Perturbations of planar quasilinear differential systems

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)(r^α|u^′|^β−au^′)^′, which includes p-Laplacian and k-Hessian.