On multi-bump semi-classical bound states of nonlinear Schrödinger equations with electromagnetic fields

Abstract

We consider the existence and asymptotic behavior of standing wave solutions to nonlinear Schrödinger equations with electromagnetic fields: ih∂ψ/∂t = (h/i∇-A(x))2 ψ+W(x)ψ-f(|ψ|2)ψ ℝ×ω.ω⊂ℝN is a domain which may be bounded or unbounded. For h > 0 small we obtain the existence of multi-bump bound states ψh(x, t) = e -iEt/huh(x) where uh concentrates simultaneously at possibly degenerate, non-isolated local minima of W as h→0. We require that W≥E and allow the possibility that {x∈ω: W(x) = E} ≠ ø. Moreover, we describe the asymptotic behavior of uh as h→0.