Normalized Solutions for Nonautonomous Schrödinger Equations on a Suitable Manifold

In this paper, we prove the existence of normalized ground state solutions for the following Schrödinger equation -Δu-a(x)f(u)=λu,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}$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u-a(x)f(u)=\lambda u, &{} x\in {\mathbb {R}}^N; \\ u\in H^1({\mathbb {R}}^N), \end{array} \right. \end{aligned}$$\end{document}and give a better representation of its geometrical structure, 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}, λ∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in {\mathbb {R}}$$\end{document}, a∈C(RN,[0,∞))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a\in {\mathcal {C}}({\mathbb {R}}^N, [0, \infty ))$$\end{document} with 0<a∞:=lim|y|→∞a(y)≤a(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<a_{\infty }:=\lim _{|y|\rightarrow \infty }a(y)\le a(x)$$\end{document} and f∈C(R,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in {\mathcal {C}}({\mathbb {R}},{\mathbb {R}})$$\end{document} satisfies general assumptions. In particular, we propose a new approach to recover the compactness for a minimizing sequence on a suitable manifold, and overcome the essential difficulties due to the nonconstant potential a.