On the existence of solutions for a class of fourth order elliptic equations of Kirchhoff type with variable exponent
In this paper, we consider a class of fourth order elliptic equations of Kirchhoff type with variable exponent ( ∆2 p(x) u − M R Ω 1 p(x) |∇u| p(x) dx ∆p(x)u = λf(x, u) in Ω, u = ∆u = 0 on ∂Ω, where Ω ⊂ R N , N ≥ 3, is a smooth bounded domain, M(t) = a + btκ , a, κ > 0, b ≥ 0, λ is...
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| Format: | Article |
| Language: | English |
| Published: |
ATNAA
2019-03-01
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| Series: | Advances in the Theory of Nonlinear Analysis and its Applications |
| Subjects: | |
| Online Access: | https://dergipark.org.tr/en/download/article-file/678135 |
| Summary: | In this paper, we consider a class of fourth order elliptic equations of Kirchhoff type with variable exponent
(
∆2
p(x)
u − M
R
Ω
1
p(x)
|∇u|
p(x) dx
∆p(x)u = λf(x, u) in Ω,
u = ∆u = 0 on ∂Ω,
where Ω ⊂ R
N , N ≥ 3, is a smooth bounded domain, M(t) = a + btκ
, a, κ > 0, b ≥ 0, λ is a positive
parameter, ∆2
p(x)
u = ∆(|∆u|
p(x)−2∆u) is the operator of fourth order called the p(x)-biharmonic operator,
∆p(x)u = div
|∇u|
p(x)−2∇u
is the p(x)-Laplacian, p : Ω → R is a log-Hölder continuous function and
f : Ω × R → R is a continuous function satisfying some certain conditions. Using Ekeland’s variational
principle combined with variational techniques, an existence result is established in an appropriate function
spac |
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| ISSN: | 2587-2648 2587-2648 |