Existence and nonuniqueness of solutions for a class of asymptotically linear nonperiodic Schrödinger equations

In this paper, we consider the existence and nonuniqueness of solutions for the following Schrödinger equations with the nonlinear term being asymptotically linear at infinity, -Δu+V(x)u=f(x,u)forx∈RN,u(x)→0as|x|→∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u=f(x,u)&{}\text{ for } x\in {\mathbb {R}}^{N},\\ u(x)\rightarrow 0&{}\text{ as } |x|\rightarrow \infty . \end{array} \right. \end{aligned}$$\end{document}We introduce a new condition on V(x) and obtain a new compact embedding theorem. Some new asymptotically linear conditions on f(x, u) are introduced which are quite different from the previous ones in the references. An existence theorem is obtained using the Generalized Mountain Pass theorem. Furthermore, we obtain the existence of infinitely many solutions for above asymptotically linear Schrödinger equations by the Variant Fountain theorem, which has been considered by only few authors.