Regularity of the extremal solutions associated to elliptic systems

Abstract

We examine the two elliptic systems given by [(G)_{\lambda,\gamma} \quad -\Delta u = \lambda f'(u) g(v), \quad -\Delta v = \gamma f(u) g'(v) \quad in $ \Omega$,] and [(H)_{\lambda,\gamma} \quad -\Delta u = \lambda f(u) g'(v), \quad -\Delta v = \gamma f'(u) g(v) \quad in $ \Omega$},] with zero Dirichlet boundary conditions and where $ \lambda,\gamma$ are positive parameters. We show that for arbitrary nonlinearities $f$ and $g$ that the extremal solutions associated with $ (G)_{\lambda,\gamma}$ are bounded provided $ \Omega$ is a convex domain in $ \mathbb R^N$ where $ N \le 3$. In the case of a radial domain we show the extremal solutions are bounded provided $ N <10$. The extremal solutions associated with $ (H)_{\lambda,\gamma}$ are bounded in the case where $ f$ is arbitrary, $ g(v)=(v+1)^q$ where $ 1 <q<\infty$ and where $ \Omega$ is a bounded convex domain in $ \mathbb R^N$, $ N \le 3$. Results are also obtained in higher dimensions for $ (G)_{\lambda,\gamma}$ and $(H)_{\lambda,\gamma}$ for the case of explicit nonlinearities of the form $ f(u)=(u+1)^p$ and $ g(v)=(v+1)^q$.