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

Abstract

In this paper we analyse the Lane-Emden system \begin{equation} \left\{ \begin{alignedat}{3} -Δu = & \, \frac{λf(x)}{(1-v)^2} & \quad \text{in} & \quadΩ\\ -Δv = & \, \frac{μg(x)}{(1-u)^2} & \quad \text{in} & \quadΩ\\ 0\leq u &, v < 1 & \quad \text{in} & \quad Ω\\ u = v & = \, 0 & \text{on} & \quad \partialΩ\\ \end{alignedat} \right.\tag{$S_{\lambda ,\mu }$} \end{equation} where $\lambda$ and $\mu$ are positive parameters and $\Omega$ is a smooth bounded domain of ${\mathbb{R}}^N$ $(N\ge 1)$. 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 $N\le 7$. Our proof is based on a delicate combination involving maximum principle and $L^p$ estimates for semi-stable solutions of $(S_{\lambda ,\mu }$).