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 $\Omega \subset {\mathbb{R}}^N$ is a smooth bounded domain, $N>3$, ${\Delta }_{p(x)}^{3}u=\mathrm{div}(\Delta (|\nabla \Delta u{|}^{p(x)-2}\nabla \Delta u))$ is the $p(x)$-triharmonic operator, $p,q,r\in C(\overline{\Omega })$, $1<p(x)<\frac{N}{3}$ for all $x\in \overline{\Omega }$, $M(s)=a-b{s}^{\gamma }$, $a,b,\gamma >0$, $\lambda >0$, $g:\Omega \times \mathbb{R}\to \mathbb{R}$ 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.

References in corpus (4)

• A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids
• On a $p(\cdot )$-biharmonic problem with no-flux boundary condition
• Existence of solutions for systems arising in electromagnetism
• Multiplicity of solutions for a class of fractional $p(x,\cdot )$-Kirchhoff type problems without the Ambrosetti-Rabinowitz condition