Abstract
This paper deals with a class of fourth order elliptic equations of Kirchhoff type with variable exponent
Δ
p
(
x
)
2
u
−
M
(
∫
Ω
1
p
(
x
)
|
∇
u
|
p
(
x
)
d
x
)
Δ
p
(
x
)
u
+
|
u
|
p
(
x
)
−
2
u
=
λ
f
(
x
,
u
)
+
μ
g
(
x
,
u
)
in
Ω
,
u
=
Δ
u
=
0
on
∂
Ω
,
$$\begin{array}{}
\left\{\begin{array}{}
\Delta^2_{p(x)}u-M\bigg(\displaystyle\int\limits_\Omega\frac{1}{p(x)}|\nabla u|^{p(x)}\,\text{d} x \bigg)\Delta_{p(x)} u + |u|^{p(x)-2}u = \lambda f(x,u)+\mu g(x,u) \quad \text{ in }\Omega,\\
u=\Delta u = 0 \quad \text{ on } \partial\Omega,
\end{array}\right.
\end{array}$$
where
p
−
:=
inf
x
∈
Ω
¯
p
(
x
)
>
max
1
,
N
2
,
λ
>
0
$\begin{array}{}
\displaystyle
p^{-}:=\inf_{x \in \overline{\Omega}} p(x) \gt \max\left\{1, \frac{N}{2}\right\}, \lambda \gt 0
\end{array}$
and μ ≥ 0 are real numbers, Ω ⊂ ℝ
N
(N ≥ 1) is a smooth bounded domain,
Δ
p
(
x
)
2
u
=
Δ
(
|
Δ
u
|
p
(
x
)
−
2
Δ
u
)
$\begin{array}{}
\displaystyle
\Delta_{p(x)}^2u=\Delta (|\Delta u|^{p(x)-2} \Delta u)
\end{array}$
is the operator of fourth order called the p(x)-biharmonic operator, Δ
p(x)
u = div(|∇u|
p(x)–2∇u) is the p(x)-Laplacian, p : Ω → ℝ is a log-Hölder continuous function, M : [0, +∞) → ℝ is a continuous function and f, g : Ω × ℝ → ℝ are two L
1-Carathéodory functions satisfying some certain conditions. Using two kinds of three critical point theorems, we establish the existence of at least three weak solutions for the problem in an appropriate space of functions.
Vector-Valued Entire Functions of Several Variables: Some Local Properties