Asymptotic non-degeneracy of the multiple blow-up solutions to the Gel'fand problem in two space dimensions (Q536523)

The authors 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 \(\lambda =\{\lambda_n\}_{n\in\mathbb N}\), \(\lambda_n\downarrow 0\), and blowing up at \(m\) points \(k_1,\dots, k_m\) in \(\Omega\). Under certain non-degeneracy assumption on some suitable finite-dimensional function (related to \(k_1,\dots,k_m\)) it is shown that \(u_n\) is non-degenerate for \(n\) large enough.