TY - JOUR

T1 - Perturbations of planar quasilinear differential systems

AU - Itakura, Kenta

AU - Onitsuka, Masakazu

AU - Tanaka, Satoshi

N1 - Funding Information:
Masakazu Onitsuka was supported by JSPS KAKENHI Grant Number 20K03668 . Satoshi Tanaka was supported by JSPS KAKENHI Grant Number 26400182 , 19K03595 and 17H01095 .
Publisher Copyright:
© 2020 The Authors

PY - 2021/1/15

Y1 - 2021/1/15

N2 - The quasilinear differential system x′=ax+b|y|p⁎−2y+k(t,x,y),y′=c|x|p−2x+dy+l(t,x,y) is considered, where a, b, c and d are real constants with b2+c2>0, p and p⁎ are positive numbers with (1/p)+(1/p⁎)=1, and k and l are continuous for t≥t0 and small x2+y2. 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)(rα|u′|β−au′)′, which includes p-Laplacian and k-Hessian.

AB - The quasilinear differential system x′=ax+b|y|p⁎−2y+k(t,x,y),y′=c|x|p−2x+dy+l(t,x,y) is considered, where a, b, c and d are real constants with b2+c2>0, p and p⁎ are positive numbers with (1/p)+(1/p⁎)=1, and k and l are continuous for t≥t0 and small x2+y2. 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)(rα|u′|β−au′)′, which includes p-Laplacian and k-Hessian.

KW - Asymptotic behavior

KW - Characteristic equation

KW - Eigenvalue

KW - Perturbation

KW - Quasilinear

KW - Quasilinear elliptic equation

UR - http://www.scopus.com/inward/record.url?scp=85091205404&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85091205404&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2020.08.024

DO - 10.1016/j.jde.2020.08.024

M3 - Article

AN - SCOPUS:85091205404

SN - 0022-0396

VL - 271

SP - 216

EP - 253

JO - Journal of Differential Equations

JF - Journal of Differential Equations

ER -