The Gelfand problem for the Infinity Laplacian

Abstract

We study the asymptotic behavior as $p\to \infty$ of the Gelfand problem \[ -Δ_{p} u=λ\,e^{u}\ \textrm{in}\ Ω\subset\mathbb{R}^n,\quad u=0 \ \textrm{on}\ \partialΩ. \] Under an appropriate rescaling on $u$ and $\lambda$, we prove uniform convergence of solutions of the Gelfand problem to solutions of \[ \min\left\{|\nabla{}u|-Λ\,e^{u}, -Δ_{\infty}u\right\}=0\ \textrm{in}\ Ω,\quad u=0\ \text{on}\ \partialΩ. \] We discuss existence, non-existence, and multiplicity of solutions of the limit problem in terms of $\Lambda$.

Dedicated to the memory of Ireneo Peral, with love and admiration