Singularly perturbed Neumann problems with potentials

Abstract

The main purpose of this paper is to study the existence of single-peaked solutions of the Neumann problem $$ \cases -\varepsilon^2 \text{\rm div} \left(J(x)\nabla u\right)+V(x)u=u^p & \text{in }\Omega, \\ \displaystyle \dfrac{\partial u}{\partial \nu}=0 & \text{on }\partial\Omega, \endcases $$ where $\Omega$ is a smooth bounded domain of $\{\mathbb R}^N$, $N\ge 3$, $1< p< (N+2)/(N-2)$ and $J$ and $V$ are positive bounded scalar value potentials. We will show that, for the existence of concentrating solutions, one has to check if at least one between $J$ and $V$ is not constant on $\partial \Omega$. In this case the concentration point is determined by $J$ and $V$ only. In the other case the concentration point is determined by an interplay among the derivatives of $J$ and $V$ calculated on $\partial \Omega$ and the mean curvature $H$ of $\partial \Omega$.