Losev A.G., Sazonov A.P. On the asymptotic behavior of some semilinear equations’ solutions on the model riemannian manifolds

Losеv Alеxandеr Gеorgiеvich

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

Sazonov Alеksеy Pavlovich

Assistent Teacher, Department of Mathematical Analysis and Function Theory
Volgograd State University
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Prospect Universitetsky, 100, 400062 Volgograd, Russian Federation

Abstract. T. Kusano, M. Naito [12] studied the positive solutions of the equation (1) in Rn and received some conditions for the existence of positive radially symmetric solutions of the equation, which are cited in the paper (Theorem A and Theorem B). The aim of this work is to study the positive solutions of a semilinear elliptic equation (1) on model Riemannian manifolds.

    The first result obtained in the study of solutions of (1) is fairly obvious and is based on the properties of manifolds of the parabolic type.

Theorem. Let the manifold Mq is such that

Then any non-negative solution of the equation (1) is identically zero.
    Next, we consider the manifold of the hyperbolic type, i. e. we assume that

We denote

    One of the main results is the statement that generalizes the Theorem of T. Kusano, M. Naito [12] (Theorem A in the paper).

Theorem. Assume that

Then, for any α > 0, the equation (1) has positive radially symmetric solution on Mq, such that u(0)= α.

    The assertion of Theorem A can be derived as the corollary of this theorem.
    In addition, in this paper we find an upper bound for the solutions of the equation (1).

Theorem. If the equation (1) has a positive radially symmetric solution, then the following estimate is valid:

Now we note a condition of positivity of the Ф(r).

Lemma. Let the function q(r) is convex. If

then

    Also in the paper we obtained the conditions under which the equation (1) has no positive radially symmetric solutions.

Theorem. Let the function q(r) is convex. Assume that

and

Then the equation (1) has no positive radially symmetric solution on Mq .

Theorem. Let the function q(r) is convex. Assume that

and

Then the equation (1) has no positive radially symmetric solutions on Mq.
Theorem. Let the function q(r) is convex. Assume that

where 0 < a ≤ 1

 

and

 

Then the equation (1) has no positive radially symmetric solutions on Mq.
    The assertion of Theorem B can be derived as the corollary of these theorems.

 Key words: semilinear elliptic equations, theorems of Liouville, model Riemannian manifolds, radially symmetric solutions, problem of Cauchy.

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On the asymptotic behavior of some semilinear equations’ solutions on the model riemannian manifolds by Losev A.G., Sazonov A.P. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Science Journal of Volgograd State University. Mathematics. Physics. №2 (19) 2013 pp. 36-56

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