On the least energy solutions for semilinear Schrodinger equation with electromagnetic fields involving critical growth and indefinite potentials

In this paper, we are concerned with the following semilinear Schrodinger equation with electromagnetic fields and critical growth -(. + iA(x)) 2u + (.a(x) -d) u = | u| 2 *-2u, x. RN for sufficiently large., where N = 4, a(x) = 0 and its zero set is not empty, 2 * is the critical Sobolev exponent, d > 0 is a constant such that the operator -(.+ iA(x)) 2+.a(x)-d might be indefinite but is non-degenerate. Using variational method and modified Nehari-Pankov method, we prove the equation admits a least energy solution which localizes near the potential well a -1(0) . The results we obtain here extend the corresponding results for the Schrodinger equation which involves critical growth but does not involve electromagnetic fields.