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

• Main Authors: Nasreddine Megrez, K. Sreenadh, Brahim Khaldi
• Format: Article
• Language: English
• Published: Texas State University 2012-12-01
• Series: Electronic Journal of Differential Equations
• Subjects: Gradient system; exponential nonlinearity; multiplicity
• Online Access: http://ejde.math.txstate.edu/Volumes/2012/236/abstr.html

In this article, we consider the problem $$displaylines{ -Delta u = lambda u^{q} + f_1(u,v) quad hbox{in } Omegacr -Delta v = lambda v^{q} + f_{2} (u,v) quad hbox{in } Omegacr u, v > 0 quad hbox{in } Omega cr u = v = 0 quad hbox{on } partialOmega, }$$ 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 $lambdain(0,Lambda)$, and no solution for $lambda>Lambda$.