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Metodi variazionali e topologici nello studio delle equazioni di Schrödinger nonlineari agli stati stazionari

Silvia Cingolani — 2001

Bollettino dell'Unione Matematica Italiana

In the present paper we survey some recents results concerning existence of semiclassical standing waves solutions for nonlinear Schrödinger equations. Furthermore, from Maxwell's equations we derive a nonlinear Schrödinger equation which represents a model of propagation of an electromagnetic field in optical waveguides.

Some results on critical groups for a class of functionals defined on Sobolev Banach spaces

Silvia CingolaniGiuseppina Vannella — 2001

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We present critical groups estimates for a functional f defined on the Banach space W 0 1 , p Ω , Ω bounded domain in R N , 2 < p < , associated to a quasilinear elliptic equation involving p -laplacian. In spite of the lack of an Hilbert structure and of Fredholm property of the second order differential of f in each critical point, we compute the critical groups of f in each isolated critical point via Morse index.

Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions

Silvia CingolaniLouis JeanjeanSimone Secchi — 2009

ESAIM: Control, Optimisation and Calculus of Variations

In this work we consider the magnetic NLS equation ( i - A ( x ) ) 2 u + V ( x ) u - f ( | u | 2 ) u = 0 in N ( 0 . 1 ) where N 3 , A : N N is a magnetic potential, possibly unbounded, V : N is a multi-well electric potential, which can vanish somewhere, f is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution u : N to (0.1), under conditions on the nonlinearity which are nearly optimal.

Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions

Silvia CingolaniLouis JeanjeanSimone Secchi — 2008

ESAIM: Control, Optimisation and Calculus of Variations

In this work we consider the magnetic NLS equation ( i - A ( x ) ) 2 u + V ( x ) u - f ( | u | 2 ) u = 0 in N ( 0 . 1 ) where N 3 , A : N N is a magnetic potential, possibly unbounded, V : N is a multi-well electric potential, which can vanish somewhere, is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution u : N to (0.1), under conditions on the nonlinearity which are nearly optimal.

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