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

Abstract

This paper is concerned with the following quasilinear Schrödinger equations: { − div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) ∣ ∇ u ∣ 2 + V ( x ) u = K ( x ) f ( u ) , x ∈ R N , u ∈ D 1 , 2 ( R N ) , where N ≥ 3 and V , K 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 D 1 , 2 ( R N ) 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.