MULTIPLE POSITIVE SOLUTIONS OF SEMI-POSITONE STURM–LIOUVILLE BOUNDARY VALUE PROBLEMS

Abstract

This paper treats the existence of multiple positive solutions of the semi-positone Sturm–Liouville boundary value problem. \begin{eqnarray*} &\lambda (p(t)y'(t))'+g(t)f(t,y(t))=0 \qquad\mbox{almost everywhere on } [R_{0},R_{1}],\\ &\alpha z(R_{0})-\beta p(R_{0})z'(R_{0})=0,\\ &\gamma z(R_{1})+\delta p(R_{1})z'(R_{1})=0, \end{eqnarray*} where $g\in L_{+}^{\infty}[R_{0}, R_{1}]$ and $f$ is allowed to take negative values (that is, $f$ is semi-positone). When $\lambda=1$, new results on the existence of one or two nonzero positive solutions are obtained. These results generalize previous results for positone cases (that is, $f\ge 0$) to the semi-positone cases. We illustrate our results with an explicit example which has two nonzero positive solutions. These results are used to deduce results on intervals of eigenvalues for which there exist one or two nonzero positive eigenfunctions. Applications of these eigenvalue results are provided.