Existence of solutions for a quasilinear Schrödinger equation with vanishing potentials

In this paper we investigate the existence of positive solutions for the quasilinear Schrödinger equation:−Δu+V(x)u−Δ(u2)u=g(u), in RN, where N⩾3, g has a quasicritical growth and V is a nonnegative potential, which can vanish at infinity.