Existence and multiplicity of solutions for nonlinear Schrِdinger equations with magnetic field and Hartree type nonlinearities

We study the existence and multiplicity of solutions of the following nonlinear Schrödinger equation in the presence of magnetic field: { ( − i ε ∇ + A ( x ) ) 2 u + V ( x ) u = ( K ( x ) ∗ | u | p ) | u | p − 2 u , | u | ∈ H 1 ( R N ) where V : R N → R is the external potential, K : R N → R is the response function and A = ( A 1 , … , A N ) : R N → R N is the magnetic vector potential associated to an external magnetic field B , that is B = c u r l A . Under suitable assumptions on the functions V ( x ) , K ( x ) and A ( x ) , if the parameter ε is small enough, we can prove the multiplicity of bound states for the equation by variational methods.