In this paper, we prove the existence of multiple solutions for the following sixth-order p ( x )-Kirchhoff-type problem − M∫ Ω 1p(x)|∇ Δ u|p(x)dxΔ p(x)3u=λ f(x)|u|q(x)− 2u+g(x)|u|r(x)− 2u+h(x)inΩ ,u=Δ u=Δ 2u=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 ⊂ ℝ N is a smooth bounded domain, N> 3,Δ p(x)3u:=div⁡ (Δ (|∇ Δ u|p(x)− 2∇ Δ u)) $\begin{array}{} N \,\,\gt\,\, 3, {\it\Delta}_{p(x)}^3u\,\, : =\,\, \operatorname{div}\Big({\it\Delta}(|\nabla {\it\Delta} u|^{p(x)-2}\nabla {\it\Delta} u)\Big) \end{array}$ 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 − bs γ , 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.

中文翻译：

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具有权变符号函数的新一类六阶非齐次p（x）-Kirchhoff问题

\ operatorname {div} \ Big（{\ it \ Delta}（| \ nabla {\ it \ Delta} u | ^ {p（x）-2} \ nabla {\ it \ Delta} u）\ Big）\ end {array} $是p（x）三谐波算子，p，q，r∈C（Ω），1 <p（x）<N3 $ \ begin {array} {} \ displaystyle \ frac N3 \ end {array } $对于所有x∈Ω，M（s）= a − bsγ，a，b，γ> 0，λ> 0，g：Ω×ℝ→ℝ是一个非负连续函数，而f，h：Ω×ℝ →ℝ是以Ω表示的符号转换连续函数。据我们所知，本文是研究六阶p（x）-Kirchhoff型问题的标志之一，该问题具有符号变化的Kirchhoff函数。Ω×ℝ→ℝ是Ω中的符号转换连续函数。据我们所知，本文是研究六阶p（x）-Kirchhoff型问题的标志之一，该问题具有符号变化的Kirchhoff函数。Ω×ℝ→ℝ是Ω中的符号转换连续函数。据我们所知，本文是研究六阶p（x）-Kirchhoff型问题的标志之一，该问题具有符号变化的Kirchhoff函数。