Global analysis of two-parameter elliptic eigenvalue problems

Peitgen, H.-O.; Schmitt, K.

doi:10.1090/S0002-9947-1984-0735409-5

Global analysis of two-parameter elliptic eigenvalue problems
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by H.-O. Peitgen and K. Schmitt PDF

Trans. Amer. Math. Soc. 283 (1984), 57-95 Request permission

Abstract:

We consider the nonlinear boundary value problem $({\ast })Lu + \lambda f(u) = 0$, $x \in \Omega , u = \sigma \phi , x \in \partial \Omega$, where $L$ is a second order elliptic operator and $\lambda$ and $\sigma$ are parameters. We analyze global properties of solution continua of these problems as $\lambda$ and $\sigma$ vary. This is done by investigating particular sections, and special interest is devoted to questions of how solutions of the $\sigma = 0$ problem are embedded in the two-parameter family of solutions of $({\ast })$. As a natural biproduct of these results we obtain (a) a new abstract method to analyze bifurcation from infinity, (b) an unfolding of the bifurcations from zero and from infinity, and (c) a new framework for the numerical computations, via numerical continuation techniques, of solutions by computing particular one-dimensional sections.

References

J. C. Alexander, A primer on connectivity, Fixed point theory (Sherbrooke, Que., 1980) Lecture Notes in Math., vol. 886, Springer, Berlin-New York, 1981, pp. 455–483. MR 643023
A. Ambrosetti and P. Hess, Positive solutions of asymptotically linear elliptic eigenvalue problems, J. Math. Anal. Appl. 73 (1980), no. 2, 411–422. MR 563992, DOI 10.1016/0022-247X(80)90287-5
K. J. Brown and H. Budin, On the existence of positive solutions for a class of semilinear elliptic boundary value problems, SIAM J. Math. Anal. 10 (1979), no. 5, 875–883. MR 541087, DOI 10.1137/0510082
David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443
Peter Hess, On multiple positive solutions of nonlinear elliptic eigenvalue problems, Comm. Partial Differential Equations 6 (1981), no. 8, 951–961. MR 619753, DOI 10.1080/03605308108820200

Topology

Jean Leray and Jules Schauder, Topologie et équations fonctionnelles, Ann. Sci. École Norm. Sup. (3) 51 (1934), 45–78 (French). MR 1509338
H.-O. Peitgen, Topologische Perturbationen beim globalen numerischen Studium nichtlinearer Eigenwert- und Verzweigungsprobleme, Jahresber. Deutsch. Math.-Verein. 84 (1982), no. 3, 107–162 (German). MR 666195
Heinz-Otto Peitgen and Klaus Schmitt, Perturbations topologiques globales des problèmes non linéaires aux valeurs propres, C. R. Acad. Sci. Paris Sér. A-B 291 (1980), no. 4, A271–A274 (French, with English summary). MR 591748
H.-O. Peitgen and K. Schmitt, Positive and spurious solutions of nonlinear eigenvalue problems, Numerical solution of nonlinear equations (Bremen, 1980) Lecture Notes in Math., vol. 878, Springer, Berlin-New York, 1981, pp. 275–324. MR 644335
H.-O. Peitgen and K. Schmitt, Global topological perturbations of nonlinear elliptic eigenvalue problems, Math. Methods Appl. Sci. 5 (1983), no. 3, 376–388. MR 716662, DOI 10.1002/mma.1670050125
H.-O. Peitgen, D. Saupe, and K. Schmitt, Nonlinear elliptic boundary value problems versus their finite difference approximations: numerically irrelevant solutions, J. Reine Angew. Math. 322 (1981), 74–117. MR 603027
Paul H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), 161–202. MR 320850, DOI 10.1216/RMJ-1973-3-2-161
Paul H. Rabinowitz, On bifurcation from infinity, J. Differential Equations 14 (1973), 462–475. MR 328705, DOI 10.1016/0022-0396(73)90061-2
Klaus Schmitt, Boundary value problems for quasilinear second-order elliptic equations, Nonlinear Anal. 2 (1978), no. 3, 263–309. MR 512661, DOI 10.1016/0362-546X(78)90019-6
Klaus Schmitt, A study of eigenvalue and bifurcation problems for nonlinear elliptic partial differential equations via topological continuation methods, Université Catholique de Louvain, Institut de Mathématique Pure et Appliquée, Louvain-La-Neuve, 1982. MR 684664
K. Schmitt and H. L. Smith, On eigenvalue problems for nondifferentiable mappings, J. Differential Equations 33 (1979), no. 3, 294–319. MR 543701, DOI 10.1016/0022-0396(79)90067-6
J. Smoller, A. Tromba, and A. Wasserman, Nondegenerate solutions of boundary value problems, Nonlinear Anal. 4 (1980), no. 2, 207–216. MR 563804, DOI 10.1016/0362-546X(80)90049-8
C. A. Stuart, Concave solutions of singular non-linear differential equations, Math. Z. 136 (1974), 117–135. MR 341219, DOI 10.1007/BF01214347
C. A. Swanson, Comparison and oscillation theory of linear differential equations, Mathematics in Science and Engineering, Vol. 48, Academic Press, New York-London, 1968. MR 0463570
Gordon Thomas Whyburn, Topological analysis, Princeton Mathematical Series, No. 23, Princeton University Press, Princeton, N. J., 1958. MR 0099642