Electron. J. Diff. Equ., Vol. 2012 (2012), No. 236, pp. 1-16.

Multiplicity of positive solutions for a gradient system with an exponential nonlinearity

Nasreddine Megrez, K. Sreenadh, Brahim Khaldi

Abstract:
In this article, we consider the problem
$$\displaylines{ -\Delta u = \lambda u^{q} + f_1(u,v) \quad \hbox{in } \Omega\cr -\Delta v = \lambda v^{q} + f_{2} (u,v) \quad \hbox{in } \Omega\cr u, v > 0 \quad \hbox{in } \Omega \cr u = v = 0 \quad \hbox{on } \partial\Omega, }$$
where $\Omega$ is a bounded domain in $\mathbb{R}^{2}$, $0<q<1$, and $\lambda>0$. We show that there exists a real number $\Lambda$ such that the above problem admits at least two solutions for $\lambda\in(0,\Lambda)$, and no solution for $\lambda>\Lambda$.

Submitted June 16, 2011. Published December 26, 2012.
Math Subject Classifications: 35J50, 35J57, 35J60.
Key Words: Gradient system; exponential nonlinearity; multiplicity.

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Nasreddine Megrez
Chemical and Material Engineering Department
University of Alberta
9107 - 116 Street, Edmonton (Alberta) T6G 2V4, Canada
email: nmegrez@gmail.com
  K. Sreenadh
Department of Mathematics
Indian Institute of Technology Delhi Hauz Khaz
New Delhi-16, India
email: sreenadh@gmail.com
Brahim Khaldi
Departement of Sciences, University of Bechar
PB 117, Bechar 08000, Algeria
email: khaldibra@yahoo.fr

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