## Advances in Differential Equations

- Adv. Differential Equations
- Volume 11, Number 7 (2006), 781-812.

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

Thomas Bartsch, E. Norman Dancer, and Shuangjie Peng

#### 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 $h\to\, 0$.

#### Article information

**Source**

Adv. Differential Equations Volume 11, Number 7 (2006), 781-812.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867676

**Mathematical Reviews number (MathSciNet)**

MR2236582

**Zentralblatt MATH identifier**

1146.35081

**Subjects**

Primary: 35J60: Nonlinear elliptic equations

Secondary: 35B33: Critical exponents 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

#### Citation

Bartsch, Thomas; Dancer, E. Norman; Peng, Shuangjie. On multi-bump semi-classical bound states of nonlinear Schrödinger equations with electromagnetic fields. Adv. Differential Equations 11 (2006), no. 7, 781--812.https://projecteuclid.org/euclid.ade/1355867676