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

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.