int }_{{ℝ}^N}}{\left|\nabla u\right|}^2+u^2\text{d}x\right)}^{\frac{1}{2}}$ .

2. Proof of Theorem 1.1

To study (1.1), we need the following doubly weighted Hardy-Littlewood-Sobolev inequality proved in [21] .

Proposition 2.1. (Doubly Weighted Hardy-Littlewood-Sobolev Inequality) Let $t,r>1$ and $0<\mu <N$ with $0\le \alpha +\beta \le \mu$ , $\frac{1}{t}+\frac{\alpha +\beta -\mu }{N}+\frac{1}{r}=1$ , $\alpha <\frac{N}{t^{\prime }}$ , $\beta <\frac{N}{r^{\prime }}$ , $f\in L^t\left({ℝ}^N\right)$ and $h\in L^r\left({ℝ}^N\right)$ , where $\frac{1}{t}+\frac{1}{t^{\prime }}=1$ and $\frac{1}{r}+\frac{1}{r^{\prime }}=1$ . Then there exists a constant $C\left(\alpha ,\beta ,\mu ,N,t,r\right)>0$ which is independent of $f\mathrm{,}h$ such that

${\displaystyle {\int }_{{ℝ}^N}}\left(I_{\mu }\ast \frac{f\left(x\right)}{{\left|x\right|}^{\alpha }}\right)\frac{h\left(x\right)}{{\left|x\right|}^{\beta }}\text{d}x\le C\left(\alpha \mathrm{,}\beta \mathrm{,}\mu \mathrm{,}N\mathrm{,}t\mathrm{,}r\right){\Vert f\Vert }_t{\Vert h\Vert }_r\mathrm{.}$

From Proposition 2.1, we easily get the following remark.

Remark 2.1. Let $0<\mu <N$ , $0\le \alpha \le \frac{\mu }{2}$ , and $p>1$ . Assume that $u\in L^{\frac{2Np}{N+\mu -2\alpha }}\left({ℝ}^N\right)$ , then there exists $C\left(\alpha ,\mu ,N\right)>0$ such that

${\displaystyle {\int }_{{ℝ}^N}}\left(I_{\mu }\ast \frac{{\left|u\right|}^p}{{\left|x\right|}^{\alpha }}\right)\frac{{\left|u\right|}^p}{{\left|x\right|}^{\alpha }}\text{d}x\le C\left(\alpha \mathrm{,}\mu \mathrm{,}N\right){\Vert u\Vert }_{\frac{2Np}{N+\mu -2\alpha }}^{2p}\mathrm{.}$ (2.1)

Consider a cut-off function $\phi \in C_0^{\infty }\left({ℝ}^N\mathrm{,}\left[\mathrm{0,1}\right]\right)$ such that $\phi \left(x\right)=1$ if $\left|x\right|\le 1$ , $\phi \left(x\right)=0$ if $\left|x\right|\ge 2$ . For any fixed $\lambda >0$ , set

${\tilde{u}}_{\lambda }\left(x\right)=\phi \left(\lambda x\right)x\cdot \nabla u\left(x\right)\text{and}{\tilde{v}}_{\lambda }\left(x\right)=\phi \left(\lambda x\right)x\cdot \nabla v\left(x\right)\mathrm{.}$

Lemma 2.1. Let $u\mathrm{,}v\ge 0$ and $\left(u\mathrm{,}v\right)\in \left(C^2\left({ℝ}^N\backslash \left\{0\right\}\right)\cap L^{\frac{2Np}{N+{\mu }_1-2\alpha }}\left({ℝ}^N\right)\right)\times \left(C^2\left({ℝ}^N\backslash \left\{0\right\}\right)\cap L^{\frac{2Nq}{N+{\mu }_2-2\beta }}\left({ℝ}^N\right)\right)$ , then

$\underset{\lambda \to 0}{lim}{\displaystyle {\int }_{{ℝ}^N}}\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^{p-1}{\tilde{v}}_{\lambda }}{{\left|x\right|}^{\alpha }}\text{d}x=\frac{-N-{\mu }_1+2\alpha }{2p}{\displaystyle {\int }_{{ℝ}^N}}\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^p}{{\left|x\right|}^{\alpha }}\text{d}x\mathrm{,}$

and

$\underset{\lambda \to 0}{lim}{\displaystyle {\int }_{{ℝ}^N}}\left(I_{{\mu }_2}\ast \frac{u^q}{{\left|x\right|}^{\beta }}\right)\frac{u^{q-1}{\tilde{u}}_{\lambda }}{{\left|x\right|}^{\beta }}\text{d}x=\frac{-N-{\mu }_2+2\beta }{2q}{\displaystyle {\int }_{{ℝ}^N}}\left(I_{{\mu }_2}\ast \frac{u^q}{{\left|x\right|}^{\beta }}\right)\frac{u^q}{{\left|x\right|}^{\beta }}\text{d}x\mathrm{.}$

Proof. A direct computation, one has

$\begin{array}{l}{\displaystyle {\int }_{{ℝ}^N}}\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^{p-1}{\tilde{v}}_{\lambda }}{{\left|x\right|}^{\alpha }}\text{d}x\\ ={\displaystyle {\int }_{{ℝ}^N}}{\displaystyle {\int }_{{ℝ}^N}}{\left|x-y\right|}^{{\mu }_1-N}{\left|y\right|}^{-\alpha }{v}^p\left(y\right){\left|x\right|}^{-\alpha }{v}^{p-1}\left(x\right)\phi \left(\lambda x\right)\left[x\cdot \nabla v\left(x\right)\right]\text{d}x\text{d}y\\ ={\displaystyle {\int }_{{ℝ}^N}}{\displaystyle {\int }_{{ℝ}^N}}{\left|x-y\right|}^{{\mu }_1-N}{\left|y\right|}^{-\alpha }{v}^p\left(y\right){\left|x\right|}^{-\alpha }\phi \left(\lambda x\right)\left[x\cdot \nabla \left(\frac{v^p\left(x\right)}{p}\right)\right]\text{d}x\text{d}y\\ =\frac{1}{2}{\displaystyle {\int }_{{ℝ}^N}}{\displaystyle {\int }_{{ℝ}^N}}{\left|x-y\right|}^{{\mu }_1-N}{\left|x\right|}^{-\alpha }{\left|y\right|}^{-\alpha }{v}^p\left(y\right)\phi \left(\lambda x\right)\left[x\cdot \nabla \left(\frac{v^p\left(x\right)}{p}\right)\right]\\ +{\left|x-y\right|}^{{\mu }_1-N}{\left|x\right|}^{-\alpha }{\left|y\right|}^{-\alpha }{v}^p\left(x\right)\phi \left(\lambda y\right)\left[y\cdot \nabla \left(\frac{v^p\left(y\right)}{p}\right)\right]\text{d}x\text{d}y\end{array}$

$\begin{array}{l}=-\frac{1}{2}{\displaystyle {\int }_{{ℝ}^N}}{\displaystyle {\int }_{{ℝ}^N}}[\left({\mu }_1-N\right){\left|x-y\right|}^{{\mu }_1-N}{\left|x\right|}^{-\alpha }{\left|y\right|}^{-\alpha }{v}^p\left(y\right)\phi \left(\lambda x\right)\frac{v^p\left(x\right)}{p}\frac{x\left(x-y\right)}{{\left|x-y\right|}^2}\\ +\left({\mu }_1-N\right){\left|x-y\right|}^{{\mu }_1-N}{\left|x\right|}^{-\alpha }{\left|y\right|}^{-\alpha }{v}^p\left(x\right)\phi \left(\lambda y\right)\frac{v^p\left(y\right)}{p}\frac{-y\left(x-y\right)}{{\left|x-y\right|}^2}\\ -\alpha {\left|x-y\right|}^{{\mu }_1-N}{\left|x\right|}^{-\alpha }{\left|y\right|}^{-\alpha }{v}^p\left(y\right)\phi \left(\lambda x\right)\frac{v^p\left(x\right)}{p}\\ -\alpha {\left|x-y\right|}^{{\mu }_1-N}{\left|x\right|}^{-\alpha }{\left|y\right|}^{-\alpha }{v}^p\left(x\right)\phi \left(\lambda y\right)\frac{v^p\left(y\right)}{p}\end{array}$

$\begin{array}{l}+\lambda {\left|x-y\right|}^{{\mu }_1-N}{\left|x\right|}^{-\alpha }{\left|y\right|}^{-\alpha }{v}^p\left(y\right)\left[x\cdot \nabla \phi \left(\lambda x\right)\right]\frac{v^p\left(x\right)}{p}\\ +\lambda {\left|x-y\right|}^{{\mu }_1-N}{\left|x\right|}^{-\alpha }{\left|y\right|}^{-\alpha }{v}^p\left(x\right)\left[y\cdot \nabla \phi \left(\lambda y\right)\right]\frac{v^p\left(y\right)}{p}\\ +N{\left|x-y\right|}^{{\mu }_1-N}{\left|x\right|}^{-\alpha }{\left|y\right|}^{-\alpha }{v}^p\left(y\right)\phi \left(\lambda x\right)\frac{v^p\left(x\right)}{p}\\ +N{\left|x-y\right|}^{{\mu }_1-N}{\left|x\right|}^{-\alpha }{\left|y\right|}^{-\alpha }{v}^p\left(x\right)\phi \left(\lambda y\right)\frac{v^p\left(y\right)}{p}]\text{d}x\text{d}y\end{array}$

$\begin{array}{l}=\frac{N-{\mu }_1}{2p}{\displaystyle {\int }_{{ℝ}^N}}{\displaystyle {\int }_{{ℝ}^N}}\frac{1}{{\left|x-y\right|}^{N-{\mu }_1}}\frac{v^p\left(x\right)}{{\left|x\right|}^{\alpha }}\frac{v^p\left(y\right)}{{\left|y\right|}^{\alpha }}\frac{\left(x-y\right)\cdot \left(x\phi \left(\lambda x\right)-y\phi \left(\lambda y\right)\right)}{{\left|x-y\right|}^2}\text{d}x\text{d}y\\ +\frac{\alpha }{p}{\displaystyle {\int }_{{ℝ}^N}}{\displaystyle {\int }_{{ℝ}^N}}\frac{1}{{\left|x-y\right|}^{N-{\mu }_1}}\frac{v^p\left(x\right)}{{\left|x\right|}^{\alpha }}\frac{v^p\left(y\right)}{{\left|y\right|}^{\alpha }}\phi \left(\lambda x\right)\text{d}x\text{d}y\\ -\frac{N}{p}{\displaystyle {\int }_{{ℝ}^N}}{\displaystyle {\int }_{{ℝ}^N}}\frac{1}{{\left|x-y\right|}^{N-{\mu }_1}}\frac{v^p\left(x\right)}{{\left|x\right|}^{\alpha }}\frac{v^p\left(y\right)}{{\left|y\right|}^{\alpha }}\phi \left(\lambda x\right)\text{d}x\text{d}y\\ -\frac{\lambda }{p}{\displaystyle {\int }_{{ℝ}^N}}{\displaystyle {\int }_{{ℝ}^N}}\frac{1}{{\left|x-y\right|}^{N-{\mu }_1}}\frac{v^p\left(x\right)}{{\left|x\right|}^{\alpha }}\frac{v^p\left(y\right)}{{\left|y\right|}^{\alpha }}\left[x\cdot \nabla \phi \left(\lambda x\right)\right]\text{d}x\text{d}y.\end{array}$

Using the Lebesgue dominated convergence theorem, we get the first equality. Similarly, we obtain the second equality.

Lemma 2.2. Under the assumptions of Theorem 1.1, the following identity holds

$\begin{array}{c}4{\displaystyle {\int }_{{ℝ}^N}}uv\text{d}x=\left[\frac{N+{\mu }_1-2\alpha }{p}-\left(N-2\right)\right]{\displaystyle {\int }_{{ℝ}^N}}\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^p}{{\left|x\right|}^{\alpha }}\text{d}x\\ +\left[\frac{N+{\mu }_2-2\beta }{q}-\left(N-2\right)\right]{\displaystyle {\int }_{{ℝ}^N}}\left(I_{{\mu }_2}\ast \frac{u^q}{{\left|x\right|}^{\beta }}\right)\frac{u^q}{{\left|x\right|}^{\beta }}\text{d}x\mathrm{.}\end{array}$

Proof. A direct computation, we have

$v\Delta {\tilde{u}}_{\lambda }={\lambda }^2\left(x\cdot \nabla u\right)v\Delta \phi \left(\lambda x\right)+2\lambda v\nabla \phi \left(\lambda x\right)\cdot \nabla \left(x\cdot \nabla u\right)+\phi \left(\lambda x\right)v\Delta \left(x\cdot \nabla u\right)\mathrm{,}$

$\phi \left(\lambda x\right)v\Delta \left(x\cdot \nabla u\right)=\phi \left(\lambda x\right)v\left[2\Delta u+x\cdot \nabla \left(\Delta u\right)\right]\mathrm{,}$

and

$\begin{array}{l}\frac{\partial }{{\partial }_{x_i}}\left[\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^p}{{\left|x\right|}^{\alpha }}{x}_i\right]\\ =\left({\mu }_1-N\right){\displaystyle {\int }_{{ℝ}^N}}\frac{I_{{\mu }_1}\left(x-y\right)v^p\left(y\right)v^p\left(x\right)x_i\left(x_i-y_i\right)}{{\left|x-y\right|}^2{\left|x\right|}^{\alpha }{\left|y\right|}^{\alpha }}\text{d}y\\ +p\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^{p-1}}{{\left|x\right|}^{\alpha }}{v}_{x_i}{x}_i-\alpha \left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^p}{{\left|x\right|}^{\alpha }}\frac{x_i^2}{{\left|x\right|}^2}+\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^p}{{\left|x\right|}^{\alpha }}.\end{array}$ (2.2)

Since $\left(u\mathrm{,}v\right)$ solves (1.1), we have $-\Delta u+u=\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^{p-1}}{{\left|x\right|}^{\alpha }}$ , combing this with (2.2), we obtain

$\begin{array}{l}{\displaystyle {\int }_{{ℝ}^N}}v\Delta {\tilde{u}}_{\lambda }\text{d}x\\ ={\lambda }^2{\displaystyle {\int }_{{ℝ}^N}}\left(x\cdot \nabla u\right)v\Delta \phi \left(\lambda x\right)\text{d}x+2\lambda {\displaystyle {\int }_{{ℝ}^N}}v\nabla \phi \left(\lambda x\right)\cdot \nabla \left(x\cdot \nabla u\right)\text{d}x\\ +2{\displaystyle {\int }_{{ℝ}^N}}\phi \left(\lambda x\right)uv\text{d}x-2{\displaystyle {\int }_{{ℝ}^N}}\phi \left(\lambda x\right)\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^p}{{\left|x\right|}^{\alpha }}\text{d}x\\ +{\displaystyle {\int }_{{ℝ}^N}}v{\tilde{u}}_{\lambda }\left(x\right)\text{d}x-{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{{ℝ}^N}}\phi \left(\lambda x\right)\frac{\partial }{{\partial }_{x_i}}\left[\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^p}{{\left|x\right|}^{\alpha }}{x}_i\right]\text{d}x\\ +{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{{ℝ}^N}}\phi \left(\lambda x\right)\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^{p-1}}{{\left|x\right|}^{\alpha }}{v}_{x_i}{x}_i\text{d}x+N{\displaystyle {\int }_{{ℝ}^N}}\phi \left(\lambda x\right)\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^p}{{\left|x\right|}^{\alpha }}\text{d}x.\end{array}$ (2.3)

Analogously,

$\begin{array}{l}{\displaystyle {\int }_{{ℝ}^N}}u\Delta {\tilde{v}}_{\lambda }\text{d}x\\ ={\lambda }^2{\displaystyle {\int }_{{ℝ}^N}}\left(x\cdot \nabla v\right)u\Delta \phi \left(\lambda x\right)\text{d}x+2\lambda {\displaystyle {\int }_{{ℝ}^N}}u\nabla \phi \left(\lambda x\right)\cdot \nabla \left(x\cdot \nabla v\right)\text{d}x\\ +2{\displaystyle {\int }_{{ℝ}^N}}\phi \left(\lambda x\right)uv\text{d}x-2{\displaystyle {\int }_{{ℝ}^N}}\phi \left(\lambda x\right)\left(I_{{\mu }_2}\ast \frac{u^q}{{\left|x\right|}^{\beta }}\right)\frac{u^q}{{\left|x\right|}^{\beta }}\text{d}x\end{array}$

$\begin{array}{l}+{\displaystyle {\int }_{{ℝ}^N}}u{\tilde{v}}_{\lambda }\left(x\right)\text{d}x-{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{{ℝ}^N}}\phi \left(\lambda x\right)\frac{\partial }{{\partial }_{x_i}}\left[\left(I_{{\mu }_2}\ast \frac{u^q}{{\left|x\right|}^{\beta }}\right)\frac{u^q}{{\left|x\right|}^{\beta }}{x}_i\right]\text{d}x\\ +{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{{ℝ}^N}}\phi \left(\lambda x\right)\left(I_{{\mu }_2}\ast \frac{u^q}{{\left|x\right|}^{\beta }}\right)\frac{u^{q-1}}{{\left|x\right|}^{\beta }}{u}_{x_i}{x}_i\text{d}x+N{\displaystyle {\int }_{{ℝ}^N}}\phi \left(\lambda x\right)\left(I_{{\mu }_2}\ast \frac{u^q}{{\left|x\right|}^{\beta }}\right)\frac{u^q}{{\left|x\right|}^{\beta }}\text{d}x.\end{array}$ (2.4)

We multiply the first equation in (1.1) by ${\tilde{v}}_{\lambda }$ , and integrate over ${ℝ}^N$ , subtract from (2.3) to obtain

$\begin{array}{l}2{\displaystyle {\int }_{{ℝ}^N}}\phi \left(\lambda x\right)\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^p}{{\left|x\right|}^{\alpha }}\text{d}x-2{\displaystyle {\int }_{{ℝ}^N}}\phi \left(\lambda x\right)uv\text{d}x\\ =-{\displaystyle {\int }_{{ℝ}^N}}v\Delta {\tilde{u}}_{\lambda }\text{d}x+{\lambda }^2{\displaystyle {\int }_{{ℝ}^N}}\left(x\cdot \nabla u\right)v\Delta \phi \left(\lambda x\right)\text{d}x\\ +2\lambda {\displaystyle {\int }_{{ℝ}^N}}v\nabla \phi \left(\lambda x\right)\cdot \nabla \left(x\cdot \nabla u\right)\text{d}x\\ -{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{{ℝ}^N}}\phi \left(\lambda x\right)\frac{\partial }{{\partial }_{x_i}}\left[\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^p}{{\left|x\right|}^{\alpha }}{x}_i\right]\text{d}x\\ +{\displaystyle {\int }_{{ℝ}^N}}v{\tilde{u}}_{\lambda }\left(x\right)\text{d}x+N{\displaystyle {\int }_{{ℝ}^N}}\phi \left(\lambda x\right)\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^p}{{\left|x\right|}^{\alpha }}\text{d}x\\ +{\displaystyle {\int }_{{ℝ}^N}}{\tilde{v}}_{\lambda }\Delta u\text{d}x-{\displaystyle {\int }_{{ℝ}^N}}u{\tilde{v}}_{\lambda }\text{d}x+2{\displaystyle {\int }_{{ℝ}^N}}\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^{p-1}}{{\left|x\right|}^{\alpha }}{\tilde{v}}_{\lambda }\text{d}x.\end{array}$ (2.5)

Similarly,

$\begin{array}{l}2{\displaystyle {\int }_{{ℝ}^N}}\phi \left(\lambda x\right)\left(I_{{\mu }_2}\ast \frac{u^q}{{\left|x\right|}^{\beta }}\right)\frac{u^q}{{\left|x\right|}^{\beta }}\text{d}x-2{\displaystyle {\int }_{{ℝ}^N}}\phi \left(\lambda x\right)uv\text{d}x\\ =-{\displaystyle {\int }_{{ℝ}^N}}u\Delta {\tilde{v}}_{\lambda }\text{d}x+{\lambda }^2{\displaystyle {\int }_{{ℝ}^N}}\left(x\cdot \nabla v\right)u\Delta \phi \left(\lambda x\right)\text{d}x\\ +2\lambda {\displaystyle {\int }_{{ℝ}^N}}u\nabla \phi \left(\lambda x\right)\cdot \nabla \left(x\cdot \nabla v\right)\text{d}x\\ -{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{{ℝ}^N}}\phi \left(\lambda x\right)\frac{\partial }{{\partial }_{x_i}}\left[\left(I_{{\mu }_2}\ast \frac{u^q}{{\left|x\right|}^{\beta }}\right)\frac{u^q}{{\left|x\right|}^{\beta }}{x}_i\right]\text{d}x\\ +{\displaystyle {\int }_{{ℝ}^N}}u{\tilde{v}}_{\lambda }\left(x\right)\text{d}x+N{\displaystyle {\int }_{{ℝ}^N}}\phi \left(\lambda x\right)\left(I_{{\mu }_2}\ast \frac{u^q}{{\left|x\right|}^{\beta }}\right)\frac{u^q}{{\left|x\right|}^{\beta }}\text{d}x\\ +{\displaystyle {\int }_{{ℝ}^N}}{\tilde{u}}_{\lambda }\Delta v\text{d}x-{\displaystyle {\int }_{{ℝ}^N}}v{\tilde{u}}_{\lambda }\text{d}x+2{\displaystyle {\int }_{{ℝ}^N}}\left(I_{{\mu }_2}\ast \frac{u^q}{{\left|x\right|}^{\beta }}\right)\frac{u^{q-1}}{{\left|x\right|}^{\beta }}{\tilde{u}}_{\lambda }\text{d}x.\end{array}$ (2.6)

Recalling that $u\left(x\right)\mathrm{,}v\left(x\right)\to 0$ when $\left|x\right|\to \infty$ and $\phi \in C_0^{\infty }\left({ℝ}^N\mathrm{,}\left[\mathrm{0,1}\right]\right)$ , by divergence theorem and Fubini theorem, we obtain

$-{\displaystyle {\int }_{{ℝ}^N}}v\Delta {\tilde{u}}_{\lambda }\text{d}x+{\displaystyle {\int }_{{ℝ}^N}}{\tilde{v}}_{\lambda }\Delta u\text{d}x-{\displaystyle {\int }_{{ℝ}^N}}u\Delta {\tilde{v}}_{\lambda }\text{d}x+{\displaystyle {\int }_{{ℝ}^N}}{\tilde{u}}_{\lambda }\Delta v\text{d}x=0$

and

$\begin{array}{l}-{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{{ℝ}^N}}\phi \left(\lambda x\right)\frac{\partial }{{\partial }_{x_i}}\left[\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^p}{{\left|x\right|}^{\alpha }}{x}_i\right]\text{d}x-{\displaystyle \underset{i=1}{\overset{N}{\sum }}}{\displaystyle {\int }_{{ℝ}^N}}\phi \left(\lambda x\right)\frac{\partial }{{\partial }_{x_i}}\left[\left(I_{{\mu }_2}\ast \frac{u^q}{{\left|x\right|}^{\beta }}\right)\frac{u^q}{{\left|x\right|}^{\beta }}{x}_i\right]\text{d}x\\ =\lambda {\displaystyle {\int }_{{ℝ}^N}}\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^p}{{\left|x\right|}^{\alpha }}\left[x\cdot \nabla \phi \left(\lambda x\right)\right]\text{d}x+\lambda {\displaystyle {\int }_{{ℝ}^N}}\left(I_{{\mu }_2}\ast \frac{u^q}{{\left|x\right|}^{\beta }}\right)\frac{u^q}{{\left|x\right|}^{\beta }}\left[x\cdot \nabla \phi \left(\lambda x\right)\right]\text{d}x\mathrm{.}\end{array}$

Consequently, adding (2.5) and (2.6), by Lemma 2.2, it follows that

$\begin{array}{l}2{\displaystyle {\int }_{{ℝ}^N}}\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^p}{{\left|x\right|}^{\alpha }}\text{d}x+2{\displaystyle {\int }_{{ℝ}^N}}\left(I_{{\mu }_2}\ast \frac{u^q}{{\left|x\right|}^{\beta }}\right)\frac{u^q}{{\left|x\right|}^{\beta }}\text{d}x-4{\displaystyle {\int }_{{ℝ}^N}}uv\text{d}x\\ =N{\displaystyle {\int }_{{ℝ}^N}}\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^p}{{\left|x\right|}^{\alpha }}\text{d}x+\frac{-N-{\mu }_1+2\alpha }{p}{\displaystyle {\int }_{{ℝ}^N}}\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^p}{{\left|x\right|}^{\alpha }}\\ +N{\displaystyle {\int }_{{ℝ}^N}}\left(I_{{\mu }_2}\ast \frac{u^q}{{\left|x\right|}^{\beta }}\right)\frac{u^q}{{\left|x\right|}^{\beta }}\text{d}x+\frac{-N-{\mu }_2+2\beta }{q}{\displaystyle {\int }_{{ℝ}^N}}\left(I_{{\mu }_2}\ast \frac{u^q}{{\left|x\right|}^{\beta }}\right)\frac{u^q}{{\left|x\right|}^{\beta }},\end{array}$

which is equivalent to

$\begin{array}{c}4{\displaystyle {\int }_{{ℝ}^N}}uv\text{d}x=\left[\frac{N+{\mu }_1-2\alpha }{p}-\left(N-2\right)\right]{\displaystyle {\int }_{{ℝ}^N}}\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^p}{{\left|x\right|}^{\alpha }}\text{d}x\\ +\left[\frac{N+{\mu }_2-2\beta }{q}-\left(N-2\right)\right]{\displaystyle {\int }_{{ℝ}^N}}\left(I_{{\mu }_2}\ast \frac{u^q}{{\left|x\right|}^{\beta }}\right)\frac{u^q}{{\left|x\right|}^{\beta }}\text{d}x\mathrm{.}\end{array}$

The proof of Lemma 2.2 is complete.

Proof of Theorem 1.1: If $\begin{array}{c}\left(u\mathrm{,}v\right)\in \left(C^2\left({ℝ}^N\backslash \left\{0\right\}\right)\cap H^1\left({ℝ}^N\right)\cap L^{\frac{2Np}{N+{\mu }_1-2\alpha }}\left({ℝ}^N\right)\right)\\ \times \left(C^2\left({ℝ}^N\backslash \left\{0\right\}\right)\cap H^1\left({ℝ}^N\right)\cap L^{\frac{2Nq}{N+{\mu }_2-2\beta }}\left({ℝ}^N\right)\right)\end{array}$ is a pair of non-negative solutions of (1.1), suppose that $\frac{N+{\mu }_1-2\alpha }{p}+\frac{N+{\mu }_2-2\beta }{q}\le 2\left(N-2\right)$ , by Lemma 2.2, there holds

$0\le 4{\displaystyle {\int }_{{ℝ}^N}}uv\text{d}x=\left(\frac{N+{\mu }_1-2\alpha }{p}+\frac{N+{\mu }_2-2\beta }{q}-2N+4\right){\displaystyle {\int }_{{ℝ}^N}}\left(I_{{\mu }_1}\ast \frac{v^p}{{\left|x\right|}^{\alpha }}\right)\frac{v^p}{{\left|x\right|}^{\alpha }}\text{d}x\le \mathrm{0,}$

which indicates $v=0$ . Similarly, we can prove that $u=0$ .

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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