Electron. J. Differential Equations, Vol. 2016 (2016), No. 328, pp. 1-22.

Regularly varying solutions with intermediate growth for cyclic differential systems of second order

Jaroslav Jaros, Kusano Takasi, Tomoyuki Tanigawa

Abstract:
In this article, we study the existence and accurate asymptotic behavior as $t \to \infty$ of positive solutions with intermediate growth for a class of cyclic systems of nonlinear differential equations of the second order
$$ (p_i(t)|x_{i}'|^{\alpha_i -1}x_{i}')' + q_{i}(t)|x_{i+1}|^{\beta_i-1}x_{i+1} = 0, \quad i = 1,\ldots,n, \; x_{n+1} = x_1, $$
where $\alpha_i$ and $\beta_i$, $i = 1,\dots,n$, are positive constants such that $\alpha_1{\dots}\alpha_n >\beta_1{\dots}\beta_n$ and $p_i, q_i: [a,\infty) \to (0,\infty)$ are continuous regularly varying functions (in the sense of Karamata). It is shown that the situation in which the system possesses regularly varying intermediate solutions can be completely characterized, and moreover that the asymptotic behavior of such solutions is governed by the unique formula describing their order of growth (or decay) precisely. The main results are applied to some classes of partial differential equations with radial symmetry including metaharmonic equations and systems involving $p$-Laplace operators on exterior domains.

Submitted December 30, 2015. Published December 22, 2016.
Math Subject Classifications: 34C11, 26A12.
Key Words: Systems of differential equations; positive solutions; asymptotic behavior; regularly varying functions.

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Jaroslav Jaros
Department of Mathematical Analysis and Numerical Mathematics
Faculty of Mathematics, Physics and Informatics
Comenius University
842 48 Bratislava, Slovakia
email: jaros@fmph.uniba.sk
  Kusano Takasi
Department of Mathematics
Faculty of Science, Hiroshima University
Higashi Hiroshima 739-8526, Japan
email: kusanot@zj8.so-net.ne.jp
  Tomoyuki Tanigawa
Department of Mathematics
Faculty of Education, Kumamoto University
Kumamoto 860-8555, Japan
email: tanigawa@educ.kumamoto-u.ac.jp

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