Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-19T08:48:48.117Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact multiplicity results for boundary value problems with nonlinearities generalising cubic

Published online by Cambridge University Press:  14 November 2011

Philip Korman ,
Yi Li  and
Tiancheng Ouyang
Philip Korman
Affiliation:
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, U.S.A.
Yi Li
Affiliation:
Department of Mathematics, University of Rochester, Rochester, NY 14627, U.S.A.
Tiancheng Ouyang
Affiliation:
Department of Mathematics, Brigham Young University, Provo, Utah 84602, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

Using techniques of bifurcation theory we present two exact multiplicity results for boundary value problems of the type

The first result concerns the case when the nonlinearity is independent of x and behaves like a cubic in u. The second one deals with a class of nonlinearities with explicit x dependence.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 3 , 1996 , pp. 599 - 616
Copyright
Copyright © Royal Society of Edinburgh 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Crandall, M. G. and Rabinowitz, P. H.. Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Rational Meek Anal. 52 (1973), 161–80.CrossRefGoogle Scholar
2Gardner, R. and Peletier, L. A.. The set of positive solutions of semilinear equations in large balls. Proc. Roy. Soc. Edinburgh Sect. A 104 (1986), 53–72.CrossRefGoogle Scholar
3Gidas, B., Ni, W.-M. and Nirenberg, L.. Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68 (1979) 209–43.CrossRefGoogle Scholar
4Korman, P. and Ouyang, T.. Exact multiplicity results for two classes of boundary-value problems. Differential Integral Equations 6 (1993), 1507–17.CrossRefGoogle Scholar
5Korman, P. and Ouyang, T.. Multiplicity results for two classes of boundary-value problems. SIAM J. Math. Anal. 26 (1995), 180–9.CrossRefGoogle Scholar
6Korman, P. and Ouyang, T.. Solution curves for two classes of boundary-value problems. Nonlin. Anal. TMA (to appear).Google Scholar
7Matano, H.. Asymptotic behavior and stability of solutions of semilinear diffusion equations. Publ. RIMS, Kyoto Univ. 15 (1979) 401–54.CrossRefGoogle Scholar
8Rabinowitz, P. H.. Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics 65 (Providence, R.I.: American Mathematical Society, 1986).CrossRefGoogle Scholar
9Schaaf, R.. Global Solution Branches of Two Point Boundary Value Problems, Lecture Notes in Mathematics 1458 (Berlin: Springer, 1990).CrossRefGoogle Scholar
10Smoller, J. and Wasserman, A.. Global bifurcation of steady-state solutions. J. Differential Equations 39(1981), 269–90.CrossRefGoogle Scholar
11Wang, S.-H.. A correction for a paper by J. Smoller and A. Wasserman. J. Differential Equations 11 (1989), 199–202.CrossRefGoogle Scholar
12Wang, S.-H. and Kazarinoff, N. D.. Bifurcation and stability of positive solutions of two-point boundary value problem. J. Austral. Math. Soc. Ser. A 52 (1992), 334–42.CrossRefGoogle Scholar
13Wang, S.-H. and Kazarinoff, N. D.. Bifurcation and steady-state solutions of a scalar reaction–diffusion equation in one space variable. J. Austral. Math. Soc. Ser. A 52 (1992), 343–55.CrossRefGoogle Scholar