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A new class of multiple nonlocal problems with two parameters and variable-order fractional
p
(
β
)
p(\cdot)
p
(
β
)
-Laplacian
Authors
Mostafa Allaoui
Mohamed Karim Hamdani
Lamine Mbarki
Publication date
9 September 2023
Publisher
View
on
arXiv
Abstract
In the present manuscript, we focus on a novel tri-nonlocal Kirchhoff problem, which involves the
p
(
x
)
p(x)
p
(
x
)
-fractional Laplacian equations of variable order. The problem is stated as follows: \begin{eqnarray*} \left\{ \begin{array}{ll} M\Big(\sigma_{p(x,y)}(u)\Big)(-\Delta)^{s(\cdot)}_{p(\cdot)}u(x) =\lambda |u|^{q(x)-2}u\left(\int_\O\frac{1}{q(x)} |u|^{q(x)}dx \right)^{k_1}+\beta|u|^{r(x)-2}u\left(\int_\O\frac{1}{r(x)} |u|^{r(x)}dx \right)^{k_2} \quad \mbox{in }\Omega, \\ u=0 \quad \mbox{on }\partial\Omega, \end{array} \right. \end{eqnarray*} where the nonlocal term is defined as
Ο
p
(
x
,
y
)
(
u
)
=
β«
Ξ©
Γ
Ξ©
1
p
(
x
,
y
)
β£
u
(
x
)
β
u
(
y
)
β£
p
(
x
,
y
)
β£
x
β
y
β£
N
+
s
(
x
,
y
)
p
(
x
,
y
)
β
d
x
β
d
y
.
\sigma_{p(x,y)}(u)=\int_{\Omega\times \Omega}\frac{1}{p(x,y)}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{N+s(x,y)p(x,y)}} \,dx\,dy.
Ο
p
(
x
,
y
)
β
(
u
)
=
β«
Ξ©
Γ
Ξ©
β
p
(
x
,
y
)
1
β
β£
x
β
y
β£
N
+
s
(
x
,
y
)
p
(
x
,
y
)
β£
u
(
x
)
β
u
(
y
)
β£
p
(
x
,
y
)
β
d
x
d
y
.
Here,
Ξ©
β
R
N
\Omega\subset\mathbb{R}^{N}
Ξ©
β
R
N
represents a bounded smooth domain with at least
N
β₯
2
N\geq2
N
β₯
2
. The function
M
(
s
)
M(s)
M
(
s
)
is given by
M
(
s
)
=
a
β
b
s
Ξ³
M(s) = a - bs^\gamma
M
(
s
)
=
a
β
b
s
Ξ³
, where
a
β₯
0
a\geq 0
a
β₯
0
,
b
>
0
b>0
b
>
0
, and
Ξ³
>
0
\gamma>0
Ξ³
>
0
. The parameters
k
1
k_1
k
1
β
,
k
2
k_2
k
2
β
,
Ξ»
\lambda
Ξ»
and
Ξ²
\beta
Ξ²
are real parameters, while the variables
p
(
x
)
p(x)
p
(
x
)
,
s
(
β
)
s(\cdot)
s
(
β
)
,
q
(
x
)
q(x)
q
(
x
)
, and
r
(
x
)
r(x)
r
(
x
)
are continuous and can change with respect to
x
x
x
. To tackle this problem, we employ some new methods and variational approaches along with two specific methods, namely the Fountain theorem and the symmetric Mountain Pass theorem. By utilizing these techniques, we establish the existence and multiplicity of solutions for this problem separately in two distinct cases: when
a
>
0
a>0
a
>
0
and when
a
=
0
a=0
a
=
0
. To the best of our knowledge, these results are the first contributions to research on the variable-order
p
(
x
)
p(x)
p
(
x
)
-fractional Laplacian operator.Comment: 21 page
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oai:arXiv.org:2309.04879
Last time updated on 06/10/2023