The Gelfand Problem in Tubular Domains

We construct stable solutions of $\mathrm{\Delta }u+\lambda e^u=0$ with Dirichlet boundary conditions in small tubular domains (i.e. geodesic $\epsilon$–neighbourhoods of a curve $\mathrm{\Lambda }$ embedded in ${\mathbb{R}}^n$), adapting the arguments of Pacard-Pacella-Sciunzi. We also show unicity of these solutions, in particular, we show that the stable branch of the bifurcation diagram is similar to the well-known nose-shaped diagram of the standard Gelfand problem in the unit ball. In this work, $\mathrm{\Lambda }$ can be replaced by any compact smooth manifold embedded in ${\mathbb{R}}^n$.