@article{Muthunayake2023, 
author = {Amila Muthunayake and Cac Phan and Ratnasingham Shivaji},
title = {An infinite semipositone problem with a reversed S-shaped bifurcation curve},
year = {2023},
journal = {Electronic Research Archive},
volume = {31},
number = {2},
pages = {1147-1156},
keywords = {two-point boundary value problems, infinite semipositone reaction terms, positive solutions, multiplicity results, reversed S-shaped bifurcation curves},
url = {https://www.sciopen.com/article/10.3934/era.2023058},
doi = {10.3934/era.2023058},
abstract = {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.}
}