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Oscillation Criteria for Semilinear Equations in General Domains

Published online by Cambridge University Press:  20 November 2018

W. Allegretto
W. Allegretto*
Affiliation:
University of Alberta
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Oscillation theory for nonlinear ordinary differential equations has been extensively developed in recent years by several authors. We refer the reader to the recent paper by Noussair and Swanson, [2], where an extensive bibliography may be found. The situation is somewhat different for the case of second order partial differential equations, an area which has recently been virtually untouched, except for the establishment of criteria which depend on a comparison with suitable linear equations and therefore are essentially linear in nature. A bibliography of such results may also be found in [2]. Of a more general nature have been the paper by the author [1], and the more recent work of Noussair and Swanson, [2]. Although the methods employed and the equations considered in [1] and [2] are different, both papers obtained results only for a class of exterior domains of Rn, of which the typical example is the complement of a bounded sphere.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 2 , 01 June 1976 , pp. 137 - 144
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Allegretto, W., Oscillation criteria for quasilinear equations, Canad. J. Math. 26 (1974), 931–947.CrossRefGoogle Scholar
2. Noussair, E. and Swanson, C. A., Oscillation theory for semilinear Schrödinger equations and inequalities, submitted for publication.CrossRefGoogle Scholar
3. Leighton, W., The detection of the oscillation of solutions of a second order linear differential equation, Duke Math. J. 17 (1950), 57–61.CrossRefGoogle Scholar
4. Protter, M. and Weinberger, H., Maximum principles in differential equations (Prentice Hall, New York, 1967).Google Scholar
5. Swanson, C. A., Strong oscillation of elliptic equations in general domains, Can. Math. Bull. 16 (1973), 105–110.CrossRefGoogle Scholar