On the eigenvalues of the <inline-formula><tex-math id="M1">$ p(x)- $</tex-math></inline-formula>triharmonic problem

In this work, we deal with the eigenvalues of the $ p(x) $-triharmonic Navier problem$ \begin{equation*} \begin{gathered} -\Delta^3_{p(x)}u = \lambda |u|^{p(x)-2}u, \quad\quad\quad\text{in }\Omega, \\ \quad\quad \Delta^2u = \Delta u = u = 0, \quad \quad\text{on }\partial \Omega. \end{gathered} \end{equation*} $We establish the existence of infinitely many eigenvalues for the sixth-order problem and investigate their properties. Specifically, we prove that the set of eigenvalues is nonempty and infinite, with the supremum of the eigenvalues being infinity. Additionally, we provide sufficient conditions for the infimum of the eigenvalues to be zero.