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An infinite semipositone problem with a reversed S-shaped bifurcation curve
Electronic Research Archive 2023, 31(2): 1147-1156
Published: 15 February 2023
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We study positive solutions to the two point boundary value problem:

L u = u = λ { A u γ + M [ u α + u δ ] } ; ( 0 , 1 ) u ( 0 ) = 0 = u ( 1 )

where A < 0, α ( 0 , 1 ) , δ > 1 , γ ( 0 , 1 ) are constants and λ > 0 , M > 0 are parameters. We prove that the bifurcation diagram ( λ vs u ) 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 λ where there exist at least two positive solutions. Here, when M 1, we prove the existence of a range of λ 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.

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