The Gelfand Problem in Tubular Domains

2019/03/05 by Prado, Francisco José Vial
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1903.01833

Abstract

We construct stable solutions of Δu + λe^u=0 with Dirichlet boundary conditions in small tubular domains (i.e. geodesic ε--neighbourhoods of a curve Λ embedded in ℝⁿ), 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, Λ can be replaced by any compact smooth manifold embedded in ℝⁿ.

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