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

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 $mathbb{R}^{2}$, $00$. 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 $lambda>Lambda$.