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

## TL;DR

This paper investigates the existence, classification, and detailed behavior of solutions to a nonlinear elliptic PDE modeling an electrostatic-elastic membrane under external pressure, focusing on singularities at the origin in two dimensions.

## Contribution

It provides a complete analysis of solution existence, classifies singularities at the origin, and refines the understanding of their behavior depending on external pressure and singularity type.

## Key findings

- Existence and nonexistence criteria for positive solutions.
- Classification of isotropic and anisotropic singularities at the origin.
- Refined asymptotic behavior of solutions near singularities depending on pressure.

## Abstract

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.