An infinite semipositone problem with a reversed S-shaped bifurcation curve

Abstract

We study positive solutions to the two point boundary value problem: $ \begin{equation*} \begin{matrix}Lu = -u'' = \lambda \bigg\{\dfrac{A}{u^\gamma}+M\big[u^\alpha+u^\delta\big]\bigg\} \; ;\; (0, 1) \\ u(0) = 0 = u(1)\; \; \; \; \; \; \; \; \; \; \end{matrix} \end{equation*} $ where $ A < 0 $, $ \alpha \in (0, 1), \delta > 1, \gamma \in (0, 1) $ are constants and $ \lambda > 0, M > 0 $ are parameters. We prove that the bifurcation diagram $ (\lambda \text{ vs } \|u\|_\infty) $ for positive solutions is at least a reversed S-shaped curve when $ M\gg1 $. Recent results in the literature imply that for $ M\gg1 $ there exists a range of $ \lambda $ where there exist at least two positive solutions. Here, when $ M\gg1 $, we prove the existence of a range of $ \lambda $ for which there exist at least three positive solutions and that the bifurcation diagram is at least a reversed S-shaped curve. Further, via a quadrature method and Python computations, for $ M\gg1 $, we show that the bifurcation diagram is exactly a reversed S-shaped curve. Also, when the operator $ L $ is replaced by a $ p $-Laplacian operator with $ p > 1 $, as well as $ p $-$ q $ Laplacian operator with $ p = 4 $ and $ q = 2 $, we show that the bifurcation diagram is again an exactly reversed S-shaped curve when $ M\gg1 $.

Keywords

• two-point boundary value problems
• infinite semipositone reaction terms
• positive solutions
• multiplicity results
• reversed s-shaped bifurcation curves