Abstract
The purpose of the present paper is to study the existence of solutions for the following nonhomogeneous singular Kirchhoff problem involving the
p
(
x
)
-biharmonic operator:
{
M
(
t
)
(
Δ
p
(
x
)
2
u
+
a
(
x
)
|
u
|
p
(
x
)
−
2
u
)
=
g
(
x
)
u
−
γ
(
x
)
∓
λ
f
(
x
,
u
)
,
in
Ω
,
Δ
u
=
u
=
0
,
on
∂
Ω
,
where
Ω
⊂
R
N
,
(
N
≥
3
)
be a bounded domain with
C
2
boundary,
λ
is a positive parameter,
γ
:
Ω
‾
⟶
(
0
,
1
)
be a continuous function,
p
∈
C
(
Ω
‾
)
with
1
<
p
−
:
=
inf
x
∈
Ω
p
(
x
)
≤
p
+
:
=
sup
x
∈
Ω
p
(
x
)
<
N
2
, as usual,
p
∗
(
x
)
=
N
p
(
x
)
N
−
2
p
(
x
)
,
g
∈
L
p
∗
(
x
)
p
∗
(
x
)
+
γ
(
x
)
−
1
(
Ω
)
. We assume that
M
(
t
)
is a continuous function with
t
:
=
∫
Ω
1
p
(
x
)
(
|
Δ
u
|
p
(
x
)
+
a
(
x
)
|
u
|
p
(
x
)
)
d
x
,
and assumed to verify assertions
(M1)
-
(M3)
in Sect.
3
, moreover
f
(
x
,
u
)
are assumed to satisfy assumptions
(f1)
-
(f6)
. In the proofs of our results we use variational techniques and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces.