Abstract
In the paper we consider solutions of the equation Δu−c(x)u=0,c(x)≥0, on complete Riemannian manifolds constituted as follows: the exterior of some compact set is isometric to the direct product of the semiaxis by some compact manifold with the metricds 2=h 2(r)dr 2+g 2(r)dθ2. Necessary and sufficient conditions under which bounded solutions of the equation have a limit independent of θ asr→∞ are obtained and also conditions under which the two-sided Liouville theorem is valid.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. A. Grigor'yan and N. S. Nadirashvili, “Liouville theorems and exterior boundary value problems,”Izv. Vyssh. Uchebn. Zaved. Mat. [Soviet Math. (Iz. VUZ)], No. 5, 25–33 (1987).
A. G. Losev, “Some Liouville theorems on Riemannian manifolds of special form,”Izv. Vyssh. Uchebn. Zaved. Mat. [Soviet Math. (Iz. VUZ)], No. 12, 15–24 (1991).
L. Sario, M. Nakai, C. Wang and L. O. Chung,Classification Theory of Riemannian Manifolds, Vol. 605, Lecture Notes in Math., Springer-Verlag, New York (1977).
M. Berger, P. Ganduchon, and E. Mazet,Le Spectre d'une variété Riemannienne, Vol. 194, Lecture Notes in Math., Springer-Verlag, New York (1971).
Author information
Authors and Affiliations
Additional information
Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 215–221, February, 1999.
Rights and permissions
About this article
Cite this article
Losev, A.G. The behavior of bounded solutions to the equation Δu−c(x)u=0 on riemannian manifolds of special type. Math Notes 65, 175–180 (1999). https://doi.org/10.1007/BF02679814
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02679814