Asymptotic non-degeneracy of the multiple blow-up solutions to the Gel'fand problem in two space dimensions

Abstract

We consider a sequence of solutions $u_n$ of the problem $$-\Delta u=\lambda e^u \quad \text{in $\Omega$}, \qquad u=0 \quad \text{on $\partial\Omega$},$$ with $\lambda=\{\lambda_n\}_{n\in\mathbb N}$ and blowing up at $m$ points $\kappa_1,\dots,\kappa_m$ in $\Omega$. Under some non-degeneracy assumption on some suitable finite-dimensional function (related to $\kappa_1,\dots,\kappa_m$) we show that $u_n$ is non-degenerate for $n$ large enough.