On a class of quasilinear Schrödinger equations with vanishing potentials and mixed nonlinearities

In this paper, we study the following generalized quasilinear Schrödinger equations with mixed nonlinearity {−div(g2(u)∇u)+g(u)g′(u)|∇u|2+V(x)u=K(x)f(u)+λξ(x)g(u)|G(u)|p−2G(u),x∈RN,u∈D1,2(RN),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {\begin{array}{*{20}{c}} { - div({g^2}(u)\nabla u) + g(u)g'(u){{\left| {\nabla u} \right|}^2} + V(x)u = K(x)f(u) + \lambda \xi (x)g(u){{\left| {G(u)} \right|}^{p - 2}}G(u), x \in {\mathbb{R}^N},} \\ {u \in {\mathcal{D}^{1,2}}({\mathbb{R}^N}),} \end{array}} \right.$$\end{document} where N ≥ 3, V, K are nonnegative continuous functions and f is a continuous function with a quasicritical growth. Using a change of variable as G(u)=∫0ug(t)dt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G(u) = \int_0^u {g(t)\rm{dt}}$$\end{document}, the above quasilinear equation is reduced to a semilinear one. Under some suitable assumptions, we prove that the above equation has at least one nontrivial solution by working in weighted Sobolev spaces and employing the variational methods.