On the moving plane method for nonlocal problems in bounded domains

TL;DR

This paper applies the moving plane method to analyze symmetry and monotonicity of positive solutions for nonlocal fractional Laplacian problems with Hardy potential in bounded domains, using comparison principles.

Contribution

It extends the moving plane method to nonlocal problems involving Hardy potential in bounded domains, establishing symmetry and monotonicity results.

Findings

Positive solutions exhibit symmetry and monotonicity.

The method applies to fractional Laplacian with Hardy potential.

Comparison principles are crucial for the analysis.

Abstract

We consider a nonlocal problem involving the fractional laplacian and the Hardy potential, in bounded smooth domains. Exploiting the moving plane method and some weak and strong comparison principles, we deduce symmetry and monotonicity properties of positive solutions under zero Dirichlet boundary conditions.