Regularity of stable solutions of a Lane-Emden type system

Abstract

We examine the system given by \hfill -\Delta u = \lambda (v+1)^p \qquad \Omega \hfill -\Delta v = \gamma (u+1)^\theta \qquad \Omega, \hfill u = v =0 \qquad \quad \partial \Omega, 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(\theta+1)}{p\theta -1} (\sqrt{\frac{p \theta (p+1)}{\theta +1}} + \sqrt{\frac{p \theta (p+1)}{\theta +1} - \sqrt{\frac{p \theta (p+1)}{\theta +1}}})]