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 $(0,0,\overline{z})$.