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

TL;DR

This paper establishes the existence of multiple solutions for a complex sixth-order $p(x)$-Kirchhoff problem involving sign-changing weights, expanding the understanding of such high-order nonlinear PDEs with variable exponents.

Contribution

It is one of the first studies addressing sixth-order $p(x)$-Kirchhoff problems with sign-changing functions, introducing new methods for such nonlinear PDEs.

Findings

Multiple solutions are proven to exist for the problem.

The problem involves sign-changing weight functions and variable exponents.

This work extends the theory of high-order nonlinear PDEs with nonstandard growth.

Abstract

We prove the existence of multiple solutions for the 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 \ \ \mbox{on} \ \partial\Omega,$ where $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain, $N > 3$, $\Delta_{p(x)}^3u = \operatorname{div}\Big(\Delta(|\nabla \Delta u|^{p(x)-2}\nabla \Delta u)\Big)$ is the $p(x)$-triharmonic operator, $p,q,r \in C(\overline\Omega)$, $1< p(x) < \frac N3$ for all $x\in \overline\Omega$, $M(s) = a - bs^\gamma$, $a,b,\gamma>0$, $\lambda>0$, $g: \Omega \times \mathbb{R} \to \mathbb{R}$ is a nonnegative continuous function while $f,h : \Omega \times \mathbb{R} \to \mathbb{R}$ are sign-changing continuous functions in $\Omega$. 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.