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 $q_i(t)$ are regularly varying functions, and analyze positive strongly increasing solutions of system $(*)$ in the framework of regular variation. We show that the situation for the existence of regularly varying solutions of positive indices for $(*)$ 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 $(*)$ 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.