Concentrating solutions of the Liouville equation with Robin boundary condition

Abstract

We construct solutions of the Liouville equation u + 2eu =0 inΩ with Ω a smooth bounded domain in R2, with Robin boundary condition ∂u ∂ν + λu =0 on∂Ω. The solutions constructed exhibit concentration as → 0 and simultaneously as λ → +∞, at points that get close to the boundary, and shows that in general the set of solutions of this problems exhibits a richer structure than the problem with Dirichlet boundary conditions.