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Global branches of positive weak solutions of semilinear elliptic problems over nonsmooth domains

Published online by Cambridge University Press:  14 November 2011

Timothy J. Healey ,
Hansjörg Kielhöfer  and
Charles A. Stuart
Timothy J. Healey
Affiliation:
Department of Theoretical & Applied Mechanics and Center for Applied Mathematics, Cornell University, Ithaca, NY 14853, U.S.A.
Hansjörg Kielhöfer
Affiliation:
Institut für Mathematik, Universität Augsburg, Universitätsstraße 8, D-8900 Augsburg, Germany
Charles A. Stuart
Affiliation:
Départment de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
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Abstract

We consider the nonlinear eigenvalue problem posed by a parameter-dependent semilinear second-order elliptic equation on a bounded domain with the Dirichlet boundary condition. The coefficients of the elliptic operator are bounded measurable functions and the boundary of the domain is only required to be regular in the sense of Wiener. The main results establish the existence of an unbounded branch of positive weak solutions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 2 , 1994 , pp. 371 - 388
Copyright
Copyright © Royal Society of Edinburgh 1994

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References

1Alexander, J. C. and Yorke, J. A.. The implicit function theorem and global methods of cohomology. J. Fund. Anal. 21 (1976), 330–339.Google Scholar
2Berestycki, H., Nirenberg, L. and Varadhan, S. R. S. The principal eigenvalue and maximum principle for second order elliptic operators in general domains. Comm. Pure Appl. Math, (to appear).Google Scholar
3Brezis, H.. Analyse fonctionelle (Paris: Masson, 1983).Google Scholar
4Crandall, M. G. and Rabinowitz, P. H.. Bifurcation from simple eigenvalues. J. Fund. Anal. 8 (1971), 321–340.Google Scholar
5Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order, 2nd edn (Berlin: Springer, 1983).Google Scholar
6Healey, T. J. and Kielh, H.öfer. Preservation of nodal structure on global bifurcating solution branches of elliptic equations with symmetry. J. Differential Equations (to appear).Google Scholar
7Kielh, H.öfer. Multiple eigenvalue bifurcation for Fredholm operators. J. Reine Angew. Math. 358 (1985), 104–124.Google Scholar
8Knowles, J. K.. On finite anti-plane-shear for incompressible elastic materials. J. Austral. Math. Soc. Ser. B 19(1976), 400–415.Google Scholar
9Rabinowitz, P. H.. Some global results for nonlinear eigenvalue problems. J. Fund. Anal. 7 (1971), 487–513.CrossRefGoogle Scholar
10Rabinowitz, P. H.. Some aspects of nonlinear eigenvalue problems. Rocky Mountain J. Math. 3 (1973), 161–202.CrossRefGoogle Scholar
11Rosakis, P.. Compact zones of shear transformation in an anisotropic solid. J. Mech. Phys. Solids 40 (1992), 1163–1195.CrossRefGoogle Scholar
12Serrin, J.. A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43 (1971), 304–318.CrossRefGoogle Scholar
13Stampacchia, G.. Le problöme de Dirichlet pour les equations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965), 189–258.Google Scholar