  The aim of this paper is investigating the existence of weak solutions of the
quasilinear elliptic model problem \[ \left\{\begin{array}{lr} - \divg
(A(x,u)\, |\nabla u|^{p-2}\, \nabla u) + \dfrac1p\, A_t(x,u)\, |\nabla u|^p\ =\
f(x,u) & \hbox{in $\Omega$,}\\ u\ = \ 0 & \hbox{on $\partial\Omega$,}
\end{array} \right. \] where $\Omega \subset \R^N$ is a bounded domain, $N\ge
2$, $p > 1$, $A$ is a given function which admits partial derivative $A_t(x,t)
= \frac{\partial A}{\partial t}(x,t)$ and $f$ is asymptotically $p$-linear at
infinity.
  Under suitable hypotheses both at the origin and at infinity, and if
$A(x,\cdot)$ is even while $f(x,\cdot)$ is odd, by using variational tools, a
cohomological index theory and a related pseudo--index argument, we prove a
multiplicity result if $p > N$ in the non--resonant case.
