Perturbations of planar quasilinear differential systems

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:271

Article

Itakura, Kenta¹  Onitsuka, Masakazu²  Tanaka, Satoshi³ 

[1] Matsue Yamamoto Met Co Ltd, Hokuryo Cho 30, Matsue, Shimane 6900816, Japan

[2] Okayama Univ Sci, Fac Sci, Dept Appl Math, Ridai Cho 1-1, Okayama 7000005, Japan

[3] Tohoku Univ, Math Inst, Aoba Ku, Aoba 6-3, Sendai, Miyagi 9808578, Japan

关键词: Quasilinear;    Perturbation;    Characteristic equation;    Eigenvalue;    Asymptotic behavior;    Quasilinear elliptic equation;   

DOI  :  10.1016/j.jde.2020.08.024

来源: Elsevier

【 摘 要 】

The quasilinear differential system x' = ax + b vertical bar y vertical bar(p)*(-2)y + k(t, x, y), y' = c vertical bar x vertical bar(p-2)x + dy + l(t, x, y) is considered, where a, b, c and dare 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 equivalent to l equivalent to 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 not equal 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(-(gamma-1))(r(alpha)vertical bar u'vertical bar(beta-a)u')', which includes p-Laplacian and k-Hessian. (C) 2020 The Authors. Published by Elsevier Inc.

【 授权许可】

Free