Opuscula Math. 35, no. 1 (2015), 47-69

http://dx.doi.org/10.7494/OpMath.2015.35.1.47

Opuscula Mathematica

# Strongly increasing solutions of cyclic systems of second order differential equations with power-type nonlinearities

Abstract. We consider \(n\)-dimensional cyclic systems of second order differential equations \[(p_i(t)|x_{i}'|^{\alpha_i -1}x_{i}')' = q_{i}(t)|x_{i+1}|^{\beta_i-1}x_{i+1},\] \[\quad i = 1,\ldots,n, \quad (x_{n+1} = x_1) \tag{\(\ast\)}\] under the assumption that the positive constants \(\alpha_i\) and \(\beta_i\) satisfy \(\alpha_1{\ldots}\alpha_n \gt \beta_1{\ldots}\beta_n\) and \(p_i(t)\) and \(q_i(t)\) are regularly varying functions, and analyze positive strongly increasing solutions of system (\(\ast\)) in the framework of regular variation. We show that the situation for the existence of regularly varying solutions of positive indices for (\(\ast\)) can be characterized completely, and moreover that the asymptotic behavior of such solutions is governed by the unique formula describing their order of growth precisely. We give examples demonstrating that the main results for (\(\ast\)) can be applied to some classes of partial differential equations with radial symmetry to acquire accurate information about the existence and the asymptotic behavior of their radial positive strongly increasing solutions.

Keywords: systems of differential equations, positive solutions, asymptotic behavior, regularly varying functions.

Mathematics Subject Classification: 34C11, 26A12.

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- Jaroslav Jaroš
- Comenius University, Faculty of Mathematics, Physics and Informatics, Department of Mathematical Analysis and Numerical Mathematics, 842 48 Bratislava, Slovakia

- Kusano Takaŝi
- Hiroshima University, Faculty of Science, Department of Mathematics, Higashi-Hiroshima 739-8526, Japan

- Communicated by Alexander Domoshnitsky.

- Received: 2014-02-14.
- Revised: 2014-05-20.
- Accepted: 2014-05-31.
- Published online: 2014-11-12.