Semiclassical solutions of Schrodinger equations with magnetic fields and critical nonlinearities

We study the following nonlinear Schrodinger equations (-i epsilon del + A(x))(2)w + V(x)w = W(x)g(vertical bar w vertical bar)w; (0.1) (-i epsilon del + A(x))(2)w + V(x)w = W(x)g(vertical bar w vertical bar + vertical bar w vertical bar(2)*(-2))w, (0.2) for w is an element of H-1(R-N,C), where g(vertical bar w vertical bar)w is super linear and subcritical, 2* = 2N/(N - 2) if N > 2 and = infinity if N = 2, min V > 0 and inf W > 0. Under proper assumptions we explore the existence and concentration phenomena of semiclassical solutions of (0.1). The most interesting result obtained here refers to the critical case. We establish the existence and describe the concentration of semiclassical ground states of (0.2) provided either min V < tau (0) for some tau(0) > 0, or mas W > kappa(0) for some kappa(0) < 0.