Principal eigenvalues for problems with indefinite weight function on 𝑅ⁿ

Brown, K. J.; Cosner, C.; Fleckinger, J.

doi:10.1090/S0002-9939-1990-1007489-1

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 changes sign on ${R^n}$ . It is proved that such eigenvalues exist if 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 $\int_{{R^n}} {g(x)dx > 0}$ .

[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), no. 1, 112–120. MR 576277, https://doi.org/10.1016/0022-247X(80)90309-1
[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), no. 1, 91–104. MR 984228, https://doi.org/10.1007/BF00276083
[4] K. J. Brown and A. Tertikas, On the bifurcation of radially symmetric steady-state solutions arising in population genetics, submitted for publication.
[5] Maxime Bôcher, The smallest characteristic numbers in a certain exceptional case, Bull. Amer. Math. Soc. 21 (1914), no. 1, 6–9. MR 1559550, https://doi.org/10.1090/S0002-9904-1914-02560-1
[6] Jacqueline Fleckinger and Michel L. Lapidus, Eigenvalues of elliptic boundary value problems with an indefinite weight function, Trans. Amer. Math. Soc. 295 (1986), no. 1, 305–324. MR 831201, https://doi.org/10.1090/S0002-9947-1986-0831201-3
[7] W. H. Fleming, A selection-migration model in population genetics, J. Math. Biol. 2 (1975), no. 3, 219–233. MR 0403720, https://doi.org/10.1007/BF00277151
[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), no. 2, 169–175. MR 786544, https://doi.org/10.1007/BF00282330
[9] Peter Hess and Tosio Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations 5 (1980), no. 10, 999–1030. MR 588690, https://doi.org/10.1080/03605308008820162
[10] Adele Manes and Anna Maria Micheletti, Un’estensione della teoria variazionale classica degli autovalori per operatori ellittici del secondo ordine, Boll. Un. Mat. Ital. (4) 7 (1973), 285–301 (Italian, with English summary). MR 0344663
[11] Michael Reed and Barry Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0493420
[12] Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493421
[13] Hans F. Weinberger, Variational methods for eigenvalue approximation, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974. Based on a series of lectures presented at the NSF-CBMS Regional Conference on Approximation of Eigenvalues of Differential Operators, Vanderbilt University, Nashville, Tenn., June 26–30, 1972; Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 15. MR 0400004