Persistence and bifurcation of degenerate solutions. (English) Zbl 0949.47050

The author studies an equation \(F(\varepsilon,\lambda,u)=0\), where \(F:{\mathbb R}\times {\mathbb R}\times X\rightarrow Y\) is a smooth nonlinear map and \(X, Y\) are Banach spaces. It is obtained a number of bifurcation results for degenerate solutions. Several applications to elliptic boundary value problems of the form \(Lu+\lambda f(\varepsilon, u)=0\) are presented.


47J15 Abstract bifurcation theory involving nonlinear operators
35B32 Bifurcations in context of PDEs
35J60 Nonlinear elliptic equations
Full Text: DOI Link


[1] Brown, K.J.; Ibrahim, M.M.A.; Shivaji, R., S-shaped bifurcation curves, Nonlinear anal., 5, 475-486, (1981) · Zbl 0458.35036
[2] Chow, Shui Nee; Hale, J.K., Methods of bifurcation theory, (1982), Springer-Verlag New York/Berlin · Zbl 0487.47039
[3] Church, P.T.; Timourian, J.G., Global structure for nonlinear operators in differential and integral equations. I. folds; II. cusps: topological nonlinear analysis, Progr. nonlinear differential equations appl., (1997), Birkhaüser Boston · Zbl 0879.58008
[4] Crandall, M.G.; Rabinowitz, P.H., Bifurcation from simple eigenvalues, J. funct. anal., 8, 321-340, (1971) · Zbl 0219.46015
[5] Crandall, M.G.; Rabinowitz, P.H., Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. rational mech. anal., 52, 161-180, (1973) · Zbl 0275.47044
[6] Dancer, E.N., On the structure of solutions of an equation in catalysis theory when a parameter is large, J. differential equations, 37, 404-437, (1980) · Zbl 0417.34042
[7] Yihong, Du, Exact multiplicity and S-shaped bifurcation curve for some semilinear elliptic problems from combustion theory, preprint, 1998. · Zbl 0974.35036
[8] Du, Yihong; Lou, Yuan, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. differential equations, 144, 390-440, (1998) · Zbl 0970.35030
[9] Yihong, Du, and, Yuan, Lou, Proof of a conjecture for the perturbed Gelfand equation from combustion theory, preprint, 1997. · Zbl 1098.35529
[10] Hastings, S.P.; McLeod, J.B., The number of solutions to an equation from catalysis, Proc. roy. soc. Edinburgh sect. A, 101, 15-30, (1985) · Zbl 0582.34022
[11] Korman, P.; Li, Yi, On the exactness of an S-shaped bifurcation curve, Proc. amer. math. soc., 127, 1011-1020, (1999) · Zbl 0917.34013
[12] P. Korman, and, Junping, Shi, Instability and exact multiplicity of solutions of semilinear equations, submitted for publication. · Zbl 0970.34015
[13] Lions, P.-L., On the existence of positive solutions of semilinear elliptic equations, SIAM rev., 24, 441-467, (1982) · Zbl 0511.35033
[14] Ouyang, Tiancheng; Shi, Junping, Exact multiplicity of positive solutions for a class of semilinear problem, J. differential equations, 146, 121-156, (1998) · Zbl 0918.35049
[15] Ouyang, Tiancheng; Shi, Junping, Exact multiplicity of positive solutions for a class of semilinear problem, II, J. differential equations, 158, 94-151, (1999) · Zbl 0947.35067
[16] Junping, Shi, Ph.D. Dissertation, Brigham Young University, 1998.
[17] Smoller, J.; Wasserman, A., Global bifurcation of steady-state solutions, J. differential equations, 39, 269-290, (1981) · Zbl 0425.34028
[18] Wang, Shin Hwa, On S-shaped bifurcation curves, Nonlinear anal., 22, 1475-1485, (1994) · Zbl 0803.34013
[19] Wang, Shin Hwa; Lee, Fu Ping, Bifurcation of an equation from catalysis theory, Nonlinear anal., 23, 1167-1187, (1994) · Zbl 0815.34017
[20] Yosida, Kosaku, Functional analysis, Grundlehren der mathematischen wissenschaften, 123, (1978), Springer-Verlag Berlin/New York · Zbl 0365.46001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.