Abstract
This paper investigates a class of nonlocal variable-order fractional $p(\cdot)$-Kirchhoff type equations with nonsmooth nonlinearities
{
M
(
∫
R
2
N
|
u
(
x
)
−
u
(
y
)
|
p
(
x
,
y
)
p
(
x
,
y
)
|
x
−
y
|
N
+
s
(
x
,
y
)
p
(
x
,
y
)
d
x
d
y
)
(
−
Δ
)
p
(
⋅
)
s
(
⋅
)
u
(
x
)
∈
∂
F
(
x
,
u
)
in
Ω
,
u
=
0
,
on
R
N
∖
Ω
,
where
Ω
⊂
R
N
is a bounded smooth domain, and
∂
F
(
x
,
u
)
is the partial generalized gradient of
F
(
x
,
⋅
)
at the point
u
. Using nonsmooth critical point theory, Krasnoselskii genus theory and variational methods, we establish the existence of multiple nontrivial solutions under suitable conditions on the functions
p
,
s
,
M
and
F
. Our findings extend the scope of degenerate
p
(
⋅
)
fractional cases to encompass nonsmooth scenarios.