Bound states of nonlinear Schrödinger equations with magnetic fields

Abstract

We study bound states of the following nonlinear Schrödinger equation in the presence of a magnetic field: where A: ℝN → ℝN, V: ℝN → ℝ and g: ℝN × ℝ → [0, ∞). We prove that if V is bounded below with the set {x ∈ ℝ N: V(x) < b} ≠ having finite measure for some b > 0, inf V ≤ 0, and g satisfies some growth conditions, then for any integer m when tsh{cyrillic} > 0 is sufficiently small the problem has m geometrically different solutions. © 2010 Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag.