Figure 1.
Global bifurcation of bifurcation curves $S_{\varepsilon }$ of (1) with varying $\varepsilon >0$
-
Previous Article
Explosive solutions of parabolic stochastic partial differential equations with lévy noise
- DCDS Home
- This Issue
-
Next Article
Long-time asymptotic solutions of convex hamilton-jacobi equations depending on unknown functions
October
2017, 37(10): 5127-5149.
doi: 10.3934/dcds.2017222
A global bifurcation theorem for a positone multiparameter problem and its application
| 1. | Fundamental General Education Center, National Chin-Yi University of Technology, Taichung 411, Taiwan |
| 2. | Center for General Education, National Formosa University, Yunlin 632, Taiwan |
| 3. | Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan |
We study the global bifurcation and exact multiplicity of positive solutions for the positone multiparameter problem
| $\left\{ \begin{align} &{{u}^{\prime \prime }}(x)+\lambda {{f}_{\varepsilon }}(u)=0\text{,}\ \ -1<x<1\text{,} \\ &u(-1)=u(1)=0\text{,} \\ \end{align} \right.$ |
where λ > 0 is a bifurcation parameter and
| $\varepsilon >0$ |
| $f_{\varepsilon }(u)$ |
| $\tilde{\varepsilon}>0$ |
| $(λ ,||u||_{∞ })$ |
| $0<\varepsilon <\tilde{\varepsilon}$ |
| $\varepsilon ≥ \tilde{\varepsilon}$ |
| $f_{\varepsilon}(u)=-\varepsilon u^{p}+bu^{2}+cu+d\ $ |
| $\varepsilon ,b,d>0,$ |
Keywords:
Global bifurcation,
multiparameter problem,
S-shaped bifurcation curve,
exact multiplicity,
positive solution.
Mathematics Subject Classification:
Primary: 34B18; Secondary: 74G35.
Citation:
Kuo-Chih Hung, Shao-Yuan Huang, Shin-Hwa Wang. A global bifurcation theorem for a positone multiparameter problem and its application. Discrete & Continuous Dynamical Systems - A,
2017, 37
(10)
: 5127-5149.
doi: 10.3934/dcds.2017222
![]()
References:
| [1] |
K. J. Brown, M. M. A. Ibrahim and R. Shivaji,
S-shaped bifurcation curves, Nonlinear Anal., 5 (1981), 475-486.
doi: 10.1016/0362-546X(81)90096-1. |
| [2] |
S.-Y. Huang and S.-H. Wang,
Proof of a conjecture for the one-dimensional perturbed Gelfand problem from combustion theory, Arch. Rational Mech. Anal., 222 (2016), 769-825.
doi: 10.1007/s00205-016-1011-1. |
| [3] |
K. -C. Hung, S. -Y. Huang and S. -H. Wang, Proof of Inequality (11). Available form http://mx.nthu.edu.tw/sy-huang/CubicNonlinearity.Google Scholar |
| [4] |
K.-C. Hung and S.-H. Wang,
A theorem on S-shaped bifurcation curve for a positone problem with convex-concave nonlinearity and its applications to the perturbed Gelfand problem, J. Differential Equations, 251 (2011), 223-237.
doi: 10.1016/j.jde.2011.03.017. |
| [5] |
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. |
| [6] |
P. Korman and Y. Li,
On the exactness of an S-shaped bifurcation curve, Proc. Amer. Math. Soc., 127 (1999), 1011-1020.
doi: 10.1090/S0002-9939-99-04928-X. |
| [7] |
P. Korman, Y. Li and T. Ouyang,
Computing the location and the direction of bifurcation, Math. Res. Lett., 12 (2005), 933-944.
doi: 10.4310/MRL.2005.v12.n6.a13. |
| [8] |
T. Laetsch,
The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1-13.
doi: 10.1512/iumj.1971.20.20001. |
| [9] |
C. Lu,
Global bifurcation of steady-state solutions on a biochemical system, SIAM J. Math. Anal., 21 (1990), 76-84.
doi: 10.1137/0521005. |
| [10] |
J. Shi,
Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531.
doi: 10.1006/jfan.1999.3483. |
| [11] |
J. Shi, Multi-parameter bifurcation and applications, in: H. Brezis, K. C. Chang, S. J. Li, P. Rabinowitz (Eds.), ICM 2002 Satellite Conference on Nonlinear Functional Analysis: Topological Methods, Variational Methods and Their Applications, World Scientific, Singapore, 2003,211-222.
doi: 10.1142/9789812704283_0023. |
| [12] |
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. |
| [13] |
S.-H. Wang,
On S-shaped bifurcation curves, Nonlinear Anal., 22 (1994), 1475-1485.
doi: 10.1016/0362-546X(94)90183-X. |
show all references
References:
| [1] |
K. J. Brown, M. M. A. Ibrahim and R. Shivaji,
S-shaped bifurcation curves, Nonlinear Anal., 5 (1981), 475-486.
doi: 10.1016/0362-546X(81)90096-1. |
| [2] |
S.-Y. Huang and S.-H. Wang,
Proof of a conjecture for the one-dimensional perturbed Gelfand problem from combustion theory, Arch. Rational Mech. Anal., 222 (2016), 769-825.
doi: 10.1007/s00205-016-1011-1. |
| [3] |
K. -C. Hung, S. -Y. Huang and S. -H. Wang, Proof of Inequality (11). Available form http://mx.nthu.edu.tw/sy-huang/CubicNonlinearity.Google Scholar |
| [4] |
K.-C. Hung and S.-H. Wang,
A theorem on S-shaped bifurcation curve for a positone problem with convex-concave nonlinearity and its applications to the perturbed Gelfand problem, J. Differential Equations, 251 (2011), 223-237.
doi: 10.1016/j.jde.2011.03.017. |
| [5] |
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. |
| [6] |
P. Korman and Y. Li,
On the exactness of an S-shaped bifurcation curve, Proc. Amer. Math. Soc., 127 (1999), 1011-1020.
doi: 10.1090/S0002-9939-99-04928-X. |
| [7] |
P. Korman, Y. Li and T. Ouyang,
Computing the location and the direction of bifurcation, Math. Res. Lett., 12 (2005), 933-944.
doi: 10.4310/MRL.2005.v12.n6.a13. |
| [8] |
T. Laetsch,
The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1-13.
doi: 10.1512/iumj.1971.20.20001. |
| [9] |
C. Lu,
Global bifurcation of steady-state solutions on a biochemical system, SIAM J. Math. Anal., 21 (1990), 76-84.
doi: 10.1137/0521005. |
| [10] |
J. Shi,
Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531.
doi: 10.1006/jfan.1999.3483. |
| [11] |
J. Shi, Multi-parameter bifurcation and applications, in: H. Brezis, K. C. Chang, S. J. Li, P. Rabinowitz (Eds.), ICM 2002 Satellite Conference on Nonlinear Functional Analysis: Topological Methods, Variational Methods and Their Applications, World Scientific, Singapore, 2003,211-222.
doi: 10.1142/9789812704283_0023. |
| [12] |
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. |
| [13] |
S.-H. Wang,
On S-shaped bifurcation curves, Nonlinear Anal., 22 (1994), 1475-1485.
doi: 10.1016/0362-546X(94)90183-X. |
Figure 2.
The bifurcation surface $\Gamma $ with the fold curve $C_{\Gamma }$ , and the projection of $C_{\Gamma }$ onto the $(\varepsilon ,\lambda )$ -parameter plane. $B_{\Gamma }=B_{1}\cup B_{2}\cup \left\{
(\tilde{\varepsilon},\tilde{\lambda})\right\} $ is the bifurcation set
Figure 3.
The projection of the bifurcation surface $C_{\Gamma }$ onto the $(
\varepsilon ,\lambda )$ -parameter plane. $B_{\Gamma
}=B_{1}\cup B_{2}\cup \left\{ (\tilde{\varepsilon},\tilde{
\lambda})\right\} $ is the bifurcation set.
Figure 4.
The evolution of time maps $T_{\varepsilon }(\alpha
)$ for $\alpha \in (0,\beta _{\varepsilon })$ with varying $\varepsilon >0.$
Figure 5.
The graph of $\phi _{\varepsilon }(u)$ .
Figure 6.
Three possible graphs of $\theta _{\varepsilon }(u)$ . (ⅰ) $\theta _{\varepsilon }(\gamma _{
\varepsilon })\leq 0$ . (ⅱ) $\theta _{\varepsilon }(\gamma _{\varepsilon })>0>\theta _{\varepsilon }(p_{2}(\varepsilon ))$ . (ⅲ) $\theta _{\varepsilon }(\gamma _{\varepsilon })>\theta _{\varepsilon }(p_{2}(\varepsilon ))\geq 0$ .
| [1] |
Tzung-shin Yeh. S-shaped and broken s-shaped bifurcation curves for a multiparameter diffusive logistic problem with holling type-Ⅲ functional response. Communications on Pure & Applied Analysis, 2017, 16 (2) : 645-670. doi: 10.3934/cpaa.2017032 |
| [2] |
Shao-Yuan Huang, Shin-Hwa Wang. On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839 |
| [3] |
Chih-Yuan Chen, Shin-Hwa Wang, Kuo-Chih Hung. S-shaped bifurcation curves for a combustion problem with general arrhenius reaction-rate laws. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2589-2608. doi: 10.3934/cpaa.2014.13.2589 |
| [4] |
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 & Applied Analysis, 2019, 18 (6) : 3267-3284. doi: 10.3934/cpaa.2019147 |
| [5] |
Sabri Bensid, Jesús Ildefonso Díaz. Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1757-1778. doi: 10.3934/dcdsb.2017105 |
| [6] |
Shao-Yuan Huang. Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1271-1294. doi: 10.3934/cpaa.2018061 |
| [7] |
Robert Skiba, Nils Waterstraat. The index bundle and multiparameter bifurcation for discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5603-5629. doi: 10.3934/dcds.2017243 |
| [8] |
Xue Dong He, Roy Kouwenberg, Xun Yu Zhou. Inverse S-shaped probability weighting and its impact on investment. Mathematical Control & Related Fields, 2018, 8 (3&4) : 679-706. doi: 10.3934/mcrf.2018029 |
| [9] |
Kuan-Ju Huang, Yi-Jung Lee, Tzung-Shin Yeh. Classification of bifurcation curves of positive solutions for a nonpositone problem with a quartic polynomial. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1497-1514. doi: 10.3934/cpaa.2016.15.1497 |
| [10] |
Alejandro Allendes, Alexander Quaas. Multiplicity results for extremal operators through bifurcation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 51-65. doi: 10.3934/dcds.2011.29.51 |
| [11] |
Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045 |
| [12] |
Yukio Kan-On. Bifurcation structures of positive stationary solutions for a Lotka-Volterra competition model with diffusion II: Global structure. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 135-148. doi: 10.3934/dcds.2006.14.135 |
| [13] |
Shao-Yuan Huang. Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3443-3462. doi: 10.3934/dcds.2019142 |
| [14] |
Sébastien Biebler. Lattès maps and the interior of the bifurcation locus. Journal of Modern Dynamics, 2019, 15: 95-130. doi: 10.3934/jmd.2019014 |
| [15] |
Inmaculada Antón, Julián López-Gómez. Global bifurcation diagrams of steady-states for a parabolic model related to a nuclear engineering problem. Conference Publications, 2013, 2013 (special) : 21-30. doi: 10.3934/proc.2013.2013.21 |
| [16] |
Olga Kharlampovich and Alexei Myasnikov. Tarski's problem about the elementary theory of free groups has a positive solution. Electronic Research Announcements, 1998, 4: 101-108. |
| [17] |
Lora Billings, Erik M. Bollt, David Morgan, Ira B. Schwartz. Stochastic global bifurcation in perturbed Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 123-132. doi: 10.3934/proc.2003.2003.123 |
| [18] |
Todd Young. A result in global bifurcation theory using the Conley index. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 387-396. doi: 10.3934/dcds.1996.2.387 |
| [19] |
M. R. S. Kulenović, Orlando Merino. Global bifurcation for discrete competitive systems in the plane. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 133-149. doi: 10.3934/dcdsb.2009.12.133 |
| [20] |
Pablo Aguirre, Bernd Krauskopf, Hinke M. Osinga. Global invariant manifolds near a Shilnikov homoclinic bifurcation. Journal of Computational Dynamics, 2014, 1 (1) : 1-38. doi: 10.3934/jcd.2014.1.1 |
2018 Impact Factor: 1.143
Tools
Metrics
Other articles
by authors
[Back to Top]