A uniqueness result for some singular semilinear elliptic equations

TL;DR

This paper proves the uniqueness of nonnegative solutions to a class of singular semilinear elliptic equations with zero boundary conditions, extending understanding of such equations in mathematical analysis.

Contribution

It establishes a new uniqueness result for solutions of singular semilinear elliptic equations with specific boundary conditions.

Findings

Solutions are unique for the considered class of equations.

The result applies to equations with nonnegative solutions and integrable source terms.

The proof covers cases with singular nonlinearities involving negative powers of the solution.

Abstract

Given $\Omega$ a bounded open subset of $\mathbb{R}^N$, we consider nonnegative solutions to the singular semilinear elliptic equation $-\Delta\,u\,=\,\frac{f}{u^{\beta}}$ in $H^1_{loc}(\Omega)$, under zero Dirichlet boundary conditions. For $\beta>0$ and $f\in L^1(\Omega)$, we prove that the solution is unique.