Symmetry and monotonicity of positive solutions for an integral system with negative exponents

Zhao Liu

Vol. 300 (2019), No. 2, 419–430

DOI: 10.2140/pjm.2019.300.419

Abstract

We study the following integral system with negative exponents:

$$\left\{\begin{array}{c}u\left(x\right)={\int \; <!--nolimits-->}_{{ℝ}^n}|x-y{|}^{\alpha -n}{v}^{-p}\left(y\right)dy,\hfill \\ v\left(x\right)={\int \; <!--nolimits-->}_{{ℝ}^n}|x-y{|}^{\alpha -n}{u}^{-q}\left(y\right)dy,\hfill \end{array}\right.$$

with $\alpha >n$, $p,q>0$ and $\frac{1}{p-1}+\frac{1}{q-1}=\frac{\alpha -n}{n}$. Such a nonlinear integral system is related to the study of the best constant of the reversed Hardy–Littlewood–Sobolev type inequality. Motivated by work of Dou, Guo and Zhu (Adv. Math. 312 (2017) 1–45) where they used the improved method of moving planes, we prove that each pair of positive measurable solutions is radially symmetric and monotonic increasing about some point. Our result is an extension of the work for $\alpha <n$, $p,q<0$ of Chen, Li and Ou (Comm. Partial Differential Equations 30:1–3 (2005), 59–65).

Keywords

Hardy–Littlewood–Sobolev inequality, method of moving plane, negative exponent

Mathematical Subject Classification 2010

Primary: 45G15, 46E30

Milestones

Received: 26 February 2018

Revised: 11 September 2018

Accepted: 12 September 2018

Published: 30 July 2019