  This paper proves the existence of nontrivial solution for two classes of
quasilinear systems of the type \begin{equation*} \left\{\; \begin{aligned}
-\Delta_{\Phi_{1}} u&=F_u(x,u,v)+\lambda R_u(x,u,v)\;\text{ in } \Omega& \\
-\Delta_{\Phi_{2}} v&=-F_v(x,u,v)-\lambda R_v(x,u,v)\;\text{ in } \Omega& \\
u=v&=0\;\text{ on } \partial\Omega& \end{aligned} \right. \end{equation*} where
$\lambda > 0$ is a parameter, $\Omega$ is a bounded domain in $\mathbb{R}^N$($N
\geq 2$) with smooth boundary $\partial \Omega$. The first class we drop the
$\Delta_2$-condition of the functions $\tilde{\Phi}_i$($i=1,2$) and assume that
$F$ has a double criticality. For this class, we use a linking theorem without
the Palais-Smale condition for locally Lipschitz functionals combined with a
concentration-compactness lemma for nonreflexive Orlicz-Sobolev space. The
second class, we relax the $\Delta_2$-condition of the functions
${\Phi}_i$($i=1,2$). For this class, we consider $F=0$ and $\lambda=1$ and
obtain the proof based on a saddle-point theorem of Rabinowitz without the
Palais-Smale condition for functionals Frechet differentiable combined with
some properties of the weak$^*$ topology.
