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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
DOI: https://doi.org/10.1090/S0002-9939-1972-0306733-2
MathSciNet review: 0306733
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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$.


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  • [1] C. 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 (1968/69), 411-421. MR 41 #2330. MR 0257680 (41:2330)
  • [2] -, On a class of non-linear elliptic boundary value problems, J. Math. Mech. 19 (1969/70), 351-356. MR 39 #7280. MR 0245974 (39:7280)
  • [3] C. V. Coffman, Spectral theory of monotone Hammerstein operators, Pacific J. Math. 36 (1971), 303-322. MR 0281067 (43:6786)
  • [4] R. E. Edwards, Functional analysis. Theory and applications, Holt, Rinehart and Winston, New York, 1965. MR 36 #4308. MR 0221256 (36:4308)
  • [5] J. B. Keller and S. Antman (Editors), Bifurcation theory and nonlinear eigenvalue problems, Benjamin, New York, 1969. MR 39 #2555. MR 0241213 (39:2555)
  • [6] M. A. Krasnosel'skiĭ, Topological methods in the theory of nonlinear integral equations, GITTL, Moscow, 1956; English transl., Macmillan, New York, 1964. MR 20 #3464; MR 28 #2414. MR 0159197 (28:2414)
  • [7] S. I. Pohožaev, On the eigenfunctions of the equation $ \Delta u + \lambda f(u) = 0$, Dokl. Akad. Nauk SSSR 165 (1965), 36-39 = Soviet Math. Dokl. 6 (1965), 1408-1411. MR 33 #411. MR 0192184 (33:411)
  • [8] M. M. Vainberg, Variational methods for the study of nonlinear operators, HoldenDay, San Francisco, Calif., 1964. MR 31 #638. MR 0176364 (31:638)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0306733-2
Keywords: Bifurcation theory, bifurcation point, principle of linearization, "bootstrap'' method, integral equation
Article copyright: © Copyright 1972 American Mathematical Society

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