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

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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

Abstract
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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 and 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.
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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.
[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.
[5]	T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1-13.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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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.
[2]	M.G. Crandall and P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.
[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.
[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.
[5]	T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1-13.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

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