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

Abstract

Abstract 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), $$\textstyle\begin{cases} -u''(t)+q(t)u(t)=\lambda h(t)f(u(t)), \quad t\in (0,2\pi ), \\ u(0)=u(2\pi ), \quad\quad u'(0)=u'(2\pi ), \end{cases} $$ where q∈C([0,2π],[0,+∞)) $q\in C([0,2\pi ], [0, +\infty ))$ with q≢0 $q\not \equiv 0$, f∈C(R,R) $f\in C(\mathbb{R},\mathbb{R})$, the weight h∈C[0,2π] $h\in C[0,2\pi ]$ is a sign-changing function, λ is a parameter. We prove the existence of three positive solutions when h(t) $h(t)$ has n positive humps separated by n+1 $n+1$ negative ones. The proof is based on the bifurcation method.

Keywords

• Three positive solutions
• Periodic boundary value problem
• Bifurcation