Symmetry of positive solutions to biharmonic Lane-Emden equation with singular set

TL;DR

This paper investigates the symmetry and monotonicity of positive solutions to a biharmonic Lane-Emden equation with singularities, using advanced symmetry methods like the moving plane and moving sphere techniques.

Contribution

It establishes symmetry and monotonicity properties for solutions with singular sets in the biharmonic Lane-Emden equation, extending previous results to singular solutions.

Findings

Solutions exhibit symmetry with respect to certain hyperplanes.

Monotonicity properties are established for solutions.

The methods used can handle solutions with singular sets.

Abstract

In this paper, we are devoted to studying the positive weak, punctured or distributional solutions to the biharmonic Lane-Emden equation \begin{equation*} \Delta^{2} u=u^{p} \quad \quad \text{in} \ \mathbb{R}^{N}\setminus Z, \end{equation*} where $N\geq5$, $1<p\leq\frac{N+4}{N-4}$, and the singular set $Z$ represents a closed and proper subset of $ \left\lbrace x_{1}=0\right\rbrace $. The symmetry and monotonicity properties of the singular solutions will be given by taking advantage of the moving plane method and the approach of moving spheres.