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

This paper is concerned with the following quasilinear Schrodinger equations: {-div(g(2)(u)del u) + g(u) g'(u)vertical bar del u vertical bar(2) + V(x)u = K(x)f(u), x is an element of R-N, u is an element of D-1,D-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,D-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. (C) 2016 Elsevier Ltd. All rights reserved.