The Gelfand problem for the Infinity Laplacian

We study the asymptotic behavior as $p\to \mathrm{\infty }$ of the Gelfand problem

$$\{\begin{array}{rlrl}-& {\mathrm{\Delta }}_{p}u=\lambda e^u& & \text{in}\mathrm{\Omega }\subset {\mathbb{R}}^n\\ & u=0& & \text{on}\mathrm{\partial }\mathrm{\Omega }.\end{array}$$

Under an appropriate rescaling on $u$ and $\lambda$, we prove uniform convergence of solutions of the Gelfand problem to solutions of

$$\{\begin{array}{rlrl}& min\left\{|\mathrm{\nabla }u|-\mathrm{\Lambda }{e}^u,-{\mathrm{\Delta }}_{\mathrm{\infty }}u\right\}=0& & \text{in}\mathrm{\Omega },\\ & u=0& & \text{on}\mathrm{\partial }\mathrm{\Omega }.\end{array}$$

We discuss existence, non-existence, and multiplicity of solutions of the limit problem in terms of $\mathrm{\Lambda }$.