GROUND STATE SOLUTIONS OF MAGNETIC SCHRODINGER EQUATIONS WITH EXPONENTIAL GROWTH

In this paper, we investigate the following nonlinear magnetic Schrodinger equation with exponential growth: (-i & nabla; + A(x))(2)u + V (x)u = f(x, |u|(2))u in R-2, where V is the electric potential and A is the magnetic potential. We prove the existence of ground state solutions both in the indefinite case with subcritical exponential growth and in the definite case with critical exponential growth. In order to overcome the difficulty brings from the presence of magnetic field, by using subtle estimates and establishing a new energy estimate inequality in complex field, we weaken the Ambrosetti-Rabinowitz type condition and the strict monotonicity condition, which are commonly used in the indefinite case. Furthermore, in the definite case, we introduce a Moser type function involving magnetic potential and some new analytical techniques, which can also be applied to related magnetic elliptic equations. Our results extend and complement the present ones in the literature.