Regularity of stable solutions of a Lane-Emden type system

Abstract

We examine the system given by \hfill -Δu = λ(v+1)^p \qquad Ω \hfill -Δv = γ(u+1)^θ\qquad Ω, \hfill u = v =0 \qquad \quad \partial Ω, where $\lambda ,\gamma$ are positive parameters and where $1<p\le \theta$ and where $\Omega$ is a smooth bounded domain in $R^N$. We show the extremal solutions associated with the above system are bounded provided [\frac{N}{2} < 1 + \frac{2(θ+1)}{pθ-1} (\sqrt{\frac{p θ(p+1)}{θ+1}} + \sqrt{\frac{p θ(p+1)}{θ+1} - \sqrt{\frac{p θ(p+1)}{θ+1}}})]