Nonexistence of positive solutions for a system of semilinear fractional Laplacian problem

Abstract

In this paper, we consider a system of semilinear equations involving the fractional Laplacian in the Euclidean space $\mathbb{R}^n$: \begin{equation*} \begin{cases} (-\Delta)^{\alpha/2}u(x)=f(x_n)v^p(x)\\ (-\Delta)^{\alpha/2}v(x)=g(x_n)u^q(x) \end{cases} \end{equation*} in the subcritical case $1 < p,q\le \frac{n+\alpha}{n-\alpha}$ where $\alpha \in (0,\,2)$. Instead of investigating the above system directly, we discuss its equivalent integral system: \begin{equation*} \begin{cases} u(x)=\int_{\mathbb{R}^n} G_{\infty}(x,y)f(y_n)v^p(y)dy\\ v(y)=\int_{\mathbb{R}^n} G_{\infty}(x,y)g(x_n)u^q(x)dx , \end{cases} \end{equation*} where $G_{\infty}(x, y)$ is the Green's function associated with the fractional Laplacian in $\mathbb{R}^n$. Under natural structure condition on $f$ and $g$, we indicate the nonexistence of the positive solutions to the above integral system according to the method of moving spheres in integral form and the classic Hardy-Littlewood-Sobolev inequality.