Local uniqueness of $m$-bubbling sequences for the Gel'fand equation

TL;DR

This paper proves the local uniqueness of m-bubbling solutions for the Gel'fand equation in two dimensions, under certain conditions on the domain and the function h, for small epsilon.

Contribution

It establishes the local uniqueness of m-bubbling solutions for the Gel'fand problem, a result not previously known for this class of equations.

Findings

Uniqueness holds for small epsilon

Results depend on assumptions on h and domain

Applicable to m-bubbling solutions in 2D

Abstract

We consider the Gel'fand problem, $$ \begin{cases} \Delta w_{\varepsilon}+\varepsilon^2 h e^{w_{\varepsilon}}=0\quad&\mbox{in}\quad\Omega, w_{\varepsilon}=0\quad&\mbox{on}\quad\partial\Omega, \end{cases} $$ where $h$ is a nonnegative function in ${\Omega\subset\mathbb{R}^2}$. Under suitable assumptions on $h$ and $\Omega$, we prove the local uniqueness of $m-$bubbling solutions for any $\varepsilon>0$ small enough.