Differential and Integral Equations

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

Teresa D'Aprile

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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


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