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

Abstract

We prove the existence of multiple solutions for the following sixth-order $p(x)$-Kirchhoff-type problem: $-M(\int_Ω\frac{1}{p(x)}|\nabla Δu|^{p(x)}dx)Δ^3_{p(x)} u = λf(x)|u|^{q(x)-2}u + g(x)|u|^{r(x)-2}u + h(x) \ \ \mbox{on} \ Ω$ and $ \ u=Δu=Δ^2 u=0 \ \ \mbox{on} \ \partialΩ,$ where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $N > 3$, $Δ_{p(x)}^3u = \operatorname{div}\Big(Δ(|\nabla Δu|^{p(x)-2}\nabla Δu)\Big)$ is the $p(x)$-triharmonic operator, $p,q,r \in C(\overlineΩ)$, $1< p(x) < \frac N3$ for all $x\in \overlineΩ$, $M(s) = a - bs^γ$, $a,b,γ>0$, $λ>0$, $g: Ω\times \mathbb{R} \to \mathbb{R}$ is a nonnegative continuous function while $f,h : Ω\times \mathbb{R} \to \mathbb{R}$ 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.