Research article

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

  • Received: 30 September 2021 Revised: 09 April 2022 Accepted: 12 April 2022 Published: 21 December 2022
  • 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 $.

    Citation: Amila Muthunayake, Cac Phan, Ratnasingham Shivaji. An infinite semipositone problem with a reversed S-shaped bifurcation curve[J]. Electronic Research Archive, 2023, 31(2): 1147-1156. doi: 10.3934/era.2023058

    Related Papers:

  • 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 $.



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    [1] D. D. Hai, R. Shivaji, Existence and multiplicity of positive radial solutions for singular superlinear elliptic systems in the exterior of a ball, J. Differ. Equ., 266 (2019), 2232–2243. https://doi.org/10.1016/j.jde.2018.08.027 doi: 10.1016/j.jde.2018.08.027
    [2] K. J. Brown, M. M. A. Ibrahim, R. Shivaji, S-Shaped bifurcation curves, Nonlinear Anl., 5 (1981), 475–486.
    [3] T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1–13.
    [4] E. Ko, E. K. Lee, R. Shivaji, On S-shaped and reversed S-shaped bifurcation curves for singular problems, Electron. J. Qual. Theo., 31 (2011), 1–12. https://doi.org/10.14232/ejqtde.2011.1.31 doi: 10.14232/ejqtde.2011.1.31
    [5] U. Das, A. Muthunayake, R. Shivaji, Existence results for a class of p-q Laplacian semipositone boundary value problems, Electron. J. Qual. Theo., 88 (2020), 1–7. https://doi.org/10.14232/ejqtde.2020.1.88 doi: 10.14232/ejqtde.2020.1.88
    [6] A. Castro, R. Shivaji, Non-negative solutions for a class of non-positone problems, P. Roy. Soc. Edinb. A, 108 (1988), 291–302. https://doi.org/10.1017/S0308210500014670 doi: 10.1017/S0308210500014670
    [7] A. C. Lazer, P. J. McKenna, On a singular nonlinear elliptic boundary value problem, P. Am. Math. Soc., 111 (1991), 720–730. https://doi.org/10.2307/2048410 doi: 10.2307/2048410
    [8] Z. Zhang, On a Dirichlet problem with a singular nonlinearity, J. Math. Anal. Appl., 194 (1995), 103–113. https://doi.org/10.1006/jmaa.1995.1288 doi: 10.1006/jmaa.1995.1288
    [9] Y. Miaoxin, J. Shi, On a singular nonlinear semilinear elliptic problem, P. Roy. Soc. Edinb. A, 128 (1998), 1389–1401. https://doi.org/10.1017/S0308210500027384 doi: 10.1017/S0308210500027384
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