Abstract

We consider the following singularly perturbed elliptic problem $$ \begin{aligned} \epsilon ^2 \Delta u -u + f&=0, \ u>0 \ \text{in} \ \Omega,\\ \epsilon \frac{\partial u}{\partial \nu } + \lambda u &=0 \ \text{on} \ \partial \Omega, \end{aligned} $$ where $f$ satisfies some growth conditions, $ 0 \le \lambda \le +\infty $, and $\Omega \subset \mathbb{R}^N$ is a smooth and bounded domain. The cases $\lambda =0$ and $\lambda = +\infty $ have been studied by many authors in recent years. We show that, there exists a generic constant $\lambda _{*} >1$ such that, as $\epsilon \rightarrow 0$, the least energy solution has a spike near the boundary if $\lambda \le \lambda _{*} $, and has an interior spike near the innermost part of the domain if $\lambda > \lambda _{*} $. Central to our study is the corresponding problem on the half space.