The Gelfand problem for the Infinity Laplacian

Abstract

We study the asymptotic behavior as $ p\to\infty $ of the Gelfand problem $ \begin{equation*} \left\{ \begin{aligned} -&\Delta_{p} u = \lambda\,e^{u}&& \text{in}\ \Omega\subset \mathbb{R}^n\\ &u = 0 && \text{on}\ \partial\Omega. \end{aligned} \right. \end{equation*} $ Under an appropriate rescaling on $ u $ and $ \lambda $, we prove uniform convergence of solutions of the Gelfand problem to solutions of $ \left\{ \begin{aligned} &\min\left\{|\nabla{}u|-\Lambda\,e^{u}, -\Delta_{\infty}u\right\} = 0&& \text{in}\ \Omega,\\ &u = 0\ &&\text{on}\ \partial\Omega. \end{aligned} \right. $ We discuss existence, non-existence, and multiplicity of solutions of the limit problem in terms of $ \Lambda $.

Keywords

• fully nonlinear
• infinity laplacian
• gelfand problem