On Lane–Emden Systems with Singular Nonlinearities and Applications to MEMS

Abstract

Abstract In this paper we analyze the Lane–Emden system - Δ ⁢ u = λ ⁢ f ⁢ ( x ) ( 1 - v ) 2 ⁢  in  ⁢ Ω , - Δ ⁢ v = μ ⁢ g ⁢ ( x ) ( 1 - u ) 2 ⁢  in  ⁢ Ω , 0 ≤ u , v < 1 ⁢  in  ⁢ Ω , u = v = 0 ⁢  on  ⁢ ∂ ⁡ Ω , -\Delta u=\frac{\lambda f(x)}{(1-v)^{2}}\text{ in }\Omega,\quad-\Delta v=\frac% {\mu g(x)}{(1-u)^{2}}\text{ in }\Omega,\quad 0\leq u,v<1\text{ in }\Omega,% \quad u=v=0\text{ on }\partial\Omega, where λ and μ are positive parameters and Ω is a smooth bounded domain of ℝ N ( N ≥ 1 ) {\mathbb{R}^{N}(N\geq 1)} . Here we prove the existence of a critical curve Γ which splits the positive quadrant of the ( λ , μ ) {(\lambda,\mu)} -plane into two disjoint sets 𝒪 1 {\mathcal{O}_{1}} and 𝒪 2 {\mathcal{O}_{2}} such that the Lane–Emden system has a smooth minimal stable solution ( u λ , v μ ) {(u_{\lambda},v_{\mu})} in 𝒪 1 {\mathcal{O}_{1}} , while for ( λ , μ ) ∈ 𝒪 2 {(\lambda,\mu)\in\mathcal{O}_{2}} there are no solutions of any kind. We also establish upper and lower estimates for the critical curve Γ and regularity results on this curve if N ≤ 7 {N\leq 7} . Our proof is based on a delicate combination involving a maximum principle and L p {L^{p}} estimates for semi-stable solutions of the Lane–Emden system.