DOI: 10.2478/mjpaa-2023-0028 ISSN: 2351-8227
p(x)-Kirchhoff bi-nonlocal elliptic problem driven by both p(x)-Laplacian and p(x)-Biharmonic operators
Mohsine Jennane, My Driss Morchid Alaoui - Applied Mathematics
- Control and Optimization
- Numerical Analysis
- Analysis
Abstract
We investigate the existence of non-trivial weak solutions for the following p(x)-Kirchhoff bi-nonlocal elliptic problem driven by both p(x)-Laplacian and p(x)-Biharmonic operators
{
M
(
σ
)
(
Δ
p
(
x
)
2
u
-
Δ
p
(
x
)
u
)
=
λ
ϑ
(
x
)
|
u
|
q
(
x
)
-
2
u
(
∫
Ω
ϑ
(
x
)
q
(
x
)
|
u
|
q
(
x
)
d
x
)
r
in
Ω
,
u
∈
W
2
,
p
(
.
)
(
Ω
)
∩
W
0
1
,
p
(
.
)
(
Ω
)
,
\left\{ {\matrix{ {M\left( \sigma \right)\left( {\Delta _{p\left( x \right)}^2u - {\Delta _{p\left( x \right)}}u} \right) = \lambda \vartheta \left( x \right){{\left| u \right|}^{q\left( x \right) - 2}}u{{\left( {\int_\Omega {{{\vartheta \left( x \right)} \over {q\left( x \right)}}{{\left| u \right|}^{q\left( x \right)}}dx} } \right)}^r}\,{\rm{in}}\,\Omega ,} \hfill \cr {u \in {W^{2,p\left( . \right)}}\left( \Omega \right) \cap W_0^{1,p\left( . \right)}\left( \Omega \right),} \hfill \cr } } \right.
under some suitable conditions on the continuous functions
p,
q, the non-negative function
ϑ and
M(
σ), where
σ
:
=
∫
Ω
|
Δ
u
|
p
(
x
)
p
(
x
)
+
|
∇
u
|
p
(
x
)
p
(
x
)
d
x
.
\sigma : = \int_\Omega {{{{{\left| {\Delta u} \right|}^{p\left( x \right)}}} \over {p\left( x \right)}} + {{{{\left| {\nabla u} \right|}^{p\left( x \right)}}} \over {p\left( x \right)}}dx.}
Our main results is obtained by employing variational techniques and the well-known symmetric mountain pass lemma.