Concentration on Surfaces for a Singularly Perturbed Neumann Problem in Three-Dimensional Domains

Abstract

We consider the following singularly perturbed elliptic problem $$ \varepsilon^2\triangle\tilde{u}-\tilde{u}+\tilde{u}^p=0, \ \tilde{u}>0\quad \mbox{in} \ Ω,\ \ \ \frac{\partial\tilde{u}}{\partial \mathbf{n}}=0 \quad \mbox{on}\ \partialΩ, $$ where $\Omega$ is a bounded domain in ${\mathbb{R}}^3$ with smooth boundary, $\epsilon$ is a small parameter, $\mathbf{n}$ denotes the inward normal of $\partial \Omega$ and the exponent $p>1$. Let $\Gamma$ be a hypersurface intersecting $\partial \Omega$ in the right angle along its boundary $\partial \Gamma$ and satisfying a {\em non-degenerate condition}. We establish the existence of a solution $u_{\epsilon }$ concentrating along a surface $\tilde{\Gamma }$ close to $\Gamma$, exponentially small in $\epsilon$ at any positive distance from the surface $\tilde{\Gamma }$, provided $\epsilon$ is small and away from certain {\em critical numbers}. The concentrating surface $\tilde{\Gamma }$ will collapse to $\Gamma$ as $\epsilon \to 0$.