  In this paper, we prove the existence of infinitely many solutions for a
class of quasilinear elliptic $m(x)$-polyharmonic Kirchhoff equations where the
nonlinear function has a quasicritical growth at infinity and without assuming
the Ambrosetti and Rabinowitz type condition. The new aspect consists in
employing the notion of a Schauder basis to verify the geometry of the
symmetric mountain pass theorem. Furthermore, for the case $m(x)\equiv Const$,
we introduce a positive quantity $\lambda_M$ similar to the first eigenvalue of
the $m$-polyharmonic operator to find a mountain pass solution, and also to
discuss the sublinear case under large growth conditions at infinity and at
zero. Our results are an improvement and generalization of the corresponding
results obtained by Colasuonno-Pucci (Nonlinear Analysis: Theory, Methods and
Applications, $2011$) and Bae-Kim (Mathematical Methods in the Applied
Sciences, $2020$).
