| Summary: | 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.
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