# New class of sixth-order nonhomogeneous p(x)-Kirchhoff problems with sign-changing weight functions

Authors
• 1 Tunisia Military School of Aeronautical Specialities, Tunisia , (Tunisia)
• 2 University of Sfax, Faculty of Science of Sfax, Tunisia , (Tunisia)
• 3 Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi , (Vietnam)
• 4 University of Ljubljana, Slovenia , (Slovenia)
• 5 Institute of Mathematics, Physics and Mechanics, Slovenia , (Slovenia)
Type
Published Article
Journal
Publisher
De Gruyter
Publication Date
Mar 24, 2021
Volume
10
Issue
1
Pages
1117–1131
Identifiers
DOI: 10.1515/anona-2020-0172
Source
De Gruyter
Keywords
In this paper, we prove the existence of multiple solutions for the following sixth-order p(x)-Kirchhoff-type problem −M∫Ω1p(x)|∇Δu|p(x)dxΔp(x)3u=λf(x)|u|q(x)−2u+g(x)|u|r(x)−2u+h(x)inΩ,u=Δu=Δ2u=0,on∂Ω, $$\begin{array}{} \displaystyle \begin{cases} -M\left( \int\limits_{\it\Omega} \frac{1}{p(x)}|\nabla {\it\Delta} u|^{p(x)}dx\right){\it\Delta}^3_{p(x)} u = \lambda f(x)|u|^{q(x)-2}u + g(x)|u|^{r(x)-2}u + h(x) &\mbox{in}\quad {\it\Omega}, \\[0.3em] u = {\it\Delta} u = {\it\Delta}^2 u = 0, \quad &\mbox{on}\quad \partial{\it\Omega}, \end{cases} \end{array}$$ where Ω ⊂ ℝN is a smooth bounded domain, N>3,Δp(x)3u:=div⁡(Δ(|∇Δu|p(x)−2∇Δu)) $\begin{array}{} N \,\,\gt\,\, 3, {\it\Delta}_{p(x)}^3u\,\, : =\,\, \operatorname{div}\Big({\it\Delta}(|\nabla {\it\Delta} u|^{p(x)-2}\nabla {\it\Delta} u)\Big) \end{array}$ is the p(x)-triharmonic operator, p, q, r ∈ C(Ω), 1 < p(x) < N3 $\begin{array}{} \displaystyle \frac N3 \end{array}$ for all x ∈ Ω, M(s) = a − bsγ, a, b,γ > 0, λ > 0, g : Ω × ℝ → ℝ is a nonnegative continuous function while f, h : Ω × ℝ → ℝ are sign-changing continuous functions in Ω. To the best of our knowledge, this paper is one of the first contributions to the study of the sixth-order p(x)-Kirchhoff type problems with sign changing Kirchhoff functions.