Multi-bump Bound States of Schrödinger Equations with a Critical Frequency

Abstract

We consider existence and qualitative properties of standing wave solutions $\Psi(x,t) = e^{-iEt/h}u(x)$ to the nonlinear Schrödinger equation $ih\frac{\partial\psi} {\partial t} = -\frac{h^2}{2m}\Delta\psi+W(x)\psi-|\psi|^{p-1}\psi = 0$ with E being a critical frequency in the sense that inf $_{x\in\mathbb{R}^N}W(x)=E$. We verify that if the zero set of W − E has several isolated points x _i ($i=1,\ldots,m$) near which W − E is almost exponentially flat with approximately the same behavior, then for h  >  0 small enough, there exists, for any integer k, $1\leq k\leq m$, a standing wave solution which concentrates simultaneously on $\{x_j|j=1,\ldots,k\}$, where $\{x_j|j=1,\ldots,k\}$ is any given subset of $\{x_i|i=1,\ldots,m\}$. This generalizes the result of Byeon and Wang in 3 (Arch Rat Mech Anal 165: 295–316, 2002).