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 $\Omega$,] 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 $g(v)=(v+1)^q$.