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Second order elliptic equations with degenerate weight


Author: W. Allegretto
Journal: Proc. Amer. Math. Soc. 107 (1989), 989-998
MSC: Primary 35J10; Secondary 35P15, 47F05
DOI: https://doi.org/10.1090/S0002-9939-1989-0977929-4
MathSciNet review: 977929
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Abstract: We consider the eigenvalue problem: $ - \Delta u - qu = \lambda \omega u,u \in \dot{H}^{1,2}(\Omega )$, in a smooth bounded domain $ \Omega \subset {{\mathbf{R}}^n}$. We allow $ - \Delta - q$ to have negative spectrum and assume $ \omega \geq 0$ in $ \Omega ,\omega \equiv 0$ in a subdomain of $ \Omega $. Under suitable regularity conditions, we establish several results for the spectrum of this problem. In particular, we give: a min.max. formula for $ \lambda $; a precise estimate on the number of negative $ \lambda $; an estimate for the location of negative $ \lambda $. An example concludes the paper.


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DOI: https://doi.org/10.1090/S0002-9939-1989-0977929-4
Article copyright: © Copyright 1989 American Mathematical Society

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