On the existence of positive solutions to a certain class of semilinear elliptic equations

In this paper, we study the following semilinear elliptic equation Delta u=phi V(x)u-f(x,u(x))inRN,u is an element of H1(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Delta u = \varphi \left( V(x) u - f(x,u(x)) \right) \quad \text {in} \,\, \mathbf {R}<^>N, \quad u \in H<^>1( \mathbf {R}<^>N ) \end{aligned}$$\end{document}where N >= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \ge 1$$\end{document} and phi(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (s)$$\end{document}, V(x), f(x, s) are given functions. Under some conditions on phi(s),V(x),f(x,s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi (s), V(x), f(x,s)$$\end{document}, we show the existence of positive solution. In particular, we extend the result of Felmer and Ikoma (J Funct Anal 275(8):2162-2196, 2018). In Felmer and Ikoma (J Funct Anal 275(8):2162-2196, 2018), the existence of positive solution was proved by topological degree theoretic argument. In this paper, we employ the variational method.