Remarks on infinitely many solutions for a class of Schrödinger equations with sign-changing potential

In this paper, we study the existence of infinitely many nontrivial solutions for the following semilinear Schrödinger equation: {−Δu+V(x)u=f(x,u),x∈RN,u∈H1(RN),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} -\Delta u+V(x)u=f(x,u), \quad x\in\mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}), \end{cases} $$\end{document} where the potential V is continuous and is allowed to be sign-changing. By using a variant fountain theorem, we obtain the existence of infinitely many high energy solutions under the condition that the nonlinearity f(x,u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(x,u)$\end{document} is of super-linear growth at infinity. The super-quadratic growth condition imposed on F(x,u)=∫0uf(x,t)dt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$F(x,u)=\int _{0}^{u}f(x,t)\,dt$\end{document} is weaker than the Ambrosetti–Rabinowitz type condition and the similar conditions employed in the references.