On Lane-Emden systems with singular nonlinearities and applications to MEMS

In this paper we analyse the Lane-Emden system \begin{equation} \left\{ \begin{alignedat}{3} -\Delta u = & \, \frac{\lambda f(x)}{(1-v)^2} & \quad \text{in} & \quad\Omega\\ -\Delta v = & \, \frac{\mu g(x)}{(1-u)^2} & \quad \text{in} & \quad\Omega\\ 0\leq u &, v < 1 & \quad \text{in} & \quad \Omega\\ u = v & = \, 0 & \text{on} & \quad \partial\Omega\\ \end{alignedat} \right.\tag{$S_{\lambda, \mu}$} \end{equation} where $\lambda$ and $\mu$ are positive parameters and $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$ $( N \geq 1)$. Here we prove the existence of a critical curve $\Gamma$ which splits the positive quadrant of the $(\lambda,\mu)\text{-plane}$ into two disjoint sets $\mathcal{O}_1$ and $\mathcal{O}_2$ such that the problem $(S_{\lambda, \mu})$ has a smooth minimal stable solution $(u_\lambda,v_\mu)$ in $\mathcal{O}_1$, while for $(\lambda,\mu)\in\mathcal{O}_2$ there are no solutions of any kind. We also establish upper and lower estimates for the critical curve $\Gamma$ and regularity results on this curve if $N\leq 7$. Our proof is based on a delicate combination involving maximum principle and $L^p$ estimates for semi-stable solutions of $(S_{\lambda, \mu}$).