Regularly varying solutions with intermediate growth for cyclic differential systems of second order
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 $$ (p_i(t)|x_{i}'|^{\alpha_i -1}x_{i}')' + q_{i}(t)|x_{...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2016-12-01
|
| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2016/328/abstr.html |
| _version_ | 1819150323764166656 |
|---|---|
| author | Jaroslav Jaros Kusano Takasi Tomoyuki Tanigawa |
| author_facet | Jaroslav Jaros Kusano Takasi Tomoyuki Tanigawa |
| author_sort | Jaroslav Jaros |
| collection | DOAJ |
| description | In this article, we study the existence and accurate asymptotic
behavior as $t \to \infty$ of positive solutions with
intermediate growth for a class of cyclic systems of nonlinear
differential equations of the second order
$$
(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,
$$
where $\alpha_i$ and $\beta_i$, $i = 1,\dots,n$, are positive
constants such that $\alpha_1{\dots}\alpha_n >\beta_1{\dots}\beta_n$ and
$p_i, q_i: [a,\infty) \to (0,\infty)$ are continuous regularly varying
functions (in the sense of Karamata). It is shown that the situation
in which the system possesses regularly varying intermediate solutions can be
completely characterized, and moreover that the asymptotic
behavior of such solutions is governed by the unique formula
describing their order of growth (or decay) precisely.
The main results are applied to some classes of partial differential
equations with radial symmetry including metaharmonic equations and
systems involving $p$-Laplace operators on exterior domains. |
| first_indexed | 2024-12-22T14:15:41Z |
| format | Article |
| id | doaj.art-d3ef29200a314f8bb0d794271720d13f |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-22T14:15:41Z |
| publishDate | 2016-12-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-d3ef29200a314f8bb0d794271720d13f2022-12-21T18:23:07ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912016-12-012016328,122Regularly varying solutions with intermediate growth for cyclic differential systems of second orderJaroslav Jaros0Kusano Takasi1Tomoyuki Tanigawa2 Comenius Univ., Bratislava, Slovakia Hiroshima Univ., Higashi Hiroshima, Japan Kumamoto Univ., Kumamoto, Japan In this article, we study the existence and accurate asymptotic behavior as $t \to \infty$ of positive solutions with intermediate growth for a class of cyclic systems of nonlinear differential equations of the second order $$ (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.http://ejde.math.txstate.edu/Volumes/2016/328/abstr.htmlSystems of differential equationspositive solutionsasymptotic behaviorregularly varying functions |
| spellingShingle | Jaroslav Jaros Kusano Takasi Tomoyuki Tanigawa Regularly varying solutions with intermediate growth for cyclic differential systems of second order Electronic Journal of Differential Equations Systems of differential equations positive solutions asymptotic behavior regularly varying functions |
| title | Regularly varying solutions with intermediate growth for cyclic differential systems of second order |
| title_full | Regularly varying solutions with intermediate growth for cyclic differential systems of second order |
| title_fullStr | Regularly varying solutions with intermediate growth for cyclic differential systems of second order |
| title_full_unstemmed | Regularly varying solutions with intermediate growth for cyclic differential systems of second order |
| title_short | Regularly varying solutions with intermediate growth for cyclic differential systems of second order |
| title_sort | regularly varying solutions with intermediate growth for cyclic differential systems of second order |
| topic | Systems of differential equations positive solutions asymptotic behavior regularly varying functions |
| url | http://ejde.math.txstate.edu/Volumes/2016/328/abstr.html |
| work_keys_str_mv | AT jaroslavjaros regularlyvaryingsolutionswithintermediategrowthforcyclicdifferentialsystemsofsecondorder AT kusanotakasi regularlyvaryingsolutionswithintermediategrowthforcyclicdifferentialsystemsofsecondorder AT tomoyukitanigawa regularlyvaryingsolutionswithintermediategrowthforcyclicdifferentialsystemsofsecondorder |