Existence of Solutions for a Quasilinear Schrödinger Equation with Potential Vanishing

We study the following quasilinear Schrödinger equation −Δu+V(x)u−Δ(u2)u=K(x)g(u),x∈ℝ3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ - \Delta u + V(x)u - \Delta ({u^2})u = K(x)g(u),\,\,\,\,\,\,\,\,x \in {\mathbb{R}^3},$$\end{document} where the nonlinearity g(u) is asymptotically cubic at infinity, the potential V(x) may vanish at infinity. Under appropriate assumptions on K(x), we establish the existence of a nontrivial solution by using the mountain pass theorem.