On singular perturbation problems with Robin boundary condition

We consider the following singularly perturbed elliptic problem

$$\begin{array}{cc}\hfill {ϵ}^2\Delta u-u+f\left(u\right)& =0,u>0\text{in}\Omega ,\hfill \\ \hfill ϵ\frac{\partial u}{\partial \nu }+\lambda u& =0\text{on}\partial \Omega ,\hfill \end{array}$$

where $f$ satisfies some growth conditions, $0\le \lambda \le +\infty$, and $\Omega \subset {ℝ}^N$ ($N>1$) is a smooth and bounded domain. The cases $\lambda =0$ (Neumann problem) and $\lambda =+\infty$ (Dirichlet problem) have been studied by many authors in recent years. We show that, there exists a generic constant ${\lambda }_{*}>1$ such that, as $ϵ\to 0$, 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.

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