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

We prove the existence of multiple solutions for following sixth-order $p(x)$-Kirchhoff-type problem: $-M(\int_\Omega \frac{1}{p(x)}|\nabla \Delta u|^{p(x)}dx)\Delta^3_{p(x)} u = \lambda f(x)|u|^{q(x)-2}u + g(x)|u|^{r(x)-2}u h(x) \ \mbox{on} \Omega$ and $ u=\Delta u=\Delta^2 u=0 \partial\Omega,$ where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $N > 3$, $\Delta_{p(x)}^3u \operatorname{div}\Big(\Delta(|\nabla u|^{p(x)-2}\nabla u)\Big)$ $p(x)$-triharmonic operator, $p,q,r \in C(\overline\Omega)$, $1< p(x) < \frac N3$ all $x\in \overline\Omega$, $M(s) - bs^\gamma$, $a,b,\gamma>0$, $\lambda>0$, $g: \Omega \times \mathbb{R} \to \mathbb{R}$ nonnegative continuous function while $f,h : are sign-changing functions in $\Omega$. To best our knowledge, this paper one first contributions to study $p(x)$-Kirchhoff type problems with sign changing Kirchhoff functions.