
-
Previous Article
The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line
- CPAA Home
- This Issue
-
Next Article
Translating solutions of non-parametric mean curvature flows with capillary-type boundary value problems
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 |
$ \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. $ |
$ p, q\geq 0 $ |
$ p\neq q $ |
$ \lambda >0 $ |
$ L>0 $ |
$ \subset $ |
$ p $ |
$ q $ |
References:
[1] |
R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys., 87 (1982), 131–152. |
[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] |
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. |
[6] |
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. |
[7] |
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. |
[8] |
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. |
[9] |
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. |
[10] |
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. |
[11] |
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. |
[12] |
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. |
[13] |
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. |
[14] |
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. |
[15] |
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:
[1] |
R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys., 87 (1982), 131–152. |
[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] |
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. |
[6] |
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. |
[7] |
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. |
[8] |
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. |
[9] |
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. |
[10] |
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. |
[11] |
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. |
[12] |
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. |
[13] |
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. |
[14] |
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. |
[15] |
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. |
[1] |
Shao-Yuan Huang. Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1271-1294. doi: 10.3934/cpaa.2018061 |
[2] |
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 |
[3] |
Alberto Boscaggin, Francesca Colasuonno, Benedetta Noris. Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1921-1933. doi: 10.3934/dcdss.2020150 |
[4] |
Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159 |
[5] |
Tetsuya Ishiwata, Takeshi Ohtsuka. Evolution of a spiral-shaped polygonal curve by the crystalline curvature flow with a pinned tip. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5261-5295. doi: 10.3934/dcdsb.2019058 |
[6] |
Ruyun Ma, Man Xu. Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2701-2718. doi: 10.3934/dcdsb.2018271 |
[7] |
Matthias Bergner, Lars Schäfer. Time-like surfaces of prescribed anisotropic mean curvature in Minkowski space. Conference Publications, 2011, 2011 (Special) : 155-162. doi: 10.3934/proc.2011.2011.155 |
[8] |
Hongjie Ju, Jian Lu, Huaiyu Jian. Translating solutions to mean curvature flow with a forcing term in Minkowski space. Communications on Pure and Applied Analysis, 2010, 9 (4) : 963-973. doi: 10.3934/cpaa.2010.9.963 |
[9] |
Uriel Kaufmann, Humberto Ramos Quoirin, Kenichiro Umezu. A curve of positive solutions for an indefinite sublinear Dirichlet problem. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 817-845. doi: 10.3934/dcds.2020063 |
[10] |
Yves Coudène, Barbara Schapira. Counterexamples in non-positive curvature. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1095-1106. doi: 10.3934/dcds.2011.30.1095 |
[11] |
Alejandro Allendes, Alexander Quaas. Multiplicity results for extremal operators through bifurcation. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 51-65. doi: 10.3934/dcds.2011.29.51 |
[12] |
Daniela Gurban, Petru Jebelean, Cǎlin Şerban. Non-potential and non-radial Dirichlet systems with mean curvature operator in Minkowski space. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 133-151. doi: 10.3934/dcds.2020006 |
[13] |
Alessio Pomponio. Oscillating solutions for prescribed mean curvature equations: euclidean and lorentz-minkowski cases. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3899-3911. doi: 10.3934/dcds.2018169 |
[14] |
Tatsuki Mori, Kousuke Kuto, Tohru Tsujikawa, Shoji Yotsutani. Exact multiplicity of stationary limiting problems of a cell polarization model. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5627-5655. doi: 10.3934/dcds.2016047 |
[15] |
Junping Shi, Ratnasingham Shivaji. Exact multiplicity of solutions for classes of semipositone problems with concave-convex nonlinearity. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 559-571. doi: 10.3934/dcds.2001.7.559 |
[16] |
Josef DiblÍk, Rigoberto Medina. Exact asymptotics of positive solutions to Dickman equation. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 101-121. doi: 10.3934/dcdsb.2018007 |
[17] |
Klaus-Jochen Engel, Marjeta Kramar FijavŽ. Exact and positive controllability of boundary control systems. Networks and Heterogeneous Media, 2017, 12 (2) : 319-337. doi: 10.3934/nhm.2017014 |
[18] |
Jifeng Chu, Delia Ionescu-Kruse, Yanjuan Yang. Exact solution and instability for geophysical waves at arbitrary latitude. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4399-4414. doi: 10.3934/dcds.2019178 |
[19] |
Fahe Miao, Michal Fečkan, Jinrong Wang. Exact solution and instability for geophysical edge waves. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2447-2461. doi: 10.3934/cpaa.2022067 |
[20] |
Mitsunori Nara, Masaharu Taniguchi. Convergence to V-shaped fronts in curvature flows for spatially non-decaying initial perturbations. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 137-156. doi: 10.3934/dcds.2006.16.137 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]