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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

* Corresponding author: Shao-Yuan Huang

Received  September 2016 Revised  May 2017 Published  June 2017

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
$\varepsilon >0$
is an evolution parameter. Under some suitable hypotheses on
$f_{\varepsilon }(u)$
, we prove that there exists
$\tilde{\varepsilon}>0$
such that, on the
$(λ ,||u||_{∞ })$
-plane, the bifurcation curve is S-shaped for
$0<\varepsilon <\tilde{\varepsilon}$
and is monotone increasing for
$\varepsilon ≥ \tilde{\varepsilon}$
. We give an application for this problem with a class of polynomial nonlinearities
$f_{\varepsilon}(u)=-\varepsilon u^{p}+bu^{2}+cu+d\ $
of degree p≥ 3 and coefficients
$\varepsilon ,b,d>0,$
c ≥ 0. Our results generalize those in Hung and Wang (Trans. Amer. Math. Soc. 365 (2013) 1933-1956.)
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. BrownM. M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves, Nonlinear Anal., 5 (1981), 475-486. doi: 10.1016/0362-546X(81)90096-1. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

P. KormanY. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531. doi: 10.1006/jfan.1999.3483. Google Scholar

[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. Google Scholar

[12]

C.-C. TzengK.-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. Google Scholar

[13]

S.-H. Wang, On S-shaped bifurcation curves, Nonlinear Anal., 22 (1994), 1475-1485. doi: 10.1016/0362-546X(94)90183-X. Google Scholar

show all references

References:
[1]

K. J. BrownM. M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves, Nonlinear Anal., 5 (1981), 475-486. doi: 10.1016/0362-546X(81)90096-1. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

P. KormanY. 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531. doi: 10.1006/jfan.1999.3483. Google Scholar

[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. Google Scholar

[12]

C.-C. TzengK.-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. Google Scholar

[13]

S.-H. Wang, On S-shaped bifurcation curves, Nonlinear Anal., 22 (1994), 1475-1485. doi: 10.1016/0362-546X(94)90183-X. Google Scholar

Figure 1.  Global bifurcation of bifurcation curves $S_{\varepsilon }$ of (1) with varying $\varepsilon >0$
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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