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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Liouville type theorems for nonlinear elliptic equations on the whole space $ \mathbb{R}^N$


Authors: Hsini Mounir and Sayeb Wahid
Journal: Proc. Amer. Math. Soc. 140 (2012), 2731-2738
MSC (2000): Primary 34-XX, 35-XX
Published electronically: November 30, 2011
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Abstract: The aim of this paper is to study the properties of the solutions of $ \Delta _{p}u+f_{1}(u)-f_{2}(u)=0$ in all $ \mathbb{R}^{N}.$ We obtain Liouville type boundedness for the solutions. We show that $ \vert u\vert\leq (\frac {\alpha }{\beta })^{\frac {1}{m-q+1}}$ on $ \mathbb{R}^{N},$ under the assumptions $ f_{1}(u)\leq \alpha u^{p-1}$ and $ f_{2}(u)\geq \beta u^{m},$ for some $ 0<\alpha \leq \beta $ and $ m>q-1\geq p-1>0.$ If $ u$ does not change sign, we prove that $ u$ is constant.


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Additional Information

Hsini Mounir
Affiliation: Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire, 1060 Tunis, Tunisia
Email: Hsini.mounir@ipeit.rnu.tn

Sayeb Wahid
Affiliation: Département de Mathématiques, Faculté des Sciences de Tunis, Campus Universitaire, 1060 Tunis, Tunisia
Email: wahid.sayeb@yahoo.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11112-2
PII: S 0002-9939(2011)11112-2
Keywords: Liouville type results, supersolution, subsolution, comparison principle.
Received by editor(s): October 22, 2010
Received by editor(s) in revised form: February 14, 2011, and March 4, 2011
Published electronically: November 30, 2011
Communicated by: Walter Craig
Article copyright: © Copyright 2011 American Mathematical Society