Regularity of stable solutions of a Lane–Emden type system

We examine the system given by\[\begin{cases}-\Delta u & = & {\lambda(v+1)}^p & \Omega \\-\Delta v & = & {\gamma(u+1)}^{\theta} & \Omega , \\\phantom{-\Delta} u & = & v=0 & \partial \Omega ,\end{cases}\]where $\lambda, \gamma$ are positive parameters and where $1 \lt p \leq \theta$ and where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$. We show the extremal solutions associated with the above system are bounded provided\[\frac{N}{2} \lt 1 + \frac{2(\theta + 1)}{p \theta - 1}\left(\sqrt{\frac{p \theta (p+1)}{\theta +1}} +\sqrt{\frac{p \theta (p+1)}{\theta +1} - \sqrt{\frac{p \theta (p+1)}{\theta +1}}}\right) .\]In particular this shows that the extremal solutions are bounded for any $1 \lt p \leq \theta$ provided $N \leq 10$.

Keywords

extremal solution, stable solution, regularity of solutions, systems

2010 Mathematics Subject Classification

35J57

Full Text (PDF format)

Published 1 October 2015