Regularity of the extremal solutions associated to elliptic systems

Abstract

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

Minor revision. Comments are welcome. Submitted June 10, 2012. For updates please see http://www.math.ualberta.ca/~fazly/research.html