Electronic Journal of Differential Equations (Dec 2016)
Regularly varying solutions with intermediate growth for cyclic differential systems of second order
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.
