Bound states of schrödinger equations with electromagnetic fields and vanishing potentials

We study the bound states to nonlinear Schrödinger equations with electro-magnetic fields ih∂ψ∂t=(hi∇-A(x))2ψ+V(x)ψ-K(x)|p-1ψ=0,onℝ+×ℝN. Let G(x)=[V(x)]p+1p-1-N2[K(x)]-2p-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/hUh(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.