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Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity. (English) Zbl 1245.34048

The authors investigate the global bifurcation diagram and the exact multiplicity of positive solutions for the problem

u '' (x)+λf ε (u)=0,-1<x<1,u(-1)=u(1)=0,f ε (u)=-εu 3 +σu 2 -κu+ρ,

where λ,ε>0 are two bifurcation parameters, and σ,ρ>0, 0<κσρ are constants. They prove that there exists ε ˜>0 such that, on the (λ,u )-plane, the bifurcation curve is S-shaped for 0<ε<ε ˜ and is monotone increasing for εε ˜.

MSC:
34C23Bifurcation (ODE)
34B18Positive solutions of nonlinear boundary value problems for ODE
34B08Parameter dependent boundary value problems for ODE
References:
[1]K.-C. Hung, 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., in press.
[2]Hung, K. -C.; Wang, S. -H.: 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, 223-237 (2011) · Zbl 1229.34037 · doi:10.1016/j.jde.2011.03.017
[3]Korman, P.; Li, Y.; Ouyang, T.: Exact multiplicity results for boundary value problems with nonlinearities generalizing cubic, Proc. roy. Soc. Edinburgh sect. A 126, 599-616 (1996) · Zbl 0855.34022 · doi:10.1017/S0308210500022927
[4]Korman, P.; Li, Y.; Ouyang, T.: Computing the location and the direction of bifurcation, Int. math. Res. lett. 12, 933-944 (2005) · Zbl 1104.34010
[5]Laetsch, T.: The number of solutions of a nonlinear two point boundary value problem, Indiana univ. Math. J. 20, 1-13 (1970) · Zbl 0215.14602 · doi:10.1512/iumj.1970.20.20001
[6]Shi, J.: Persistence and bifurcation of degenerate solutions, J. funct. Anal. 169, 494-531 (1999) · Zbl 0949.47050 · doi:10.1006/jfan.1999.3483
[7]Shi, J.: Multi-parameter bifurcation and applications, ICM2002 satellite conference on nonlinear functional analysis: topological methods, variational methods and their applications, 211-222 (2003)
[8]Smoller, J.; Wasserman, A.: Global bifurcation of steady-state solutions, J. differential equations 39, 269-290 (1981) · Zbl 0425.34028 · doi:10.1016/0022-0396(81)90077-2
[9]Wang, S. -H.: A correction for a paper by J. Smoller and A. Wasserman, J. differential equations 77, 199-202 (1989) · Zbl 0688.34017 · doi:10.1016/0022-0396(89)90162-9