A classification of solutions of a fourth order semi-linear elliptic equation in $\mathbb R^n$

Abstract

In this paper, we classify all regular sign changing~solutions~of $$ \Delta ^2 u=u_+^{p} \,\,\,\mbox {in}\, \, \mathbb R^n\ \ \,\,u_+^{p}\in L^1(\mathbb R^n), $$ where $\Delta ^2$ denotes the biharmonic operator in $\mathbb R^n$, $1 < p\leq \frac{n}{n-4}$ and $n\geq 5$. We prove by using the procedure of moving parallel planes that such solutions are radially symmetric about some point in $\mathbb R^n$. We also present a sup+inf type inequality for regular solutions of the following equation: $$ (-\Delta )^m u=u_+^{p}\,\,\,\mbox{in}\,\,\, \Omega, $$ where $\Omega$ is a bounded domain in $\mathbb R^n$, $m\geq1$, $n\geq 2m+1$ and $p\in (1,(n+2m)/(n-2m) )$.