Multiplicity and concentration behavior of solutions for magnetic Choquard equation with critical growth

In this paper, we consider the following nonlinear Choquard equation with magnetic field {(epsilon i del-A(x))2u+V(x)u=epsilon mu-N integral RN|u(y)|2 & lowast;mu+F(|u(y)|2)|x-y|mu dy}(|u|2 & lowast;mu-2u+12 & lowast;mu f(|u|2)u)inRN,u is an element of H1(RN,C) where epsilon>0 is a small parameter,N >= 3, 0<mu<N,2 & lowast;mu=2N-mu N-2,V(x):RN -> RNandA(x):RN -> RN is a continuous potential,fis a continuous subcritical term, andFis the primitive function off. Under a local assumption onthe potentialV, by the variational methods, the penalization techniques and the Ljusternik-Schnirelmann theory, we prove the multiplicity and concentration properties of nontrivial solutions of the above problem for epsilon>0 small enough