On elliptic equations with Stein-Weiss type convolution parts

TL;DR

This paper investigates critical and subcritical elliptic equations with Stein-Weiss convolution terms, establishing existence, symmetry, and regularity of solutions using concentration-compactness and moving plane methods.

Contribution

It develops a nonlocal concentration-compactness principle and applies it to analyze solutions of Stein-Weiss type elliptic equations, including regularity and symmetry results.

Findings

Existence of solutions in critical and subcritical cases

Solutions exhibit symmetry and regularity

Development of a nonlocal concentration-compactness framework

Abstract

The aim of this paper is to study the critical elliptic equations with Stein-Weiss type convolution parts $$ \displaystyle-\Delta u =\frac{1}{|x|^{\alpha}}\left(\int_{\mathbb{R}^{N}}\frac{|u(y)|^{2_{\alpha, \mu}^{\ast}}}{|x-y|^{\mu}|y|^{\alpha}}dy\right) |u|^{2_{\alpha, \mu}^{\ast}-2}u,~~~x\in\mathbb{R}^{N}, $$ where the critical exponent is due to the weighted Hardy-Littlewood-Sobolev inequality and Sobolev embedding. We develop a nonlocal version of concentration-compactness principle to investigate the existence of solutions and study the regularity, symmetry of positive solutions by moving plane arguments. In the second part, the subcritical case is also considered, the existence, symmetry, regularity of the positive solutions are obtained.