In this paper, we investigate the following nonlinear and non-homogeneous elliptic system:where φ i (t) = a i (|t|)t(i = 1, 2) are two increasing homeomorphisms from R onto R, functions V i (i = 1, 2) and F are 1-periodic in x, and F satisfies some (φ 1 , φ 2 )-superlinear Orlicz-Sobolev conditions. By using a variant mountain pass lemma, we obtain that the system has a ground state. MSC: 35J20; 35J50; 35J55; 35A15

In this paper, we investigate the following nonlinear and non-homogeneous elliptic systemwhere Ω is a bounded domain in R N (N 2) with smooth boundary ∂Ω, functions φ i (t)t (i = 1, 2) are increasing homeomorphisms from R + onto R + . When F satisfies some (φ 1 , φ 2 )-superlinear and subcritical growth conditions at infinity, by using the mountain pass theorem we obtain that system has a nontrivial solution, and when F satisfies an additional symmetric condition, by using the symmetric mountain pass theorem, we obtain that system has infinitely many solutions. Some of our results extend and improve those corresponding results in Carvalho et al. [M. L. M. Carvalho,