Losev A.G., Mazepa E.A. On asymptotical behavior of the positive solutions some quasilinear inequalities on model riemannian manifolds

http://dx.doi.org/10.15688/jvolsu1.2013.1.5

Losеv Alеxandеr Gеorgiеvich

Doctor of Physical and Mathematical Sciences, Director, Institute of Mathematics and IT Volgograd State University
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Prospekt Universitetskij, 100, 400062 Volgograd, Russian Federation

Mazеpa Elеna Alеksееvna


Candidate of Physical and Mathematical Sciences, 
Associate Professor, Department of Fundamental Informatics and Optimal Control Volgograd State University
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Prospekt Universitetskij, 100, 400062 Volgograd, Russian Federation

 

Abstract. In this paper asymptotic behavior of positive solutions of quasilinear elliptic inequalities (1) on spherically symmetric noncompact (model) Riemannian manifolds is researched. In particular, we find conditions under which Liouville theorems on no nontrivial solutions, as well as the conditions of existence and cardinality of the set of positive solutions of the studied inequalities on the Riemannian manifolds. The results generalize similar results obtained previously by Naito. Y. and Usami H. for the Euclidean space Rn.

    We describe Riemannian manifolds. Fix the origin ORn and a smooth function q in the interval [0,∞) such that q(0) = 0 and q′(0) = 1. We define a model Riemannian manifold Mq as follows:
    (1) the set of points Mq is all Rn;
    (2) in polar coordinates (r, θ) (where r ∈ (0,∞) and θ ∈ Sn-1) Riemannian metric on Mq ∖ {O} defined as

where— the standard Riemannian metric on the sphere Sn-1

    (3) Riemannian metric at O is a smooth continuation of the metric.
    Will further assume that the function A in the inequality (1) satisfies the following conditions:

 

c(x) ≡ c(r) — continuous positive on R+ function, and the function  f  ≠ 0 such that

    We also use the following assumption on the function f:

     First, consider the case where  Introduce designation 

Theorem. Let  and manifold Mq is such that lim sup Then, if the condition (F), then positive integer solutions of the inequality (1) on Mq does not exist.
    Next, consider the case where  We prove a theorem on the non-existence of positive solutions of (1) and the conditions for the existence of a continuum of positive integer solutions of the inequality.

Key words: quasilinear elliptic inequalities, asymptotic behavior, the theorem of Liouville type, model Riemannian manifolds, cardinality of the set of solutions.

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On asymptotical behavior of the positive solutions some quasilinear inequalities on model riemannian manifolds by Losev A.G., Mazepa E.A. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Science Journal of Volgograd State University. Mathematics. Physics. №1 (18) 2013 pp. 59-69

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