Liouville type theorems for nonlinear elliptic equations on the whole space $\mathbb {R}^N$
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- by Hsini Mounir and Sayeb Wahid PDF
- Proc. Amer. Math. Soc. 140 (2012), 2731-2738 Request permission
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 $|u|\leq (\frac {\alpha }{\beta })^{\frac {1}{m-q+1}}$ 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.References
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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
- Received by editor(s): October 22, 2010
- Received by editor(s) in revised form: February 14, 2011
- Published electronically: November 30, 2011
- Communicated by: Walter Craig
- © Copyright 2011 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 140 (2012), 2731-2738
- MSC (2000): Primary 34-XX, 35-XX
- DOI: https://doi.org/10.1090/S0002-9939-2011-11112-2
- MathSciNet review: 2910761