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July  2016, 15(4): 1497-1514. doi: 10.3934/cpaa.2016.15.1497

Classification of bifurcation curves of positive solutions for a nonpositone problem with a quartic polynomial

Kuan-Ju Huang 1, , Yi-Jung Lee 1, and  Tzung-Shin Yeh 1,

1. 

Department of Applied Mathematics, National University of Tainan, Tainan 700, Taiwan, Taiwan

Received  September 2015 Revised  February 2016 Published  April 2016

We study exact multiplicity and bifurcation curves of positive solutions of the boundary value problem \begin{eqnarray} &u"(x)+\lambda (-u^4+\sigma u^3-\tau u^2+\rho u)=0, -1 < x < 1, \\ &u(-1)=u(1)=0, \end{eqnarray} where $\sigma, \tau \in \mathbb{R}$, $\rho \geq 0,$ and $\lambda >0$ is a bifurcation parameter. Then on the $(\lambda, \|u\|_\infty)$-plane, we give a classification of four qualitatively different bifurcation curves: an S-shaped curve, a broken S-shaped curve, a $\subset$-shaped curve and a monotone increasing curve.
Keywords: $S$-shaped bifurcation curve, exact multiplicity, broken $S$-shaped bifurcation curve, nonpositone problem, time map..
Mathematics Subject Classification: Primary: 34B15, 34B1.
Citation: 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
References:
[1]

I. Addou and S.-H. Wang, Exact multiplicity results for a $p$-Laplacian problem with concave-convex-concave nonlinearities, Nonlinear Anal., 53 (2003), 111-137. doi: 10.1016/S0362-546X(02)00298-5.  Google Scholar

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[3]

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

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P. Korman, Y. Li and T. Ouyang, Exact multiplicity results for boundary value problems with nonlinearities generalising cubic, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 599-616. doi: 10.1017/S0308210500022927.  Google Scholar

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T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1-13.  Google Scholar

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J. Shi, Multi-parameter bifurcation and applications, in ICM2002 Satellite Conference on Nonlinear Functional Analysis: Topological Methods, Variational Methods and Their Applications (H. Brezis, K.C. Chang, S.J. Li and P. Rabinowitz Eds.), World Scientific, Singapore, (2003), 211-222.  Google Scholar

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J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations, 39 (1981), 269-290. doi: 10.1016/0022-0396(81)90077-2.  Google Scholar

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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.  Google Scholar

[9]

S.-H. Wang, A correction for a paper by J. Smoller and A. Wasserman, J. Differential Equations, 77 (1989), 199-202. doi: 10.1016/0022-0396(89)90162-9.  Google Scholar

[10]

S.-H. Wang and N. D. Kazarinoff, Bifurcation and stability of positive solutions of a two-point boundary value problem, J. Austral. Math. Soc. Ser. A, 52 (1992), 334-342.  Google Scholar

[11]

S.-H. Wang and N. D. Kazarinoff, Bifurcation of steady-state solutions of a scalar reaction-diffusion equation in one space variable, J. Austral. Math. Soc. Ser. A, 52 (1992), 343-355.  Google Scholar

[12]

S.-H. Wang and T.-S. Yeh, S-shaped and broken S-shaped bifurcation diagrams with hysteresis for a multiparameter spruce budworm population problem in one space dimension, J. Differential Equations, 255 (2013), 812-839. doi: 10.1016/j.jde.2013.05.004.  Google Scholar

show all references

References:
[1]

I. Addou and S.-H. Wang, Exact multiplicity results for a $p$-Laplacian problem with concave-convex-concave nonlinearities, Nonlinear Anal., 53 (2003), 111-137. doi: 10.1016/S0362-546X(02)00298-5.  Google Scholar

[2]

M.G. Crandall and P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.  Google Scholar

[3]

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

[4]

P. Korman, Y. Li and T. Ouyang, Exact multiplicity results for boundary value problems with nonlinearities generalising cubic, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 599-616. doi: 10.1017/S0308210500022927.  Google Scholar

[5]

T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1-13.  Google Scholar

[6]

J. Shi, Multi-parameter bifurcation and applications, in ICM2002 Satellite Conference on Nonlinear Functional Analysis: Topological Methods, Variational Methods and Their Applications (H. Brezis, K.C. Chang, S.J. Li and P. Rabinowitz Eds.), World Scientific, Singapore, (2003), 211-222.  Google Scholar

[7]

J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations, 39 (1981), 269-290. doi: 10.1016/0022-0396(81)90077-2.  Google Scholar

[8]

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.  Google Scholar

[9]

S.-H. Wang, A correction for a paper by J. Smoller and A. Wasserman, J. Differential Equations, 77 (1989), 199-202. doi: 10.1016/0022-0396(89)90162-9.  Google Scholar

[10]

S.-H. Wang and N. D. Kazarinoff, Bifurcation and stability of positive solutions of a two-point boundary value problem, J. Austral. Math. Soc. Ser. A, 52 (1992), 334-342.  Google Scholar

[11]

S.-H. Wang and N. D. Kazarinoff, Bifurcation of steady-state solutions of a scalar reaction-diffusion equation in one space variable, J. Austral. Math. Soc. Ser. A, 52 (1992), 343-355.  Google Scholar

[12]

S.-H. Wang and T.-S. Yeh, S-shaped and broken S-shaped bifurcation diagrams with hysteresis for a multiparameter spruce budworm population problem in one space dimension, J. Differential Equations, 255 (2013), 812-839. doi: 10.1016/j.jde.2013.05.004.  Google Scholar

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