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June  2019, 39(6): 3443-3462. doi: 10.3934/dcds.2019142

## Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications

 Center for General Education, Ming Chi University of Technology, New Taipei City 24301, Taiwan

Received  August 2018 Published  February 2019

In this paper, we study the classification and evolution of bifurcation curves of positive solutions for one-dimensional Minkowski-curvature problem
 \left\{ \begin{array}{*{35}{l}} \begin{align} & -{{\left( {{u}^{\prime }}/\sqrt{1-{{u}^{\prime }}^{2}} \right)}^{\prime }}=\lambda f(u),\ \text{in }\left( -L,L \right), \\ & u(-L)=u(L)=0, \\ \end{align} \\\end{array} \right.
where
 $\lambda >0$
is a bifurcation parameter,
 $L>0$
is an evolution parameter,
 $f\in C[0, \infty )\cap C^{2}(0, \infty )$
and there exists
 $\beta >0$
such that
 $\left( \beta -z\right) f(z)>0$
for
 $z\neq \beta$
. In particular, we find that the bifurcation curve
 $S_{L}$
is monotone increasing for all
 $L>0$
when
 $f(u)/u$
is of Logistic type, and is either
 $\subset$
-shaped or S-shaped for large
 $L>0$
when
 $f(u)/u$
is of weak Allee effect type. Finally, we can apply these results to obtain the global bifurcation diagrams in some important applications including ecosystem model.
Citation: Shao-Yuan Huang. Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3443-3462. doi: 10.3934/dcds.2019142
##### References:
 [1] R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys., 87 (1982), 131-152.  doi: 10.1007/BF01211061. [2] D. Butler, R. Shivaji and A. Tuck, S-shaped bifurcation curves for logistic growth and weak Allee effect growth models with grazing on an interior patch, Proceedings of the Ninth MSU-UAB Conference on Differential Equations and Computational Simulations, 15–25, Electron. J. Differ. Equ. Conf., 20, Texas State Univ., San Marcos, TX, 2013. [3] I. Coelho, C. Corsato, F. Obersnel and P. Omari, Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation, Adv. Nonlinear Stud., 12 (2012), 621-638.  doi: 10.1515/ans-2012-0310. [4] C. Corsato, Mathematical analysis of some differential models involving the Euclidean or the Minkowski mean curvature operator, PhD thesis, University of Trieste, 2015. Available at https://www.openstarts.units.it/bitstream/10077/11127/1/PhD_Thesis _Corsato.pdf. [5] R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, Vol. 2: Mainly electromagnetism and matter. Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1964. [6] S.-Y. Huang, Classification and evolution of bifurcation curves for the one-dimensional Minkowski-curvature problem and its applications, J. Differential Equations, 264 (2018), 5977-6011.  doi: 10.1016/j.jde.2018.01.021. [7] S.-Y. Huang, Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application, Commun. Pure Appl. Anal., 17 (2018), 1271-1294.  doi: 10.3934/cpaa.2018061. [8] S.-Y. Huang and S.-H. Wang, An evolutionary property of the bifurcation curves for a positone problem with cubic nonlinearity, Taiwanese J. Math., 20 (2016), 639-661.  doi: 10.11650/tjm.20.2016.6563. [9] S.-Y. Huang, Some proofs in a paper "Bifurcation diagrams of one-dimensional Minkowski-curvature problem and its applications", available from http://mx.nthu.edu.tw/~symbol126sy-huang/Proofs2018. [10] K.-C. Hung, S.-Y. Huang and S.-H. Wang, A global bifurcation theorem for a positone multiparameter problem and its application, Discrete Contin. Dyn. Syst., 37 (2017), 5127-5149.  doi: 10.3934/dcds.2017222. [11] K.-C. Hung and S.-H. Wang, Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications, Trans. Amer. Math. Soc., 365 (2013), 1933-1956.  doi: 10.1090/S0002-9947-2012-05670-4. [12] E. Lee, S. Sasi and R. Shivaji, S-shaped bifurcation curves in ecosystems, J. Math. Anal. Appl., 381 (2011), 732-741.  doi: 10.1016/j.jmaa.2011.03.048. [13] R. Ma and Y. Lu, Multiplicity of positive solutions for second order nonlinear Dirichlet problem with one-dimension Minkowski-curvature operator, Adv. Nonlinear Stud., 15 (2015), 789-803.  doi: 10.1515/ans-2015-0403. [14] P. M. McCabe, J. A. Leach and D. J. Needham, The evolution of travelling waves in fractional order autocatalysis with decay. Ⅰ. Permanent from travelling waves, SIAM J. Appl. Math., 59 (1998), 870-899.  doi: 10.1137/S0036139996312594. [15] T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problems, Journal of Differential Equations, 146 (1998), 121-156.  doi: 10.1006/jdeq.1998.3414. [16] E. Poole, B. Roberson and B. Stephenson, Weak Allee effect, grazing, and S-shaped bifurcation curves, Involve, 5 (2012), 133-158.  doi: 10.2140/involve.2012.5.133. [17] J. Shi and R. Shivaji, Persistence in reaction diffusion models with weak Allee effect, J. Math. Biol., 52 (2006), 807-829.  doi: 10.1007/s00285-006-0373-7. [18] C.-C. Tzeng, K.-C. Hung and S.-H. Wang, Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity, J. Differential Equations, 252 (2012), 6250-6274.  doi: 10.1016/j.jde.2012.02.020. [19] M.-H. Wang and M. Kot, Speeds of invasion in a model with strong or weak Allee effects, Math. Biosci., 171 (2001), 83-97.  doi: 10.1016/S0025-5564(01)00048-7. [20] J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.  doi: 10.1137/S0036144599364296. [21] T.-S. Yeh, S-shaped and broken S-shaped bifurcation curves for a multiparameter diffusive logistic problem with Holling type-Ⅲ functional response, Commun. Pure Appl. Anal., 16 (2017), 645-670.  doi: 10.3934/cpaa.2017032. [22] T.-S. Yeh, Bifurcation curves of positive steady-state solutions for a reaction-diffusion problem of lake eutrophication, J. Math. Anal. Appl., 449 (2017), 1708-1724.  doi: 10.1016/j.jmaa.2016.12.063.

show all references

##### References:
 [1] R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys., 87 (1982), 131-152.  doi: 10.1007/BF01211061. [2] D. Butler, R. Shivaji and A. Tuck, S-shaped bifurcation curves for logistic growth and weak Allee effect growth models with grazing on an interior patch, Proceedings of the Ninth MSU-UAB Conference on Differential Equations and Computational Simulations, 15–25, Electron. J. Differ. Equ. Conf., 20, Texas State Univ., San Marcos, TX, 2013. [3] I. Coelho, C. Corsato, F. Obersnel and P. Omari, Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation, Adv. Nonlinear Stud., 12 (2012), 621-638.  doi: 10.1515/ans-2012-0310. [4] C. Corsato, Mathematical analysis of some differential models involving the Euclidean or the Minkowski mean curvature operator, PhD thesis, University of Trieste, 2015. Available at https://www.openstarts.units.it/bitstream/10077/11127/1/PhD_Thesis _Corsato.pdf. [5] R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, Vol. 2: Mainly electromagnetism and matter. Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1964. [6] S.-Y. Huang, Classification and evolution of bifurcation curves for the one-dimensional Minkowski-curvature problem and its applications, J. Differential Equations, 264 (2018), 5977-6011.  doi: 10.1016/j.jde.2018.01.021. [7] S.-Y. Huang, Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application, Commun. Pure Appl. Anal., 17 (2018), 1271-1294.  doi: 10.3934/cpaa.2018061. [8] S.-Y. Huang and S.-H. Wang, An evolutionary property of the bifurcation curves for a positone problem with cubic nonlinearity, Taiwanese J. Math., 20 (2016), 639-661.  doi: 10.11650/tjm.20.2016.6563. [9] S.-Y. Huang, Some proofs in a paper "Bifurcation diagrams of one-dimensional Minkowski-curvature problem and its applications", available from http://mx.nthu.edu.tw/~symbol126sy-huang/Proofs2018. [10] K.-C. Hung, S.-Y. Huang and S.-H. Wang, A global bifurcation theorem for a positone multiparameter problem and its application, Discrete Contin. Dyn. Syst., 37 (2017), 5127-5149.  doi: 10.3934/dcds.2017222. [11] K.-C. Hung and S.-H. Wang, Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications, Trans. Amer. Math. Soc., 365 (2013), 1933-1956.  doi: 10.1090/S0002-9947-2012-05670-4. [12] E. Lee, S. Sasi and R. Shivaji, S-shaped bifurcation curves in ecosystems, J. Math. Anal. Appl., 381 (2011), 732-741.  doi: 10.1016/j.jmaa.2011.03.048. [13] R. Ma and Y. Lu, Multiplicity of positive solutions for second order nonlinear Dirichlet problem with one-dimension Minkowski-curvature operator, Adv. Nonlinear Stud., 15 (2015), 789-803.  doi: 10.1515/ans-2015-0403. [14] P. M. McCabe, J. A. Leach and D. J. Needham, The evolution of travelling waves in fractional order autocatalysis with decay. Ⅰ. Permanent from travelling waves, SIAM J. Appl. Math., 59 (1998), 870-899.  doi: 10.1137/S0036139996312594. [15] T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problems, Journal of Differential Equations, 146 (1998), 121-156.  doi: 10.1006/jdeq.1998.3414. [16] E. Poole, B. Roberson and B. Stephenson, Weak Allee effect, grazing, and S-shaped bifurcation curves, Involve, 5 (2012), 133-158.  doi: 10.2140/involve.2012.5.133. [17] J. Shi and R. Shivaji, Persistence in reaction diffusion models with weak Allee effect, J. Math. Biol., 52 (2006), 807-829.  doi: 10.1007/s00285-006-0373-7. [18] C.-C. Tzeng, K.-C. Hung and S.-H. Wang, Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity, J. Differential Equations, 252 (2012), 6250-6274.  doi: 10.1016/j.jde.2012.02.020. [19] M.-H. Wang and M. Kot, Speeds of invasion in a model with strong or weak Allee effects, Math. Biosci., 171 (2001), 83-97.  doi: 10.1016/S0025-5564(01)00048-7. [20] J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.  doi: 10.1137/S0036144599364296. [21] T.-S. Yeh, S-shaped and broken S-shaped bifurcation curves for a multiparameter diffusive logistic problem with Holling type-Ⅲ functional response, Commun. Pure Appl. Anal., 16 (2017), 645-670.  doi: 10.3934/cpaa.2017032. [22] T.-S. Yeh, Bifurcation curves of positive steady-state solutions for a reaction-diffusion problem of lake eutrophication, J. Math. Anal. Appl., 449 (2017), 1708-1724.  doi: 10.1016/j.jmaa.2016.12.063.
ⅰ) monotone increasing. (ⅱ) $\subset$-shaped. (ⅲ) S-shaped
Graphs of bifurcation curve $S_{L}$ of (1) with varying $L>0$. (ⅰ) (C2) and (H2) hold. (ⅱ) either (C4) or (C6) holds
Graphs of bifurcation curve $S_{L}$ of (1) with varying $L>0$. (ⅰ) $S_{\mathring{L}}$ is monotone increasing. (ⅱ) $S_{% \mathring{L}}$ is not monotone increasing
Graphs of bifurcation curve $S_{L}$ of (1), (5). $\phi _{1}, \phi _{2}\in C(\sqrt{27} , \infty )$ satisfy $\phi _{1}(K)>\phi _{2}(K)$ and $\Phi (K, \phi _{1}(K)) = \Phi (K, \phi _{2}(K)) = 0.$
Graphs of bifurcation curve SL of (1), (6)
Graphs of bifurcation curve SL of (1), (7). (ⅰ) c = 0. (ⅱ) c > 0
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