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

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.
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.

[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.

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.

[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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