## Differential and Integral Equations

- Differential Integral Equations
- Volume 16, Number 3 (2003), 349-384.

### On a class of solutions with non-vanishing angular momentum for nonlinear Schrödinger equations

#### Abstract

In this work we consider the following class of nonlinear perturbed Schrödinger equations $$ i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^{2}}{2m}\Delta \psi+V(x,z)\psi-\gamma|\psi|^{p-2}\psi ,$$$\gamma, $ $m>0,$$(x,z)=(x_1,x_2,z)\in{{\mathbb R}}^{3}$, where $\hbar>0$, $2 <p <1+\sqrt{5}$,$\psi:{{\mathbb R}}^{3}\rightarrow{{\mathbb C}}.$ Here, the potential $V$ is bounded from below away from zero and satisfies: $V(x,z)=V(|x|,z)$ for all $(x,z)=(x_1,x_2,z)\in{{\mathbb R}}^{3}$. We are interested in finding solutions having the form $\psi(r,z,\theta,t)=\exp\left[ i(\omega\theta-Et)/\hbar\right]v(r,z),$ being $(r,z,\theta)$ the cylindrical coordinates in the space. When the parameter $\hbar$ approaches zero our solutions concentrate around a circle lying on a plane $z=\overline{z}$ with center $(0,0,\overline{z})$.

#### Article information

**Source**

Differential Integral Equations, Volume 16, Number 3 (2003), 349-384.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060675

**Mathematical Reviews number (MathSciNet)**

MR1947957

**Zentralblatt MATH identifier**

1031.35130

**Subjects**

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

Secondary: 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations

#### Citation

D'Aprile, Teresa. On a class of solutions with non-vanishing angular momentum for nonlinear Schrödinger equations. Differential Integral Equations 16 (2003), no. 3, 349--384. https://projecteuclid.org/euclid.die/1356060675