Differential and Integral Equations

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

Ridha Chammakhi, Abdellaziz Harrabi, and Abdelbaki Selmi

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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

Permanent link to this document
https://projecteuclid.org/euclid.die/1493863395

Mathematical Reviews number (MathSciNet)
MR3646464

Zentralblatt MATH identifier
06738562

Subjects
Primary: 35J60: Nonlinear elliptic equations 35J65: Nonlinear boundary value problems for linear elliptic equations 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)

Citation

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


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