  The aim of this paper is investigating the existence of one or more weak
solutions of the coupled quasilinear elliptic system of gradient type \[
(P)\qquad \left\{ \begin{array}{ll} - {\rm div} (A(x, u)\vert\nabla u\vert^{p_1
-2} \nabla u) + \frac{1}{p_1}A_u (x, u)\vert\nabla u\vert^{p_1} = G_u(x, u, v)
&\hbox{ in $\Omega$,}\\[5pt] - {\rm div} (B(x, v)\vert\nabla v\vert^{p_2 -2}
\nabla v) +\frac{1}{p_2}B_v(x, v)\vert\nabla v\vert^{p_2} = G_v\left(x, u,
v\right) &\hbox{ in $\Omega$,}\\[5pt] u = v = 0 &\hbox{ on $\partial\Omega$,}
\end{array} \right. \] where $\Omega \subset \mathbb{R}^N$ is an open bounded
domain, $p_1$, $p_2 > 1$ and $A(x,u)$, $B(x,v)$ are
$\mathcal{C}^1$-Carath\'eodory functions on $\Omega \times \mathbb{R}$ with
partial derivatives $A_u(x,u)$, respectively $B_v(x,v)$, while $G_u(x,u,v)$,
$G_v(x,u,v)$ are given Carath\'eodory maps defined on $\Omega \times
\mathbb{R}\times \mathbb{R}$ which are partial derivatives of a function
$G(x,u,v)$. We prove that, even if the coefficients make the variational
approach more difficult, under suitable hypotheses functional $\cal{J}$,
related to problem $(P)$, admits at least one critical point in the ''right''
Banach space $X$. Moreover, if $\cal{J}$ is even, then $(P)$ has infinitely
many weak bounded solutions. The proof, which exploits the interaction between
two different norms, is based on a weak version of the Cerami-Palais-Smale
condition, a ''good'' decomposition of the Banach space $X$ and suitable
generalizations of the Ambrosetti-Rabinowitz Mountain Pass Theorems.
