Bound states of schrödinger equations with electromagnetic fields and vanishing potentials

Abstract We study the bound states to nonlinear Schrodinger equations with electro-magnetic fields i h ∂ ψ ∂ t = ( h i ∇ - A ( x ) ) 2 ψ + V ( x ) ψ - K ( x ) | p - 1 ψ = 0 , on ℝ + × ℝ N . Let G ( x ) = [ V ( x ) ] p + 1 p - 1 - N 2 [ K ( x ) ] - 2 p - 1 and suppose that G(x) has k local minimum points. For h > 0 small, we find multi-bump bound states ψh (x,t) = e −lEt / h Uh (χ) with Uh concentrating at the local minimum points of G(x) simultaneously as h → 0. The potentials V(x) and K(x) are allowed to be either compactly supported or unbounded at infinity.