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

Abstract

Abstract In this paper, we prove the existence of multiple solutions for the following sixth-order p(x)-Kirchhoff-type problem − M ∫ Ω 1 p ( x ) | ∇ Δ u | p ( x ) d x Δ p ( x ) 3 u = λ f ( x ) | u | q ( x ) − 2 u + g ( x ) | u | r ( x ) − 2 u + h ( x ) in Ω , u = Δ u = Δ 2 u = 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 ) 3 u := 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) < N 3 $\begin{array}{} \displaystyle \frac N3 \end{array}$ 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.