Abstract
We consider a class of nonlinear elliptic eigenvalue problems, in an arbitrary bounded regular domain in ℝn, with multiple bending points (infinite in some cases). We associate with them a family of perturbed problems; the study of the corresponding singular perturbation enables us to extend the limiting elliptic problem into a free boundary problem. The latter also admits an infinite number of free boundary solutions in some cases of hyperspherical geometries.
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References
H. AMMANN, On the existence of positive solutions of nonlinear elliptic boundary value problems. Ind. Univ. Math. J., 21, p. 125–146 (1971).
H. AMANN, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Review 18, p. 620–709 (1976).
R. ARIS, The mathematical theory of diffusion and reaction, clarendon Press, Oxford (1975).
H.T. BANKS, Modeling and control in the biomedical sciences, Lect. Notes in Biomathematics 6, Springer-Verlag (1975).
A. BENSOUSSAN and J.L. LIONS, Application des inequations variationelles en contrôle stochastique, Dunod (volume 2, to appear).
H. BERESTYCKI and P.L. LIONS, This volume.
C.M. BRAUNER, Perturbations singulières dans des systèmes non linéaires et applications à la biochimie, Thèse, Université Paris-Sud (1975).
C.M. BRAUNER and B. NICOLAENKO, Perturbation singulière, solutions multiples et hystérésis dans un problème de biochimie, C.R. Acad. Sc. Paris, Série A, 283, p. 775–778 (1976).
C.M. BRAUNER and B. NICOLAENKO, Singular perturbation, multiple solutions and hysteresis in a nonlinear problem, Lect. Notes in Math. 594, Springer-Verlag, p. 50–76 (1977).
C.M. BRAUNER and B. NICOLAENKO, Sur une classe de problemes elliptiques non linéaires, C.R. Acad. Sc. Paris, Série A, 286, p. 1007–1010 (1978).
C.M. BRAUNER and B. NICOLAENKO, Sur des problèmes aux valeurs propres non linéaires qui se prolongent en problèmes à frontière libre, C.R. Acad. Sci. Paris, Série A, 287, p. 1105–1108 (1978), and 288, p. 125–127 (1979).
C.M. BRAUNER and B. NICOLAENKO, To appear.
C.M. BRAUNER, B. GAY, and B. NICOLAENKO, Colloque d'Analyse Numérique, Giens (1978).
H. BREZIS (Private communication).
A.J. CALLEGARI, H.B. KELLER and E.L. REISS, Membrane buckling: a study of solution multiplicity, C.P.A.M., 24, p. 499–527 (1971).
M.G. CRANDALL and P.H. RABINOWITZ, Bifurcation from simple eigenvalues, J. Funct. Anal., 8, p. 321–340 (1971).
M.G. CRANDALL and P.H. RABINOWITZ, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rat. Mech. Anal., 52, p. 161–180 (1973).
M.G. CRANDALL and P.H. RABINOWITZ, Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rat. Mech. Anal., 58, p. 207–218 (1975).
J.P. KEENER and H.B. KELLER, Positive solutions of convex nonlinear eigenvalue problems, J. Diff. Equ., 16, p. 103–125 (1974).
J. LERAY, Thèse, Paris (1934), J. Math. Pures et appl., 12, p. 1–80 (1933).
J. LERAY and J. SCHAUDER, Tapologie et équations fonctionnelles, Ann. Sci. Ecole Norm. Sup., 3, vol. 51, p. 45–78 (1934).
J.L. LIONS, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod (1969).
J.L. LIONS, Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Lect. Notes in Math. 323, Springer-Verlag (1973).
J.L. LIONS (Private communication).
F. MIGNOT and J.P. PUEL, Sur une classe de problèmes non linéaires avec non linéarité positive croissante, convexe, Colloque d'Analyse non linéaire, Rome, Mai 1978.
J.P. PUEL, Existence, comportement à l'infini et stabilité dans certains problèmes quasilinéaires elliptiques et parabolique d'ordre 2, Ann. Sc. Norm. Pisa, 3, p. 85–119 (1976).
P.H. RABINOWITZ, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7, p. 487–513 (1971).
D.H. SATTINGER, Topics in stability and bifurcation theory, Lect. Notes in Math. 309, Springer-Verlag (1973).
K. STEWARTSON, Further solutions of the Falkner-Skan equation, Proc. Camb. Phil. Soc, 50, p. 454–465 (1954).
A.N. IL'IN, A.S. KALASHNIKOV, and O.A. OLEINIK, Linear equations of the second order of parabolic type, Russian Math. Surveys, 17, No 3, p. 1–143 (1962).
J. FAVARD, Cours de Géométrie, Gauthier-Villars, Paris (1957).
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Brauner, C.M., Nicolaenko, B. (1980). On nonlinear eigenvalue problems which extend into free boundaries problems. In: Bardos, C., Lasry, J.M., Schatzman, M. (eds) Bifurcation and Nonlinear Eigenvalue Problems. Lecture Notes in Mathematics, vol 782. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0090428
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DOI: https://doi.org/10.1007/BFb0090428
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