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

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

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

where $N\ge 3$, $A:{ℝ}^N\to {ℝ}^N$ 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.

Détail

DOI : 10.1051/cocv:2008055

Classification : 35J20, 35J60
Keywords: nonlinear Schrödinger equations, magnetic fields, multi-peaks