Multi-peak solutions for magnetic NLS equations without non–degeneracy conditions S. Cingolani Dipartimento di Matematica Politecnico di Bari via Orabona 4, I–70125 Bari, Italy
[email protected] L. Jeanjean Equipe de Math´ematiques Universit´e de Franche–Comt´e (UMR CNRS 6623), 16 Route de Gray, F–25030 Besan¸con, France.
[email protected] S. Secchi ∗ Dipartimento di Matematica ed Applicazioni Universit`a di Milano–Bicocca, via Cozzi 53, I–20125 Milano, Italy. Simone.
[email protected] October 17, 2007
Abstract In the work we consider the magnetic NLS equation „ «2 ~ ∇ − A(x) u + V (x)u − f (|u|2 )u = 0 i
in RN
(1)
where N ≥ 3, A : RN → RN is a magnetic potential, possibly unbounded, V : RN → R is a multiwell electric potential, which can vanish somewhere, f is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution u : RN → C, under conditions on the nonlinearity which are nearly optimal.
1
Introduction
We study the existence of a standing wave solution ψ(x, t) = exp(−iEt/~)u(x), E ∈ R, u : RN → C to the time–dependent nonlinear Schr¨ odinger equation in the presence of an external electromagnetic field 2 ~ ∂ψ = ∇ − A(x) ψ + V (x)ψ − f (|ψ|2 )ψ, (t, x) ∈ R × RN . (2) i~ ∂t i ∗ Supported
by M.I.U.R., national project Variational methods and nonlinear differential equations.
1
Here ~ is the Planck’s constant, i the imaginary unit, A : RN → RN denotes a magnetic potential and V : RN → R an e