Three positive solutions for second-order periodic boundary value problems with sign-changing weight

In this paper, we study the global structure of positive solutions of periodic boundary value problems {−u″(t)+q(t)u(t)=λh(t)f(u(t)),t∈(0,2π),u(0)=u(2π),u′(0)=u′(2π),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textstyle\begin{cases} -u''(t)+q(t)u(t)=\lambda h(t)f(u(t)), \quad t\in (0,2\pi ), \\ u(0)=u(2\pi ), \quad\quad u'(0)=u'(2\pi ), \end{cases} $$\end{document} where q∈C([0,2π],[0,+∞))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q\in C([0,2\pi ], [0, +\infty ))$\end{document} with q≢0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q\not \equiv 0$\end{document}, 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}$f\in C(\mathbb{R},\mathbb{R})$\end{document}, the weight h∈C[0,2π]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$h\in C[0,2\pi ]$\end{document} is a sign-changing function, λ is a parameter. We prove the existence of three positive solutions when h(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$h(t)$\end{document} has n positive humps separated by n+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n+1$\end{document} negative ones. The proof is based on the bifurcation method.