Symmetry and monotonicity of positive solutions for a Choquard equation with the fractional Laplacian

Abstract

In this paper, we consider the Choquard-type equation with the fractional Laplacian $\left\{\begin{array}{ll}(-\mathrm{\Delta }{)}^{s}u(x)+au(x)=\left(\frac{1}{|x{|}^{n-\alpha }}\ast u^p\right)u^{p-1},& x\in {\mathbb{R}}^n,\\ u\left(x\right)>0,& x\in {\mathbb{R}}^n,\end{array}\right.$ where 0<s<1, $0<\alpha <n$, $1\le p<\mathrm{\infty }$, $a\ge 0$ is a constant and $n\ge 2$. We will give a variant decay at infinity and a variant narrow region principle, then combining the direct method of moving planes to prove the solutions of the above equation must be radially symmetric and monotone decreasing about some point in ${\mathbb{R}}^n$.

Keywords:

• Fractional Laplacian
• Choquard equation
• direct method of moving planes
• decay at infinity
• narrow region principle

AMS Subject Classifications:

• 35H30
• 35J25
• 35J20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This project was supported by the National Natural Science Foundation of China [grant number 11571093]; the Natural Science Foundation of Jiangsu Education Commission China [grant number 19KJB110016].