On oscillatory elliptic equations on manifolds

Authors:
A. Baider and E. A. Feldman

Journal:
Trans. Amer. Math. Soc. **258** (1980), 495-504

MSC:
Primary 58G25; Secondary 35B05, 35J15

DOI:
https://doi.org/10.1090/S0002-9947-1980-0558186-6

MathSciNet review:
558186

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Abstract: In this note we investigate the possibility of oscillatory behavior for a second-order selfadjoint elliptic operators on noncompact Riemannian manifolds (*M, g*). Let *A* be such an operator which is semibounded below and symmetric on where is a volume element on *M*. If is a function such that , we would naively say that is oscillatory (and by extension is oscillatory if it possesses such an eigenfunction ) if has an infinite number of bounded connected components. For technical reasons this is not quite adequate for a definition. However, in §1 we give the usual definition of oscillatory which is a slight generalization of the one above. Let be the number below which this phenomenon cannot occur; is the oscillatory constant for the operator *A*. In that *A* is semibounded and symmetric on , *A* has a Friedrichs extension. Let be the bottom of the continuous spectrum of the Friedrichs extension of *A*. Our main result is .

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DOI:
https://doi.org/10.1090/S0002-9947-1980-0558186-6

Article copyright:
© Copyright 1980
American Mathematical Society