Some existence and concentration results for nonlinear Schrödinger equations

Abstract

In this paper we are concerned with the existence of solutions with non-vanishing angular momentum fora class of nonlinear Schrödinger equations of the form$ i \h$$ \frac{\partial\psi}{\partial t}=-$ $\frac{ \h^2}{2m}\Delta \psi+V(x)\psi-\gamma|\psi|^{p-2}\psi,$ $\gamma>0,$ $ x\in\mathbb R^{N}$ where $\h$$ >0$, $p>2$, $\psi:\mathbb R^{N}\rightarrow\mathbb C,$ and the potential $V$ satisfies some symmetric properties. In particular the cases $N=2$ with $V$ radially symmetric and $N=3$ with $V$ having a cylindrical symmetry are discussed.Our main purpose is to study the asymptotic behaviour of such solutions in the semiclassical limit(i.e. as $\hbar \rightarrow 0^+$)when a concentration phenomenon around a point of $\mathbb R^N$ appears.