Symmetry of solutions of some semilinear elliptic equations with singular nonlinearities

TL;DR

This paper investigates the symmetry and monotonicity of positive solutions to certain singular semilinear elliptic equations in bounded domains, employing maximum principles and the Moving Plane Method.

Contribution

It establishes new maximum principles for solutions and applies the Moving Plane Method to prove symmetry and monotonicity properties.

Findings

Solutions exhibit symmetry with respect to certain planes.

Maximum principles are extended to the H^1_0 part of solutions.

Monotonicity properties are derived for solutions.

Abstract

We consider positive solutions to a singular semilinear elliptic equation in bounded smooth domains, with zero Dirichlet boundary conditions. We provide some weak and strong maximum principles for the H^1_0 part of the solution that allow to deduce symmetry and monotonicity properties of the solutions, via the Moving Plane Method.