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

Abstract

In this paper we prove the existence and concentration behavior of positive ground state solutions for quasilinear Schrodinger equations of the form −e2Δu + V(z)u − e2 [Δ(u2)]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 [Δ(u2)]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.