Existence and Multiplicity of Normalized Solutions to Biharmonic Schrödinger Equations with Subcritical Growth

This paper is concerned with the existence and multiplicity of normalized solutions to the following biharmonic Schrödinger equation: Δ2u-h(εx)|u|p-2u=λuinRN,∫RNu2dx=c,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} {\Delta }^2u-h(\varepsilon x) |u|^{p-2}u=\lambda u\quad \text{ in }\ {\mathbb {R}}^N, \\ \int _{{\mathbb {R}}^N} u^2 {\textrm{d}}x = c, \\ \end{array} \right. \end{aligned}$$\end{document}where ε,c>0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon , c>0,$$\end{document}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}2<p<2+8N,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2<p<2+\frac{8}{N},$$\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} is a Lagrangian multiplier and h:RN→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h:{\mathbb {R}}^N\rightarrow {{\mathbb {R}}}$$\end{document} is a continuous function. Under a class of reasonable assumptions on h,  we obtain the existence of ground-state normalized solutions. Meanwhile, we also prove that the number of normalized solutions is at least the number of global maximus points of h when ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} is small enough. Some recent results are generalized and improved.