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

• Published in: Boundary value problems 2018-06, Vol.2018 (1), p.1-17, Article 93
• Main Authors: He, Zhiqian, Ma, Ruyun, Xu, Man
• Format: Article
• Language: English
• Subjects: Analysis; Approximations and Expansions; Bifurcation; Bifurcations; Boundary value problems; Difference and Functional Equations; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Periodic boundary value problem; Three positive solutions
• ISSN: 1687-2770; 1687-2762; 1687-2770
• DOI: 10.1186/s13661-018-1011-1

In this paper, we study the global structure of positive solutions of periodic boundary value problems { − u ″ ( t ) + q ( t ) u ( t ) = λ h ( t ) f ( u ( t ) ) , t ∈ ( 0 , 2 π ) , u ( 0 ) = u ( 2 π ) , u ′ ( 0 ) = u ′ ( 2 π ) , where q ∈ C ( [ 0 , 2 π ] , [ 0 , + ∞ ) ) with q ≢ 0 , f ∈ C ( R , R ) , the weight h ∈ C [ 0 , 2 π ] 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.