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

Abstract

In this paper, we study the global structure of positive solutions of periodic boundary value problems \t\t\t{−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}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\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 qin C([0,2pi ], [0, +infty )) with qnot equiv 0, fin C(mathbb{R},mathbb{R}), the weight hin C[0,2pi ] is a sign-changing function, λ is a parameter. We prove the existence of three positive solutions when h(t) has n positive humps separated by n+1 negative ones. The proof is based on the bifurcation method.