Positive Solutions for a Class of Quasilinear Schrödinger Equations with Two Parameters

In this paper, we consider the following quasilinear Schrödinger equation -Δu+V(x)u+κ2Δ(u2)u=λf(u)+h(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}$$\begin{aligned} -\Delta u+V(x)u+\frac{\kappa }{2}\Delta (u^2)u=\lambda f(u)+h(u),\,\, x\in {\mathbb {R}}^N, \end{aligned}$$\end{document}where λ,κ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ,\kappa >0$$\end{document}, N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 3$$\end{document} and 2∗=2NN-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2^*=\frac{2N}{N-2}$$\end{document}. By using a change of variable, we obtain the existence of positive solutions for this problem with subcritical nonlinearities by using the mountain pass theorem and Moser iterative method. Our results extend and supplement some other related literatures.