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

Megrez, Nasreddine ; Sreenadh, K. ; Khaldi, Brahim

Electronic Journal of Differential Equations, Tome 2012 (2012).

Résumé

Summary: 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 $\mathbb{R}^{2}, 0$, 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 $\lambda>\Lambda$.

Détail

Classification : 35J50, 35J57, 35J60
Keywords: gradient system, exponential nonlinearity, multiplicity