Concentrating solutions for a magnetic Schrödinger equation with critical growth

Abstract

We deal with the following nonlinear Schrödinger equation with magnetic field and critical growth:{(εı∇−A(x))2u+V(x)u=f(|u|2)u+|u|2⁎−2u in RN,u∈H1(RN,C), where ε>0 is a small parameter, N≥3, 2⁎=2NN−2 is the critical Sobolev exponent, A∈C1(RN,RN) is a magnetic vector potential, V:RN→R is a continuous positive potential having a local minimum and f:R→R is a superlinear continuous function with subcritical growth. Using penalization techniques and variational methods, we investigate the existence and concentration of nontrivial solutions for ε>0 small enough.