Quasilinear elliptic problems under asymptotically linear conditions at infinity and at the origin

We obtain existence and multiplicity of solutions for the quasilinear Schrödinger equation -Δu+V(x)u-Δ(u2)u=g(x,u),x∈RN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta u + V(x)u - \Delta(u^2)u = g(x,u), \,\, x \in \mathbb{R}^N,$$\end{document}where V is a positive potential and the nonlinearity g(x, t) behaves like t at the origin and like t3 at infinity. In the proof, we apply a changing of variables besides variational methods. The obtained solutions belong to W1,2(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${W^{1,2}(\mathbb{R}^N)}$$\end{document} .