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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

$\left\{\begin{array}{c}{u}^{\text{'}\text{'}}\left(x\right)+\lambda {f}_{\epsilon }\left(u\right)=0,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}-1

where $\lambda ,\phantom{\rule{0.166667em}{0ex}}\epsilon >0$ are two bifurcation parameters, and $\sigma ,\phantom{\rule{0.166667em}{0ex}}\rho >0$, $0<\kappa \le \sqrt{\sigma \rho }$ are constants. They prove that there exists $\stackrel{˜}{\epsilon }>0$ such that, on the $\left(\lambda ,\parallel u{\parallel }_{\infty }\right)$-plane, the bifurcation curve is $S$-shaped for $0<\epsilon <\stackrel{˜}{\epsilon }$ and is monotone increasing for $\epsilon \ge \stackrel{˜}{\epsilon }$.

##### MSC:
 34C23 Bifurcation (ODE) 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B08 Parameter 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