  We analyze nonnegative solutions of the nonlinear elliptic problem $\Delta
u=\frac{\lambda f(x)}{u^2}+P$, where $\lambda>0$ and $P\geq0$, on a bounded
domain $\Omega$ 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 $\lambda >0$ 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 $\Omega=B_1(0)\subset \mathbb{R}^2$ 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 $u(x)$ 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.
