On a nonlinear partial differential equation with a \(p\left( .\right) \)-triharmonic operator

We study the following \(p\left( .\right) \)-triharmonic problem

$$\begin{aligned} \left\{ \begin{array}{cc} \Delta _{p(.)}^{3}u+a(x)\left| u\right| ^{p(x)-2}u=\lambda (V_{1}(x)\left| u\right| ^{q(x)-2}u-V_{2}(x)\left| u\right| ^{\alpha (x)-2}u), & \text {in }\Omega \\ \left| \nabla \Delta u\right| ^{p(x)-2}\frac{\partial u}{\partial \upsilon }+\beta (x)\left| u\right| ^{p(x)-2}u=0, & \text {on } \partial \Omega ,\end{array} \right. \end{aligned}$$

where \(\Omega \) is a smooth bounded domain in \( \mathbb {R} ^N\), and \(\lambda >0\) is a parameter. Using some variational methods and compact embedding results for variable exponent third-order Sobolev space, we obtain the existence of weak solutions for the problem.