Abstract
It is the purpose of this work to study the Choquard equation
i
u
˙
-
(
-
Δ
)
s
u
=
±
|
x
|
γ
(
I
α
∗
|
·
|
γ
|
u
|
p
)
|
u
|
p
-
2
u
in the space
H
˙
s
∩
H
˙
s
c
, where
0
<
s
c
<
s
corresponds to the scale invariant homogeneous Sobolev norm. Here, one considers to two separate cases. The first one is the classical case
s
=
1
and the second one is the fractional regime
0
<
s
<
1
with radial data. One tries to develop a local theory using a new adapted sharp Gagliardo–Nirenberg estimate. Moreover, one investigates the concentration of non-global solutions in
L
T
∗
∞
(
H
˙
s
c
)
. One needs to deal with the lack of a mass conservation, since the data are not supposed to be in
L
2
. This note gives a complementary to the previous works about the same problem in the energy space
H
1
.