This paper proves the existence of nontrivial solution for two classes of quasilinear systems of the type { − ∆ Φ 1 u = F u ( x , u , v )+ λ R u ( x , u , v ) in Ω − ∆ Φ 2 v = − F v ( x , u , v ) − λ R v ( x , u , v ) in Ω u = v = 0 on ∂ Ω where λ>0 is a parameter, Ω is a bounded domain in R N ( N≥2) with smooth boundary ∂Ω. The first class we drop the ∆ 2 -condition of the functions Φ ̵̃ 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 ∆ 2 -condition of the functions Φ i ( i=1 ,2). For this class, we consider F=0 and λ=1 and obtain the proof based on a saddle-point theorem of Rabinowitz without the Palais-Smale condition for functionals Fréchet differentiable combined with some properties of the weak ∗ topology.