Infinitely Many Weak Solutions for a Neumann Problem Involving \(p(\cdot )\)-Kirchhoff Triharmonic Operator

The following problem with Neumann boundary conditions is the subject of our study in this paper \(\begin{aligned} {\left\{ \begin{array}{ll} &{}M\left[ \int _{\Omega }\displaystyle \frac{1}{p(x)}\left( |\nabla \Delta u|^{p(x)}+a(x)|u|^{p(x)}\right) dx\right] \left( -\Delta ^{3}_{p(\cdot )}+a(x)|u|^{p(x)-2}u\right) = g(x,u) \text{ in } \Omega , \\ &{}\frac{\partial }{\partial \nu }\left( \Delta \left( |\Delta u|^{p(x)-2}\Delta u\right) \right) =\frac{\partial }{\partial \nu }\left( |\Delta u|^{p(x)-2}\Delta u\right) =\frac{\partial u}{\partial \nu }=0\quad \text{ on } \quad \partial \Omega .\end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}& M\left[{\displaystyle {\int }_{\mathrm{\Omega }}\frac{1}{p(x)}\left({|\mathrm{\nabla }\mathrm{\Delta }u|}^{p(x)},+,a,(x),{|u|}^{p(x)}\right)dx}\right]\left(-,{\mathrm{\Delta }}_{p(·)}^3,+,a,(x),{|u|}^{p(x)-2},u\right)=g(x,u)\text{in}\mathrm{\Omega },\hfill \\ \hfill & \frac{\partial }{\partial \nu }\left(\mathrm{\Delta },\left({|\mathrm{\Delta }u|}^{p(x)-2},\mathrm{\Delta },u\right)\right)=\frac{\partial }{\partial \nu }\left({|\mathrm{\Delta }u|}^{p(x)-2},\mathrm{\Delta },u\right)=\frac{\partial u}{\partial \nu }=0\text{on}\partial \mathrm{\Omega }.\hfill \end{array}\right)\end{array}$$ Where \(\Omega \) $\mathrm{\Omega }$ is a bounded domain in \({\mathbb {R}}^{N}\) ${\mathbb{R}}^N$ with smooth boundary \(\partial \Omega \) $\partial \mathrm{\Omega }$ , \(N\ge 4\) $N\ge 4$ . Employing a technical approach based on B. Ricceri’s critical points theorem and the theory of variable exponent Sobolev spaces, we establish conditions that guarantee the existence of a multitude of weak solutions for the problem featuring the \(p(\cdot )\) $p(·)$ -Kirchhoff triharmonic operator.