Positive solutions for generalized quasilinear Schrödinger equations with potential vanishing at infinity

This paper is concerned with the following quasilinear Schrödinger equations: {−div(g2(u)∇u)+g(u)g′(u)∣∇u∣2+V(x)u=K(x)f(u),x∈RN,u∈D1,2(RN), where N≥3N≥3 and VV, KK are nonnegative continuous functions. Firstly by using a change of variables, the quasilinear equation is reduced to a semilinear one, whose associated functional is still not well defined in D1,2(RN)D1,2(RN) because of the potential vanishing at infinity. However, by using a Hardy-type inequality, we can work in the weighted Sobolev space in which the functional is well defined. Using this fact together with the variational methods, we obtain a positive solution.