## Differential and Integral Equations

### 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 $p\in (1,(n+2m)/(n-2m) )$.

#### Article information

Source
Differential Integral Equations, Volume 30, Number 7/8 (2017), 569-586.

Dates
Accepted: May 2016
First available in Project Euclid: 4 May 2017

Chammakhi, Ridha; Harrabi, Abdellaziz; Selmi, Abdelbaki. A classification of solutions of a fourth order semi-linear elliptic equation in $\mathbb R^n$. Differential Integral Equations 30 (2017), no. 7/8, 569--586. https://projecteuclid.org/euclid.die/1493863395