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)

 
 

 

Principal eigenvalues for problems with indefinite weight function on $ {\bf R}\sp n$


Authors: K. J. Brown, C. Cosner and J. Fleckinger
Journal: Proc. Amer. Math. Soc. 109 (1990), 147-155
MSC: Primary 35P05; Secondary 35J25
DOI: https://doi.org/10.1090/S0002-9939-1990-1007489-1
MathSciNet review: 1007489
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the existence of positive principal eigenvalues of the problem $ - \Delta u(x) = \lambda g(x)u$ for $ x \in {R^n};u(x) \to 0$ as $ x \to \infty $ where the weight function $ g$ changes sign on $ {R^n}$. It is proved that such eigenvalues exist if $ g$ is negative and bounded away from 0 at $ \infty $ or if $ n \geq 3$ and $ \vert g(x)\vert$ is sufficiently small at $ \infty $ but do not exist if $ n = 1\,{\text{or}}\,2$ and $ \int_{{R^n}} {g(x)dx > 0} $.


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

  • [1] A. S. Bonnet, Ph.D. thesis, University Paris 6, 1988.
  • [2] K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl 75 (1980), 112-120. MR 576277 (82a:35080)
  • [3] K. J. Brown, S. S. Lin and A. Tertikas, Existence and nonexistence of steady-state solutions for a selection-migration model in population genetics, J. Math. Biol. 27 (1989), 91-104. MR 984228 (90i:92012)
  • [4] K. J. Brown and A. Tertikas, On the bifurcation of radially symmetric steady-state solutions arising in population genetics, submitted for publication.
  • [5] M. Bocher, The smallest characteristic numbers in a certain exceptional case, Bull. Amer. Math. Soc. 21 (1914), 6-9. MR 1559550
  • [6] J. Fleckinger and M. L. Lapidus, Eigenvalues of elliptic boundary value problems with an indefinite weight function, Trans. Amer. Math. Soc. 295 (1986), 305-324. MR 831201 (87j:35282)
  • [7] W. H. Fleming, A selection-migration model in population genetics, J. Math. Biol. 2 (1975), 219-233. MR 0403720 (53:7531)
  • [8] J. -P. Gossez and E. Lami Dozo, On the principal eigenvalue of a second order linear elliptic problem, Arch. Rational Mech. Anal. 89 (1985), 169-175. MR 786544 (86m:35043)
  • [9] P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with indefinite weight function, Comm. Partial Differential Equations 5 (1980), 999-1030. MR 588690 (81m:35102)
  • [10] A. Manes and A. M. Michelletti, Un' estensione delta teoria vanazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Un. Math. Ital. 7 (1973), 285-301. MR 0344663 (49:9402)
  • [11] M. Reed and B. Simon, Methods of modern mathematical physics II: Fourier analysis and self-adjointness, Academic Press, New York, San Francisco, and London, 1975. MR 0493420 (58:12429b)
  • [12] -, Methods of modern mathematical physics IV: analysis of operators, Academic Press, New York, San Francisco, and London, 1978. MR 0493421 (58:12429c)
  • [13] H. F. Weinberger, Variational methods for eigenvalue approximation, CBMS Regional Conf. Ser. in Appl. Math. 15 (1974). MR 0400004 (53:3842)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 35P05, 35J25

Retrieve articles in all journals with MSC: 35P05, 35J25


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-1007489-1
Keywords: Elliptic boundary value problems, indefinite weight function, spectral theory
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society