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

Abstract

We consider the following singularly perturbed elliptic problem $$ \varepsilon^2\triangle\tilde{u}-\tilde{u}+\tilde{u}^p=0, \ \tilde{u}>0\quad \mbox{in} \ \Omega,\ \ \ \frac{\partial\tilde{u}}{\partial \mathbf{n}}=0 \quad \mbox{on}\ \partial\Omega, $$ where $\Omega$ is a bounded domain in $\mathbb{R}^3$ with smooth boundary, $\varepsilon$ 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 $u_\varepsilon$ concentrating along a surface $\tilde{\Gamma}$ close to $\Gamma$, exponentially small in $\varepsilon$ at any positive distance from the surface $\tilde{\Gamma}$, provided $\varepsilon$ is small and away from certain {\em critical numbers}. The concentrating surface $\tilde{\Gamma}$ will collapse to $\Gamma$ as $\varepsilon\rightarrow 0$.