Regularity of the extremal solutions associated to elliptic systems

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 $\mathrm{\Omega }$,] and [(H){\lambda,\gamma} \quad -\Delta u = \lambda f(u) g’(v), \quad -\Delta v = \gamma f’(u) g(v) \quad in $\mathrm{\Omega }$},] 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 $\mathrm{\Omega }$ 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 $1<q<\mathrm{\infty }$ and where $\mathrm{\Omega }$ 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 $g(v)=(v+1{)}^q$.