Abstract
We consider the elliptic quasilinear equation
-
Δ
m
u
=
u
p
|
∇
u
|
q
{-\Delta_{m}u=u^{p}\lvert\nabla u\rvert^{q}}
in
ℝ
N
{\mathbb{R}^{N}}
,
q
≥
m
{q\geq m}
and
p
>
0
{p>0}
,
1
<
m
<
N
{1<m<N}
.
Our main result is a Liouville-type property, namely, all the positive
C
1
{C^{1}}
solutions in
ℝ
N
{\mathbb{R}^{N}}
are constant.
We also give their asymptotic behaviour; all the solutions in an exterior domain
ℝ
N
∖
B
r
0
{\mathbb{R}^{N}\setminus B_{r_{0}}}
are bounded.
The solutions in
B
r
0
∖
{
0
}
{B_{r_{0}}\setminus\{0\}}
can be extended as continuous functions in
B
r
0
{B_{r_{0}}}
.
The solutions in
ℝ
N
∖
{
0
}
{\mathbb{R}^{N}\setminus\{0\}}
has a finite limit
l
≥
0
{l\geq 0}
as
|
x
|
→
∞
{\lvert x\rvert\to\infty}
.
Our main argument is a Bernstein estimate of the gradient of a power of the solution, combined with a precise Osserman-type estimate for the equation satisfied by the gradient.