Perturbations of planar quasilinear differential systems
Perturbations of planar quasilinear differential systems
复制标题
平面拟线性微分系统的扰动
DOI：
10.1016/j.jde.2020.08.024
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发表时间：
2021
影响因子：
2.4
通讯作者：
Tanaka Satoshi
中科院分区：
文献类型：
--
作者：
Itakura Kenta;Onitsuka Masakazu;Tanaka Satoshi
考虑拟线性微分系统x ' = a x+ b| y| p(,)，其中a， b, c， d为实常数，b2 + c2 bb， p和p是正数，(1/p)+(1/p)= 1， k和l在t≥t0和小x2 + y时连续。当p= 2时，该系统被简化为线性摄动系统。证明了在原点（0,0）附近解的行为与非摄动系统解的行为非常相似，即当k≡l≡0时，系统在（0,0）附近，且k和l在某种意义上较小。强调当p≠2时，该系统在（0,0）处不能线性化，因为雅可比矩阵在（0,0）处不能定义。我们的结果将适用于具有微分算子r−(γ−1)(r α| u ' | β - a u ‘) ’的拟线性椭圆方程的径向解，其中包括p- laplace和k-Hessian。
The quasilinear differential system x′= a x+ b| y| p⁎− 2 y+ k (t, x, y), y′= c| x| p− 2 x+ d y+ l (t, x, y) is considered, where a, b, c and d are real constants with b 2+ c 2> 0, p and p⁎ are positive numbers with (1/p)+(1/p⁎)= 1, and k and l are continuous for t≥ t 0 and small x 2+ y 2. When p= 2, this system is reduced to the linear perturbed system. It is shown that the behavior of solutions near the origin (0, 0) is very similar to the behavior of solutions to the unperturbed system, that is, the system with k≡ l≡ 0, near (0, 0), provided k and l are small in some sense. It is emphasized that this system can not be linearized at (0, 0) when p≠ 2, because the Jacobian matrix can not be defined at (0, 0). Our result will be applicable to study radial solutions of the quasilinear elliptic equation with the differential operator r−(γ− 1)(r α| u′| β− a u′)′, which includes p-Laplacian and k-Hessian.