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

Cingolani, Silvia; Jeanjean, Louis; Secchi, Simone

doi:10.1051/cocv:2008055

Cingolani, Silvia ; Jeanjean, Louis ; Secchi, Simone

ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 3, pp. 653-675.

Résumé

In this work we consider the magnetic NLS equation

{(\frac{ℏ}{i} \nabla - A (x))}^{2} u + V (x) u - {f (| u |}^{2}) u = 0 in ℝ^{N} (0.1)

where

N \geq 3

A : ℝ^{N} \to ℝ^{N}

is a magnetic potential, possibly unbounded,

V : ℝ^{N} \to ℝ

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 : ℝ^{N} \to ℂ

to (0.1), under conditions on the nonlinearity which are nearly optimal.

MR 2542577 | 1 citation dans Numdam

DOI : https://doi.org/10.1051/cocv:2008055
Classification : 35J20, 35J60
Mots clés : nonlinear Schrödinger equations, magnetic fields, multi-peaks

@article{COCV_2009__15_3_653_0,
     author = {Cingolani, Silvia and Jeanjean, Louis and Secchi, Simone},
     title = {Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {653--675},
     publisher = {EDP-Sciences},
     volume = {15},
     number = {3},
     year = {2009},
     doi = {10.1051/cocv:2008055},
     mrnumber = {2542577},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2009__15_3_653_0/}
}

Cingolani, Silvia; Jeanjean, Louis; Secchi, Simone. Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 3, pp. 653-675. doi : 10.1051/cocv:2008055. http://www.numdam.org/item/COCV_2009__15_3_653_0/

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