On interacting bumps of semi-classical states of nonlinear Schrödinger equations

Abstract

We study concentrated positive bound states of the following nonlinear Schr\"odinger equation: \[ h^2 \Delta u - V(x) u + u^p=0,\ \ \ u>0, \ \ x \in R^N , \] where $ p$ is subcritical. We prove that, at a local maximum point $x_0$ of the potential function $V(x)$ and for arbitrary positive integer $K (K>1)$, there always exist solutions with $K$ interacting bumps concentrating near $x_0$. We also prove that at a nondegenerate local minimum point of $V(x) $ such solutions do not exist.