Abstract
We study some energy well-posedness issues of the Schrödinger equation with an inhomogeneous mixed nonlinearity and radial data
i
u
˙
−
(
−
Δ
)
s
u
±
|
x
|
ρ
|
u
|
p
−
1
u
±
|
u
|
q
−
1
u
=
0
,
0
<
s
<
1
,
ρ
≠
0
,
p
,
q
>
1
.
Our aim is to treat the competition between the homogeneous term
|
u
|
q
−
1
u
and the inhomogeneous one
|
x
|
ρ
|
u
|
p
−
1
u
. We simultaneously treat two different regimes,
ρ
>
0
and
ρ
<
−
2
s
. We deal with three technical challenges at the same time: the absence of a scaling invariance, the presence of the singular decaying term
|
⋅
|
ρ
, and the nonlocality of the fractional differential operator
(
−
Δ
)
s
. We give some sufficient conditions on the datum and the parameters
N
,
s
,
ρ
,
p
,
q
to have the global versus nonglobal existence of energy solutions. We use the associated ground states and some sharp Gagliardo–Nirenberg inequalities. Moreover, we investigate the
L
2
concentration of the mass-critical blowing-up solutions. Finally, in the attractive regime, we prove the scattering of energy global solutions. Since there is a loss of regularity in Strichartz estimates for the fractional Schrödinger problem with nonradial data, in this work, we assume that
u
|
t
=
0
is spherically symmetric. The blowup results use ideas of the pioneering work by Boulenger el al. (J. Funct. Anal. 271:2569–2603,
2016
).