Regularity of the extremal solutions associated to elliptic systems

Abstract

We examine the elliptic system given by \begin{eqnarray*} \qquad \left\{ \begin{array}{lcl} -\Delta u =\lambda f(v) \quad \mbox{ in } \Omega -\Delta v =\gamma f(u) \quad \mbox{ in } \Omega, u=v =0, \quad \mbox{ on } \pOm \end{array}\right. \end{eqnarray*} where $\lambda,\gamma$ are positive parameters, $\Omega$ is a smooth bounded domain in $\IR^N$ and $f$ is a $C^{2}$ positive, nondecreasing and convex function in $[0,\infty)$ such that $\frac{f(t)}{t}\rightarrow\infty$ as $t\rightarrow\infty$. Assuming $$0<\tau_{-}:=\liminf_{t\rightarrow\infty} \frac{f(t)f"(t)}{f'(t)^{2}}\leq \tau_{+}:=\limsup_{t\rightarrow\infty} \frac{f(t)f"(t)}{f'(t)^{2}}\leq 2,$$ we show that the extremal solution $(u^*, v^*)$ associated to the above system is smooth provided\\ $N<\frac{2\alpha_{*}(2-\tau_{+})+2\tau_{+}}{\tau_{+}}\max\{1,\tau_{+}\}$, where $\alpha_{*}>1$ denotes the largest root of the $2^{nd}$ order polynomial $$P_{f}(\alpha,\tau_{-},\tau_{+}):=(2-\tau_{-})^{2} \alpha^{2}- 4(2-\tau_{+})\alpha+4(1-\tau_{+}).$$ As a consequences, $u^*, v^*\in L^\infty(\Omega)$ for $N<5$. Moreover, if $\tau_{-}=\tau_{+}$, then $u^*, v^*\in L^\infty(\Omega)$ for $N<10$.