DOI: 10.1515/gmj-2026-3049 ISSN: 1072-947X
Existence and multiplicity of solutions for nonlocal fourth-order
p
(
x
)-Kirchhoff problems with Leray--Lions operators, Hardy potentials, and logarithmic
Salah Boulaaras Abstract
This paper is devoted to the existence and multiplicity of weak solutions
for a class of nonlocal fourth-order
p
(
x
)
{p(x)}
-Kirchhoff problems involving
a Leray–Lions type operator, Hardy-type singular potentials,
and logarithmic nonlinear reaction terms in the framework of variable
exponent Sobolev spaces.
More precisely, we investigate the boundary value problem
geändert
{
M
(
∫
Ω
A
(
x
,
Δ
u
)
𝑑
x
)
Δ
(
a
(
x
,
Δ
u
)
)
+
θ
(
x
)
|
u
|
s
(
x
)
-
2
u
|
x
|
2
s
(
x
)
+
c
(
x
)
|
u
|
q
(
x
)
-
2
u
=
λ
f
(
x
,
u
)
+
μ
g
(
x
,
u
)
+
β
|
u
|
r
(
x
)
-
2
u
ln
(
1
+
|
u
|
)
,
in
Ω
,
u
=
Δ
u
=
0
,
on
∂
Ω
,
see text
\left\{\begin{aligned} &\displaystyle M\Biggl{(}\int_{\Omega}A(x,\Delta u)\,dx%
\Biggr{)}\Delta(a(x,\Delta u))+\frac{\theta(x)|u|^{s(x)-2}u}{|x|^{2s(x)}}+c(x)%
|u|^{q(x)-2}u\\
&\displaystyle=\lambda f(x,u)+\mu g(x,u)+\beta|u|^{r(x)-2}u\ln(1+|u|),&&%
\displaystyle\phantom{}\text{in }\Omega,\\
\displaystyle\quad u&\displaystyle=\Delta u=0,&&\displaystyle\phantom{}\text{%
on }\partial\Omega,\end{aligned}\right.
where
Ω
⊂
ℝ
N
{\Omega\subset\mathbb{R}^{N}}
(
N
≥
2
{N\geq 2}
) is a bounded domain with
smooth boundary,
M
is a nonlocal Kirchhoff function,
a
(
x
,
⋅
)
{a(x,\cdot\,)}
is a Leray–Lions type operator satisfying suitable
structural assumptions,
and the singular weighted term
|
u
|
s
(
x
)
-
2
u
|
x
|
2
s
(
x
)
{\frac{|u|^{s(x)-2}u}{|x|^{2s(x)}}}
represents a Hardy-type potential.
The additional logarithmic perturbation
β
|
u
|
r
(
x
)
-
2
u
ln
(
1
+
|
u
|
)
{\beta|u|^{r(x)-2}u\ln(1+|u|)}
models a broad class of nonlinear reaction mechanisms exhibiting
intermediate growth between polynomial and super-polynomial
nonlinearities.
The nonlinearities
f
and
g
satisfy suitable Carathéodory
conditions, while
λ
>
0
{\lambda>0}
,
μ
≥
0
{\mu\geq 0}
, and
β
≥
0
{\beta\geq 0}
are real parameters.
Using variational methods, compact embedding theorems in variable
exponent Sobolev spaces, Hardy–Rellich inequalities,
monotonicity techniques associated with Leray–Lions operators,
and the Three Critical Points Theorems of
Bonanno–Marano and Bonanno–Candito,
we establish the existence of at least three distinct weak solutions.
The presence of the logarithmic reaction term requires a refined
variational analysis owing to its nonhomogeneous growth and the
delicate compactness properties arising in variable exponent settings.
The obtained results significantly generalize and unify several
existing multiplicity theorems for fourth-order
p
(
x
)
{p(x)}
-Kirchhoff and
p
(
x
)
{p(x)}
-biharmonic problems by simultaneously
incorporating nonlocal Kirchhoff effects,
Leray–Lions structures,
Hardy-type singular potentials,
variable exponent logarithmic nonlinearities,
and additional nonlinear perturbations within a unified
variational framework.
To the best of our knowledge,
this paper provides one of the first multiplicity results for
nonlocal fourth-order variable exponent Kirchhoff problems combining
all these features in a single mathematical model.