The quasilinear differential system
$$
x' = a x + b
y
^{p^*-2}y + k(t,x,y), \quad
y' = c
x
^{p-2}x + d y + l(t,x,y)
$$
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 $t \ge t_0$ 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 $k\equiv l\equiv 0$, 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
$p\ne 2$, 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
$r^{-(\gamma-1)}(r^\alpha
u'
^{\beta-a}u')'$, which includes
$p$-Laplacian and $k$-Hessian.
Research papers (academic journals)
