Regularity of stable solutions of a Lane-Emden type system

TL;DR

This paper investigates the boundedness of extremal solutions in a Lane-Emden type system, establishing conditions based on domain dimension and system parameters that guarantee regularity of solutions.

Contribution

It provides new criteria for the regularity of extremal solutions in a coupled nonlinear PDE system, extending previous results to more general parameter ranges.

Findings

Extremal solutions are bounded under specific dimensional and parameter conditions.

Derived explicit inequality involving domain dimension and system parameters for solution regularity.

Results contribute to understanding the regularity of solutions in nonlinear elliptic systems.

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}}})]