Solitary waves for a class of quasilinear Schrodinger equations in dimension two

In this paper we prove the existence and concentration behavior of positive ground state solutions for quasilinear Schrodinger equations of the form -epsilon(2)Delta u + V(z)u - epsilon(2)[Delta(u(2))]u = h(u) in the whole two-dimension space where e is a small positive parameter and V is a continuous potential uniformly positive. The main feature of this paper is that the nonlinear term h(u) is allowed to enjoy the critical exponential growth with respect to the Trudinger-Moser inequality and also the presence of the second order nonhomogeneous term [Delta(u(2))]u which prevents us to work in a classical Sobolev space. Using a version of the Trudinger-Moser inequality, a penalization technique and mountain-pass arguments in a nonstandard Orlicz space we establish the existence of solutions that concentrate near a local minimum of V.