Temat: positive solutions - Katalog OPAC zbiorów

Skocz do pozycji: 1.

Tytuł:: Strongly increasing solutions of cyclic systems of second order differential equations with power-type nonlinearities
Autorzy:: Jaroś, J.
Takaśi, K.
Powiązania:: https://bibliotekanauki.pl/articles/1397499.pdf
Data publikacji:: 2015
Wydawca:: Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
Tematy:: systems of differential equations
positive solutions
asymptotic behavior
regularly varying functions
Opis:: We consider n-dimensional cyclic systems of second order differential equations $(p_i(t)|x_i^\prime |^{\alpha_i - 1} x_i^\prime )^\prime = q_i(t) |x_{i+q}|^{\beta_i -1} x_{i+1}, i= 1, ..., n, \ \ (x_{n+1} = x_1) \ \ (\ast) $ under the assumption that the positive constants $\alpha_i $ and $ \beta_i $ satisfy $ \alpha_1 ... \alpha_n > \beta_1 ... \beta_n $ and $p_i(t)$ 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.
Źródło:: Opuscula Mathematica; 2015, 35, 1; 47-69
1232-9274
2300-6919
Pojawia się w:: Opuscula Mathematica
Dostawca treści:: Biblioteka Nauki

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Skocz do pozycji: 2.

Tytuł:: Positive solutions with specific asymptotic behavior for a polyharmonic problem on $\mathbb{R}^n$
Autorzy:: Dhifli, A.
Powiązania:: https://bibliotekanauki.pl/articles/1397483.pdf
Data publikacji:: 2015
Wydawca:: Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie. Wydawnictwo AGH
Tematy:: asymptotic behavior
Dirichlet problem
Schauder fixed point theorem
positive bounded solutions
Opis:: This paper is concerned with positive solutions of the semilinear polyharmonic equation $(-\Delta)^m u = a(x)u^\alpha$ on $\mathbb{R}^n$, where m and n are positive integers with n > 2m, $\alpha \in (—1,1)$. The coefncient a is assumed to satisfy $a(x) \approx (1+|x|)^{-\lambda}L(1+|x|)$ for $ x \in \mathbb{R}^n$, where $ \lambda \in [2m,\infty)$ and $L \in C^1([q,\infty))$ is positive with $ {tL^\prime (t)}/{L(t)} \rightarrow 0 $ as $ t \rightarrow \infty $ if $ \lambda = 2m$, one also assumes that $\int_1^\infty t^{-1} L(t) dt < \infty $. We prove the existence of a positive solution $u$ such that $ u(x) \approx (1 + |x|)^{-\tilde{\lambda}} L(1+|x|)$ for $ x \in \mathbb{R}^n$, with $\tilde{\lambda} := \text{min}(n-2m, {\lambda-2m}/{1-\alpha})$ and a function $ \tilde{L} $, given explicitly in terms of $L$ and satisfying the same condition as infinity. (Given positive functions $f$ and $g$ on $\mathbb{R}^n$, $f \approx g $ means that $c^{-1} g \leq f \leq cg $ for some constant c > 1.)
Źródło:: Opuscula Mathematica; 2015, 35, 1; 5-19
1232-9274
2300-6919
Pojawia się w:: Opuscula Mathematica
Dostawca treści:: Biblioteka Nauki

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