Figure 1.
Global bifurcation of bifurcation curves $S_{\varepsilon }$ of (1) with varying $\varepsilon >0$
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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 and Continuous Dynamical Systems,
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. |
| [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. |
| [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) $\theta _{\varepsilon }(\gamma _{\varepsilon })>\theta _{\varepsilon }(p_{2}(\varepsilon ))\geq 0$ .
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