  In this paper, the semilinear elliptic systems with Dirichlet boundary value
are considered \begin{align} \left\{\begin{array}{ll}
  -\Delta v=f(u) & \mathrm{in}\ \Omega,
  -\Delta u=g(v) & \mathrm{in}\ \Omega, u=0, \ v=0 & \mathrm{on}\
\partial\Omega, \end{array} \right. \end{align}
  We extend the notion of subcritical growth from polynomial growth to
N-function growth. Under N-function growth, nontrivial solutions are obtained
via Orlicz-Sobolev spaces and variational methods. It's also noteworthy that
the nonlinear term $g(v)$ does not have to satisfy the usual
Ambrosetti-Rabinowitz condition. So, in a sense, we enrich recent results of D.
~G. de Figueiredo, J. ~M. do {\'O} and B. ~Ruf [D. ~G. de Figueiredo, J.M. do
{\'O}, B. ~Ruf, An {O}rlicz-space approach to superlinear elliptic systems, J.
Funct. Anal. 224 (2005) 471--496].
