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September  2008, 7(5): 1057-1075. doi: 10.3934/cpaa.2008.7.1057

Nonexistence results of sign-changing solutions to a supercritical nonlinear problem

M. Ben Ayed 1, and  Kamal Ould Bouh 2,

1. 

Département de Mathématiques, Faculté des Sciences de Sfax, Route Soukra, Sfax, Tunisia
2. 

Département de Mathématiques, Faculté des Sciences de Sfax, Route Soukra, 3000, BP 1171, Sfax, Tunisia

Received  August 2007 Revised  January 2008 Published  June 2008

In this paper we study the nonlinear elliptic problem involving nearly critical exponent $ (P_\varepsilon ): -\Delta u= $ $ |u|^{4/(n-2)+\varepsilon}u$ in $\Omega$, $u = 0$ on $\partial \Omega $, where $\Omega $ is a smooth bounded domain in $\mathbb R^n $, $n \geq 3 $ and $\varepsilon$ is a positive real parameter. We show that, for $\varepsilon$ small, $(P_\varepsilon) $ has no sign-changing solutions with low energy which blow up at two points. Moreover, we prove that there is no sign-changing solutions which blow up at three points. We also show that $(P_\varepsilon)$ has no bubble-tower sign-changing solutions.
Keywords: Sign-changing solutions, critical exponent, bubble-tower solution..
Mathematics Subject Classification: Primary: 35J20; Secondary: 35J6.
Citation: M. Ben Ayed, Kamal Ould Bouh. Nonexistence results of sign-changing solutions to a supercritical nonlinear problem. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1057-1075. doi: 10.3934/cpaa.2008.7.1057
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