Simplicity and stability of the first eigenvalue of a (p;q) Laplacian system

This article concerns special properties of the principal eigenvalue of a nonlinear elliptic system with Dirichlet boundary conditions. In particular, we show the simplicity of the first eigenvalue of $$displaylines{ -Delta_p u = lambda |u|^{alpha-1}|v|^{eta-1}v quad hbox{in } Omega,cr -Delta_q v = lambda |u|^{alpha-1}|v|^{eta-1}u quad hbox{in } Omega,cr (u,v)in W_{0}^{1,p}(Omega) imes W_{0}^{1,q}(Omega), }$$ with respect to the exponents p and q, where $Omega$ is a bounded domain in $mathbb{R}^{N}$.