Existence and uniqueness of multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields

In this paper, we are concerned with the existence and uniqueness of multi-bump bound states of the nonlinear Schrödinger equations with electromagnetic potential iℏ∂ψ∂t=ℏi∇-A(x)2ψ+V(x)ψ-|ψ|p-2ψ,x∈RNfor sufficiently small ℏ>0ℏ>0, where i   is the imaginary unit, 20ℏ>0.