On multi-bump semi-classical bound states of nonlinear Schrödinger equations with electromagnetic fields

Abstract

We consider the existence and asymptotic behavior of standing wave solutions to nonlinear Schrödinger equations with electromagnetic fields: $ih\frac{\partial\psi}{\partial t} =\left(\frac{h}{i}\nabla-A(x)\right)^2\psi+W(x)\psi-f(|\psi|^2)\psi$ on ${{\mathbb R}}\times{\Omega}$. $\Omega\subset{{\mathbb R}}^N$ is a domain which may be bounded or unbounded. For $h>0$ small we obtain the existence of multi-bump bound states $\psi_h (x,t)=e^{-iEt/h}u_h(x)$ where $u_h$ concentrates simultaneously at possibly degenerate, non-isolated local minima of $W$ as $h\to0$. We require that $W\geq E$ and allow the possibility that $\{x\in\Omega:W(x)=E\}\not=\emptyset$. Moreover, we describe the asymptotic behavior of $u_h$ as $h\to\, 0$.