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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
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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} $.


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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