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

We study positive solutions to the two point boundary value problem:

$$\begin{array}{c}Lu=-u^{″}=\lambda \{{\displaystyle \frac{A}{u^{\gamma }}}+M[u^{\alpha }+u^{\delta }]\};(0,1)\\ u(0)=0=u(1)\end{array}$$

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 $(\lambda \text{vs}\Vert u{\Vert }_{\mathrm{\infty }})$ for positive solutions is at least a reversed S-shaped curve when $M\gg 1$. Recent results in the literature imply that for $M\gg 1$ there exists a range of $\lambda$ where there exist at least two positive solutions. Here, when $M\gg 1$, 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 $M\gg 1$, 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 $M\gg 1$.