Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 236, pp. 1-16.
Title: Multiplicity of positive solutions for a
gradient system with an exponential nonlinearity
Authors: Nasreddine Megrez (Univ. of Alberta, Canada)
K. Sreenadh (Indian Institute of Tech., New Delhi, India)
Brahim Khaldi (Univ. of Bechar, Algeria)
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 $\mathbb{R}^{2}$, $00$. 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.