In this paper, we study the following quasilinear Schrödinger equation: − Δ u + V ( x ) u − Δ ( u 2 ) u = K ( x ) g ( u ) in R N , $$-\Delta u+V(x)u-\Delta\bigl(u^{2}\bigr)u=K(x)g(u) \quad\mbox{in } \mathbb{R}^{N}, $$ where N ≥ 3 , V is a nonnegative continuous function, which can vanish at infinity, and g is a continuous function with a quasicritical growth. Under some appropriate conditions, we get the existence of a positive solution by combining the variational method with a Hardy-type inequality.