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

Abstract

We consider the Gel'fand problem, $$ \begin{cases} Δw_{\varepsilon}+\varepsilon^2 h e^{w_{\varepsilon}}=0\quad&\mbox{in}\quadΩ, w_{\varepsilon}=0\quad&\mbox{on}\quad\partialΩ, \end{cases} $$ 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 $\epsilon >0$ small enough.