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

We study positive solutions to the two point boundary value problem: Lu= -u('')= lambda{A/u(gamma) + M[u(alpha) + u(delta)]}; (0,1) u(0) = 0 = u(1) where A<0, alpha is an element of(0,1),delta > 1,gamma is an element of(0,1) are constants and lambda > 0,M > 0 are parameters. We prove that the bifurcation diagram (lambda vs parallel to u parallel to(infinity)) for positive solutions is at least a reversed S-shaped curve when M >> 1. Recent results in the literature imply that for M >> 1 there exists a range of lambda where there exist at least two positive solutions. Here, when M >> 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 >> 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 >> 1.