Abstract
AbstractWe study the existence, nonexistence and qualitative properties of the solutions to the problem $$\begin{aligned} ({\mathcal {P}}) \quad \quad \left\{ \begin{aligned} (-\Delta )^s u -\theta \frac{u}{|x|^{2s}}&=u^p - u^q \quad \text {in }\,\, {\mathbb {R}}^N\\ u&> 0 \quad \text {in }\,\, {\mathbb {R}}^N\\ u&\in {\dot{H}}^s({\mathbb {R}}^N)\cap L^{q+1}({\mathbb {R}}^N), \end{aligned} \right. \end{aligned}$$
(
P
)
(
-
Δ
)
s
u
-
θ
u
|
x
|
2
s
=
u
p
-
u
q
in
R
N
u
>
0
in
R
N
u
∈
H
˙
s
(
R
N
)
∩
L
q
+
1
(
R
N
)
,
where $$s\in (0,1)$$
s
∈
(
0
,
1
)
, $$N>2s$$
N
>
2
s
, $$q>p\ge {(N+2s)}/{(N-2s)}$$
q
>
p
≥
(
N
+
2
s
)
/
(
N
-
2
s
)
, $$\theta \in (0, \Lambda _{N,s})$$
θ
∈
(
0
,
Λ
N
,
s
)
and $$\Lambda _{N,s}$$
Λ
N
,
s
is the sharp constant in the fractional Hardy inequality. For qualitative properties of the solutions, we mean both the radial symmetry, that is obtained by using the moving plane method in a nonlocal setting on the whole $$\mathbb {R}^N$$
R
N
, and a suitable upper bound behavior of the solutions. To this last end, we use a representation result that allows us to transform the original problem into a new nonlocal problem in a weighted fractional space.
Funder
Università della Calabria
Publisher
Springer Science and Business Media LLC
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