Notes on Multiple Periodic Solutions for Second Order Hamiltonian Systems

In this paper, we study the multiplicity of periodic solutions for the second order Hamiltonian systems u¨+∇F(t,u)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{u}+\nabla F(t,u)=0$$\end{document} with the boundary condition u(0)-u(T)=u˙(0)-u˙(T)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u(0)-u(T)=\dot{u}(0)-\dot{u}(T)=0$$\end{document}, where the potential F is either subquadratic k(t)-concave or subquadratic μ(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu (t)$$\end{document}-convex. Based on the reduction method and a three-critical-point theorem due to Brezis and Nirenberg, we obtain the multiplicity results, which complement and sharply improve some related results in the literature.