Multidimensional boundary layers for a singularly perturbed Neumann problem

Abstract

We continue the study of [34], proving concentration phenomena for the equation − ε² Δu + u = u^p in a smooth bounded domain Ω ⊆ $\mathbb{R}^n$ and with Neumann boundary conditions. The exponent p is greater than or equal to 1, and the parameter ε is converging to zero. For a suitable sequence ε_j → 0, we prove the existence of positive solutions u_j concentrating at the whole boundary of Ω or at some of its components.