Some existence and concentration results for nonlinear Schrödinger equations

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December 2002, 1(4): 457-474. doi: 10.3934/cpaa.2002.1.457

Some existence and concentration results for nonlinear Schrödinger equations

1.	Dipartimento di Matematica, Universita di Bari, via E. Orabona 4, 70125 Bari, Italy

Received February 2002 Revised May 2002 Published September 2002

Abstract
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In this paper we are concerned with the existence of solutions with non-vanishing angular momentum for a 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.

Keywords: semiclassical limit, existence, Angular momentum, concentration..

Mathematics Subject Classification: 35J2.

Citation: Teresa D'Aprile. Some existence and concentration results for nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2002, 1 (4) : 457-474. doi: 10.3934/cpaa.2002.1.457

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