Abstract
We consider operator equations of the form
, (1) where A 0, B 0 are linear operators between real Banach spaces and N(λ, u) is a nonlinear operator with the property that N(λ, 0)=0 for all real λ. Assuming that λ 0, a specific value of λ, is an isolated eigenvalue of A 0 − λB 0 of multiplicity m, we study the phenomenon of bifurcation for equation (1), where it is merely assumed that N(λ, u) is Lipschitz continuous in u near u=0 with a small Lipschitz constant. It is shown that when (1) has a variational structure, for each suitable normalization of u, two non-zero solutions (λ, u) occur near (λ0,0) (m pairs occur if N is odd in u). Further results concern the existence of branches of solutions when m is odd and the asymptotic behavior of solutions in terms of the size of the Lipschitz constant. The motivation for the study and the main application of the results concerns buckling of a von Kármán plate resting on a foundation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Amann, H., Lusternik-Schnirelman theory and non-linear eigenvalue problems, Math. Ann. 199 (1972), 55–72.
Berger, M.S., Bifurcation theory and the type numbers of Marston Morse, Proc. Nat. Acad. Sc. 69 (1972), 1737–1738.
Böhme, R., Die Lösung der Verzweigungsgleichungen für nichtlineare Eigenwertprobleme, Math. Z. 127 (1972), 105–126.
Clark, D. C., Eigenvalue bifurcation for odd gradient operators, Rocky Mountain J. Math. 5 (1975), 317–336.
Crandall, M.G., & Rabinowitz, P.H., Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321–340.
Dancer, E. N., Global solution branches for positive mappings, Arch. Rational Mech. Analysis 52 (1973), 181–192.
Ize, G.A., Bifurcation theory for Fredholm operators, Thesis, New York Univ., 1974.
Kirchgässner, K., Multiple eigenvalue bifurcation for holomorphic mappings, Contributions to Nonlinear Functional Analysis, E.H. Zarantonello (edit.), Academic Press, New York, 1971.
Krasnoselski, M.A., Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan, New York, 1964.
Krasnoselski, M.A., Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.
MacBain, J., Global bifurcation theorems for non-compact operators, Bull. Amer. Math. Soc. 80 (1974), 1005–1009.
McLeod, J.B., & Sattinger, D.H., Loss of stability and bifurcation at a double eigenvalue, J. Funct. Anal. 14 (1973), 62–84.
Marino, A., & Prodi, G., La teoria di Morse per gli spazi di Hilbert, Rend. Sem. Mat. Univ. Padova 41 (1968), 43–68.
Marino, A., La biforcazione nel caso variazionale, Conf. Sem. Mat. Univ. Bari (1973), 132.
Nussbaum, R., A global bifurcation theorem with applications to functional differential equations, preprint.
Rabinowitz, P.H., Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487–513.
Rabinowitz, P. H., Variational methods for nonlinear eigenvalue problems, Eigenvalues of Nonlinear Problems, C.I.M.E., Varenna, Italy, 1974.
Rabinowitz, P.H., A bifurcation theorem for potential operators, in progress.
Reeken, M., Stability of critical points under small perturbations, Part I: Topological theory, Manuscripta Math. 7 (1972), 387–411; Part II: Analytic theory, ibid. 8 (1973), 69–92.
Sather, D., Branching of solutions of nonlinear equations, Rocky Mountain J. Math. 3 (1973), 203–250.
Stuart, C.A., Some bifurcation theory for k-set contractions, Proc. London Math. Soc. (3) 27 (1973), 531–550.
Turner, R.E.L., Transversality in nonlinear eigenvalue problems, Contributions to Nonlinear Functional Analysis, E.H. Zarantonello (edit.), Academic Press, New York, 1971.
Turner, R.E.L., Transversality and cone maps, Arch. Rational Mech. Analysis 58 (1975), 151–179.
Vainberg, M.M., & Trenogin, V.A., The methods of Lyapunov and Schmidt in the theory of nonlinear equations and their further development, Russ. Math. Surveys 17 (1962), 1–60.
Schwartz, J.T., Nonlinear Functional Analysis, Lecture Notes, New York University, New York, 1965.
Schaefer, H.H., Topological Vector Spaces, Macmillan, New York, 1966.
Dunford, N., & Schwartz, J.T., Linear Operators, Vol. I: General theory, Interscience, New York, 1958.
Vainberg, M. M., Variational Methods for the Study of Nonlinear Operators, Holden-Day, San Francisco, 1964.
Birnbaum, S., Spectral theory for a pair of closed linear transformations acting between two Banach spaces, Report No. SR 0520-99, Martin Co., 1965.
Turner, R.E.L., Eigenfunction expansions in Banach spaces, Quart. J. Math. (2) 19 (1968), 193–211.
Palais, R.S., Critical point theory and the minimax principle, Proc. Symp. Pure Math. 15, A.M.S., Providence, 1970, 185–212.
Coffman, C.V., A minimum-maximum principle for a class of nonlinear integral equations, J. Analyse Math. 22 (1969), 391–419.
Turner, R.E.L., Nonlinear eigenvalue problems with applications to elliptic equations, Arch. Rational Mech. Analysis 42 (1971), 184–193.
Berger, M.S., & Fife, P.C., On von Kármán's equations and the buckling of a thin elastic plate, Bull Amer. Math. Soc. 72 (1966), 1006–1011.
Berger, M.S., On von Kármán's equations and the buckling of a thin elastic plate, I: The clamped plate, Comm. Pure Appl. Math. 20 (1967), 687–719.
Berger, M.S., & Fife, P.C., Von Kármán's equations and the buckling of a thin elastic plate, II: Plate with general edge conditions, Comm. Pure Appl. Math. 21 (1968), 227–241.
Friedrichs, K.O., & Stoker, J.J., The nonlinear boundary value problem of the buckled plate, Amer. J. Math. 63 (1941), 839–888.
Keller, H.B., Keller, J.B., & Reiss, E.L., Buckled states of circular plates, Quart. Appl. Math. 20 (1962), 55–65.
Morosov, N.F., On the nonlinear theory of thin plates, Dokl. Akad. Nauk SSSR 114 (1957), 968–971.
Rabinowitz, P.H., Variational methods for nonlinear elliptic eigenvalue problems, Indiana Univ. Math. J. 23 (1974), 729–754.
Steinberg, H., Buckling of a circular plate resting on an elastic foundation, Report No. TSR 1063, Mathematics Research Center, Madison, 1972.
Steinberg, H., Buckling of a circular plate embedded in a nonlinear elastic foundation, Report No. TSR 1064, Mathematics Research Center, Madison, 1971.
Stoker, J.J., Nonlinear Elasticity, Thomas Nelson and Sons, London, 1968.
Wolkowisky, J.H., Existence of buckled states of circular plates, Comm. Pure Appl. Math. 20 (1967), 549–560.
Wolkowisky, J.H., Buckling of the circular plate embedded in elastic springs. An application to geophysics, Comm. Pure Appl. Math. 22 (1969), 639–667.
Ladyzhenskaya, O.A., & Ural'tseva, N.N., Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.
Lions, J.L., & Magenes, E., Problèmes aux Limites non Homogènes et Applications, Vol. I, Dunod, Paris, 1968.
Sokolnikoff, I.S., Mathematical Theory of Elasticity (Second edition), McGraw-Hill, New York, 1956.
Agmon, S., Douglis, A., & Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math. 12 (1959), 623–727.
Friedrichs, K.O., Die Randwertund Eigenwertprobleme aus der Theorie der elastischen Platten. (Anwendung der direkten Methoden der Variationrechnung.), Math. Ann. 98 (1928), 205–247.
Author information
Authors and Affiliations
Additional information
Note: This research was sponsored by the United Kingdom Science Research Council and the United States Army under Contract No. DAAG29-75-C-0024, and the National Science Foundation under Grant MPS 75-06556.
Rights and permissions
About this article
Cite this article
McLeod, J.B., Turner, R.E.L. Bifurcation for non-differentiable operators with an application to elasticity. Arch. Rational Mech. Anal. 63, 1–45 (1976). https://doi.org/10.1007/BF00280140
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00280140