AbstractThis paper is mainly concerned with the following semi-linear system involving the fractional Laplacian: $$ \textstyle\begin{cases} (-\Delta )^{\frac{\alpha }{2}}u(x)= (\frac{1}{ \vert \cdot \vert ^{\sigma }} \ast v^{p_{1}} )v^{p_{2}}(x), \quad x\in \mathbb{R}^{n}, \\ (-\Delta )^{\frac{\alpha }{2}}v(x)= (\frac{1}{ \vert \cdot \vert ^{\sigma }} \ast u^{q_{1}} )u^{q_{2}}(x), \quad x\in \mathbb{R}^{n}, \\ u(x)\geq 0,\quad\quad v(x)\geq 0, \quad x\in \mathbb{R}^{n}, \end{cases} $$
{
(
−
Δ
)
α
2
u
(
x
)
=
(
1
|
⋅
|
σ
∗
v
p
1
)
v
p
2
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x
)
,
x
∈
R
n
,
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−
Δ
)
α
2
v
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x
)
=
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1
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⋅
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σ
∗
u
q
1
)
u
q
2
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x
)
,
x
∈
R
n
,
u
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x
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≥
0
,
v
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x
)
≥
0
,
x
∈
R
n
,
where $0<\alpha \leq 2$
0
<
α
≤
2
, $n\geq 2$
n
≥
2
, $0<\sigma <n$
0
<
σ
<
n
, and $0< p_{1}, q_{1}\leq \frac{2n-\sigma }{n-\alpha }$
0
<
p
1
,
q
1
≤
2
n
−
σ
n
−
α
, $0< p_{2}, q_{2}\leq \frac{n+\alpha -\sigma }{n-\alpha }$
0
<
p
2
,
q
2
≤
n
+
α
−
σ
n
−
α
. Applying a variant (for nonlocal nonlinearity) of the direct method of moving spheres for fractional Laplacians, which was developed by W. Chen, Y. Li, and R. Zhang (J. Funct. Anal. 272(10):4131–4157, 2017), we derive the explicit forms for positive solution $(u,v)$
(
u
,
v
)
in the critical case and nonexistence of positive solutions in the subcritical cases.
Classification of nonnegative solutions to an equation involving the Laplacian of arbitrary order
