Existence of solutions for a quasilinear Schrِdinger equation with vanishing potentials

In this paper we investigate the existence of positive solutions for the quasilinear Schrödinger equation: − Δ u + V ( x ) u − Δ ( u 2 ) u = g ( u ) , in R N , where N ⩾ 3 , g has a quasicritical growth and V is a nonnegative potential, which can vanish at infinity.