Abstract
In this article, we consider the following quasilinear elliptic equation
{
−
div
(
∇
u
(
1
+
u
)
2
θ
)
−
γ
θ
|
∇
u
|
2
(
1
+
u
)
1
+
2
θ
=
u
p
,
x
∈
ℝ
N
,
u
>
0
,
lim
|
x
|
→
+
∞
u
(
x
)
=
0
,
where
θ
>
0
,
γ
∈
ℝ
, and
p
>
N
+
2
N
−
2
. We first perform a change of variables, which is more general because it generalizes the study of the cases
γ
=
1
and
γ
=
0
to all real
γ
. Then, by using the Pohozaev identity in the entire space, we show that the equation does not admit solutions in
H
1
(
ℝ
N
)
in some cases, which leads us to find solutions in a larger space. Lastly, we introduce a parameter and apply a perturbation method to get a continuum of solutions with decay
O
(
|
x
|
−
2
p
−
1
)
at infinity.