Positive solutions for a class of quasilinear Schrödinger equations with vanishing potentials

摘要

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 , $$-Delta u+V(x)u-Deltaigl(u^{2}igr)u=K(x)g(u) quadmbox{in } mathbb{R}^{N}, $$ where N ≥ 3 $Ngeq3$ , 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.