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Singular Behavior of an Electrostatic--Elastic Membrane System with an External Pressure

We analyze nonnegative solutions of the nonlinear elliptic problem $Δu=\frac{λf(x)}{u^2}+P$, where $λ>0$ and $P\geq0$, on a bounded domain $Ω$ of $\mathbb{R}^N$ ($N\geq 1$) with a Dirichlet boundary condition. This equation models an electrostatic--elastic membrane system with an external pressure $P\geq 0$, where $λ>0$ denotes the applied voltage. First, we completely address the existence and nonexistence of positive solutions. The classification of all possible singularities at $|x|=0$ for nonnegative solutions $u(x)$ satisfying $u(0)=0$ is then analyzed for the special case where $Ω=B_1(0)\subset \mathbb{R}^2$ and $f(x)=|x|^α$ with $α\geq0$. In particular, we show that for some $α,$ $u(x)$ admits only the "isotropic" singularity at $|x|=0$, and otherwise $u(x)$ may admit the "anisotropic" singularity at $|x|=0$. When $u(x)$ admits the "isotropic" singularity at $|x|=0$, the refined singularity of $u(x)$ at $|x|=0$ is further investigated, depending on whether $P>0$, by applying Fourier analysis.