The Gelfand Problem in Tubular Domains

Abstract

We construct stable solutions of $\Delta u + \lambda e^u=0$ with Dirichlet boundary conditions in small tubular domains (i.e. geodesic $\varepsilon$--neighbourhoods of a curve $\Lambda$ 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, $\Lambda$ can be replaced by any compact smooth manifold embedded in $\mathbb{R}^n$.