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We analyze nonnegative solutions of the nonlinear elliptic problem $\Deltau=\frac{\lambda f(x)}{u^2}+P$, where $\lambda>0$ and $P\geq0$, on a boundeddomain $\Omega$ of $\mathbb{R}^N$ ($N\geq 1$) with a Dirichlet boundarycondition. This equation models an electrostatic--elastic membrane system withan external pressure $P\geq 0$, where $\lambda >0$ denotes the applied voltage.First, we completely address the existence and nonexistence of positivesolutions. The classification of all possible singularities at $|x|=0$ fornonnegative solutions $u(x)$ satisfying $u(0)=0$ is then analyzed for thespecial case where $\Omega=B_1(0)\subset \mathbb{R}^2$ and $f(x)=|x|^{\alpha}$with $\alpha \geq0$. In particular, we show that for some $\alpha,$ $u(x)$admits only the "isotropic" singularity at $|x|=0$, and otherwise $u(x)$ mayadmit 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 applyingFourier analysis.

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