Positive solutions for a class of semipositone periodic boundary value problems via bifurcation theory

Abstract

In this paper, we are concerned with the existence of positive solutions of nonlinear periodic boundary value problems like \begin{equation*} \begin{split} &-u''+q(x)u=\lambda f(x,u),\qquad x\in(0,2\pi),\\ &u(0)=u(2\pi),\qquad u'(0)=u'(2\pi), \end{split} \end{equation*} where $q\in C([0,2\pi],[0,\infty))$ with $q\not \equiv 0$, $f\in C([0,2\pi]\times \mathbb{R}^+,\mathbb{R})$, $\lambda>0$ is the bifurcation parameter. By using bifurcation theory, we deal with both asymptotically linear, superlinear as well as sublinear problems and show that there exists a global branch of solutions emanating from infinity. Furthermore, we proved that for $\lambda$ near the bifurcation value, solutions of large norm are indeed positive.

Keywords

• semipositone
• positive solutions
• periodic problems
• bifurcation