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Some existence and concentration results for nonlinear Schrödinger equations

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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.

    Mathematics Subject Classification: 35J20.


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