Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the bifurcation theory of semilinear elliptic eigenvalue problems

Author: Charles V. Coffman
Journal: Proc. Amer. Math. Soc. 31 (1972), 170-176
MSC: Primary 35P99
MathSciNet review: 0306733
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A standard ``bootstrap'' method is used to show that the bifurcation problem for the semilinear eigenvalue problem $ \Delta u + \lambda f(x,u) = 0$ in $ \Omega $, $ u{\vert _{\partial \Omega }} = 0$, where $ f(x,0) \equiv 0$, and $ (\partial /\partial u)f(x,0) > 0$, and when formulated in terms of weak solutions, is a local problem, i.e. independent of the behavior of $ f$ for large $ u$. A principle of linearization for this problem is proved under mild differentiability conditions on $ f$.

References [Enhancements On Off] (What's this?)

  • [1] Charles V. Coffman, An existence theorem for a class of non-linear integral equations with applications to a non-linear elliptic boundary value problem, J. Math. Mech. 18 (1968/1969), 411–421. MR 0257680
  • [2] Charles V. Coffman, On a class of non-linear elliptic boundary value problems, J. Math. Mech. 19 (1969/1970), 351–356. MR 0245974
  • [3] Charles V. Coffman, Spectral theory of monotone Hammerstein operators, Pacific J. Math. 36 (1971), 303–322. MR 0281067
  • [4] R. E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart and Winston, New York-Toronto-London, 1965. MR 0221256
  • [5] Bifurcation theory and nonlinear eigenvalue problems, Edited by Joseph B. Keller and Stuart Antman, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0241213
  • [6] M. A. Krasnosel’skii, Topological methods in the theory of nonlinear integral equations, Translated by A. H. Armstrong; translation edited by J. Burlak. A Pergamon Press Book, The Macmillan Co., New York, 1964. MR 0159197
  • [7] S. I. Pohožaev, On the eigenfunctions of the equation Δ𝑢+𝜆𝑓(𝑢)=0, Dokl. Akad. Nauk SSSR 165 (1965), 36–39 (Russian). MR 0192184
  • [8] M. M. Vainberg, Variational methods for the study of nonlinear operators, Holden-Day, Inc., San Francisco, Calif.-London-Amsterdam, 1964. With a chapter on Newton’s method by L. V. Kantorovich and G. P. Akilov. Translated and supplemented by Amiel Feinstein. MR 0176364

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35P99

Retrieve articles in all journals with MSC: 35P99

Additional Information

Keywords: Bifurcation theory, bifurcation point, principle of linearization, "bootstrap'' method, integral equation
Article copyright: © Copyright 1972 American Mathematical Society