Abstract
AbstractIn this paper, we will focus on following nonlocal quasilinear elliptic system with singular nonlinearities: $$\begin{aligned} (S) \left\{ \begin{array}{ll} (-\Delta )_{p}^{s_1} u =\dfrac{1}{v^{\alpha _1}}+v^{\beta _1}&{} \text { in }\Omega , \\ (-\Delta )_{q}^{s_2} u =\dfrac{1}{u^{\alpha _2}}+u^{\beta _2}&{} \text { in }\Omega , \\ u,v=0 &{} \text { in } ({I\!\!R}^N\setminus \Omega ) , \\ u,v>0 &{} \text { in } \Omega , \end{array} \right. \end{aligned}$$
(
S
)
(
-
Δ
)
p
s
1
u
=
1
v
α
1
+
v
β
1
in
Ω
,
(
-
Δ
)
q
s
2
u
=
1
u
α
2
+
u
β
2
in
Ω
,
u
,
v
=
0
in
(
I
R
N
\
Ω
)
,
u
,
v
>
0
in
Ω
,
where $$\Omega \subset {I\!\!R}^N$$
Ω
⊂
I
R
N
be a smooth bounded domain, $$s_1,\,s_2\in (0,1)$$
s
1
,
s
2
∈
(
0
,
1
)
, $$\alpha _1$$
α
1
, $$\alpha _2$$
α
2
, $$\beta _1$$
β
1
, $$\beta _2$$
β
2
are suitable positive constants, $$(-\Delta )_{p}^{s_1}$$
(
-
Δ
)
p
s
1
and $$(-\Delta )_{q}^{s_2}$$
(
-
Δ
)
q
s
2
are the fractional $$p-\text {Laplacian}$$
p
-
Laplacian
and $$q-\text {Laplacian}$$
q
-
Laplacian
operators. Using approximating arguments, Rabinowitz bifurcation Theorem, and fractional Hardy inequality, we are able to show the existence of positive solution to the above system.
Publisher
Springer Science and Business Media LLC
Reference33 articles.
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