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November  2019, 18(6): 3267-3284. doi: 10.3934/cpaa.2019147

## Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity

 Center for General Education, National Formosa University, Yunlin 632, Taiwan

Received  July 2018 Revised  December 2018 Published  May 2019

In this paper, we study the global bifurcation curves and the exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem
 \left\{ \begin{array}{*{35}{l}} \begin{align} & -{{\left( {{u}^{\prime }}/\sqrt{1-{{u}^{\prime }}^{2}} \right)}^{\prime }}=\lambda \left( {{u}^{p}}-{{u}^{q}} \right),\ \ \ \text{in}\left( {-L},{L} \right),\ \\ & u(-L)=u(L)=0, \\ \end{align} \\\end{array} \right.
where
 $p, q\geq 0$
,
 $p\neq q$
,
 $\lambda >0$
is a bifurcation parameter and
 $L>0$
is an evolution parameter. We prove that the bifurcation curve is continuous and further classify its exact shape (either monotone increasing or
 $\subset$
-shaped by
 $p$
and
 $q$
). Moreover, we can achieve the exact multiplicity of positive solutions.
Citation: Shao-Yuan Huang. Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3267-3284. doi: 10.3934/cpaa.2019147
##### References:
  R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys., 87 (1982), 131–152.  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.  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.   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. G. Dai, Global bifurcation for problem with mean curvature operator on general domain, NoDEA Nonlinear Di erential Equations Appl., 24 (2017), Art. 30, 10 pp. doi: 10.1007/s00030-017-0454-x.   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.  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.   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.   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.   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.   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.   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.   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.   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.   X. Zhang and M. Feng, Bifurcation diagrams and exact multiplicity of positive solutions of one-dimensional prescribed mean curvature equation in Minkowski space, Commun. Contemp. Math.. doi: 10.1142/S0219199718500037.  show all references

##### References:
  R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys., 87 (1982), 131–152.  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.  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.   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. G. Dai, Global bifurcation for problem with mean curvature operator on general domain, NoDEA Nonlinear Di erential Equations Appl., 24 (2017), Art. 30, 10 pp. doi: 10.1007/s00030-017-0454-x.   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.  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.   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.   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.   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.   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.   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.   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.   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.   X. Zhang and M. Feng, Bifurcation diagrams and exact multiplicity of positive solutions of one-dimensional prescribed mean curvature equation in Minkowski space, Commun. Contemp. Math.. doi: 10.1142/S0219199718500037.   Graphs of $f(u)$ on $[0, \infty )$. (i) $q>p\geq 0$. (ii) $p>q\geq 0.$
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