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July/August 2017 A classification of solutions of a fourth order semi-linear elliptic equation in$\mathbb R^n$
Ridha Chammakhi, Abdellaziz Harrabi, Abdelbaki Selmi
Differential Integral Equations 30(7/8): 569-586 (July/August 2017).

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

Citation

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Ridha Chammakhi. Abdellaziz Harrabi. Abdelbaki Selmi. "A classification of solutions of a fourth order semi-linear elliptic equation in$\mathbb R^n$." Differential Integral Equations 30 (7/8) 569 - 586, July/August 2017.

Information

Accepted: 1 May 2016; Published: July/August 2017
First available in Project Euclid: 4 May 2017

zbMATH: 06738562
MathSciNet: MR3646464

Subjects:
Primary: 35J60, 35J65, 58E05

Rights: Copyright © 2017 Khayyam Publishing, Inc.

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Vol.30 • No. 7/8 • July/August 2017
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