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Bifurcation diagram of the \(p\)-Laplacian problem with generalized Allen-Cahn type nonlinearities. (English) Zbl 1416.34018

Summary: We study the exact multiplicity of (classical) positive solutions and the bifurcation diagram of the \(p\)-Laplacian problem with generalized Allen-Cahn type nonlinearities. We give a complete classification of totally six qualitatively different bifurcation diagrams.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B09 Boundary eigenvalue problems for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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[1] G. A. Afrouzi and S. H. Rasouli, Instability of positive solutions for population models involving the p-Laplacian operator, Glob. J. Pure Appl. Math., 2(2006), 51-54. · Zbl 1126.35019
[2] G. A. Afrouzi and S. H. Rasouli, Population models involving the p-Laplacian with inde…nite weight and constant yield harvesting, Chaos Solitons Fractals, 31(2007), 404-408. · Zbl 1138.35010
[3] A. Cañada, P. Drábek and J. L. Gámez, Existence of positive solutions for some problems with nonlinear di¤usion, Trans. Amer. Math. Soc., 349(1997), 4231-4249. · Zbl 0884.35039
[4] J. G. Cheng, Uniqueness results for the one-dimensional p-Laplacian, J. Math. Anal. Appl., 311(2005), 381-388. · Zbl 1082.34021
[5] J. I. Díaz, Nonlinear Partial Di¤erential Equations and Free Boundaries. Vol. I. Elliptic Equations. Research Notes in Mathematics, 106, Pitman, Boston, MA, 1985. · Zbl 0595.35100
[6] Y. Du and Z. Guo, Liouville type results and eventual ‡atness of positive solutions for p-Laplacian equations, Adv. Di¤erential Equations, 7(2002), 1479-1512. · Zbl 1075.35522
[7] Y. X. Huang and J. W.-H. So, On bifurcation and existence of positive solutions for a certain p-Laplacian system, Rocky Mountain J. Math., 25(1995), 285-297. · Zbl 0831.35017
[8] K.-C. Hung and S.-H. Wang, Bifurcation diagrams of a p-Laplacian Dirichlet problem with Allee e¤ect and an application to a di¤usive logistic equation with predation, J. Math. Anal. Appl., 37(2011), 294-309. · Zbl 1213.34039
[9] T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20(1970), 1-13. · Zbl 0215.14602
[10] A. Lakmeche and A. Hammoudi, Multiple positive solutions of the one-dimensional p-Laplacian, J. Math. Anal. Appl., 317(2006), 43-49. · Zbl 1097.34519
[11] S.-Y. Lee, S.-H. Wang and C.-P. Ye, Explicit necessary and su¢ cient conditions for the existence of a dead core solution of a p-Laplacian steady-state reactiondi¤usion problem, Discrete Contin. Dyn. Syst., Suppl., (2005), 587-596. · Zbl 1141.35381
[12] S. Lian, H. Yuan, C. Cao, W. Gao and X. Xu, On the Cauchy problem for the evolution p-Laplacian equations with gradient term and source, J. Di¤erential Equations, 235(2007), 544-585. · Zbl 1128.35059
[13] P. M. McCabe, J. A. Leach and D. J. Needham, The evolution of travelling waves in fraction order autocatalysis with decay, I. Permanent from travelling waves, SIAM J. Appl. Math., 59(1998), 870-899. · Zbl 0938.35075
[14] J. D. Murray, A simple method for obtaining approximate solutions for a class of di¤usion-kinetics enzyme problems. II. Further examples and nonsymmetric problems, Math. Biosci., 3(1968), 115-133.
[15] S. Oruganti, J. Shi and R. Shivaji, Logistic equation with the p-Laplacian and constant yield harvesting, Abstr. Appl. Anal., 2004(2004), 723-727. · Zbl 1133.35370
[16] M. Ramaswamy and R. Shivaji, Multiple positive solutions for classes of pLaplacian equations, Di¤erential Integral Equations, 17(2004), 1255-1261. · Zbl 1150.35419
[17] J. Shi and R. Shivaji, Persistence in reaction di¤usion models with weak Allee e¤ect, J. Math. Biol., 52(2006), 807-829. · Zbl 1110.92055
[18] M.-H. Wang and M. Kot, Speeds of invasion in a model with strong or weak Allee e¤ects, Math. Biosci., 171 (2001), 83-97. · Zbl 0978.92033
[19] R. L. Wheeden, A. Zygmund, Measure and Integral: An Introduction to Real Analysis, Marcel Dekker, New York, 1977. · Zbl 0362.26004
[20] J. Xin, Front propagation in heterogeneous media, SIAM Review, 42(2000), 161- 230. · Zbl 0951.35060
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