Nontrivial solutions for semilinear elliptic systems via Orlicz-Sobolev theory

In this paper, the semilinear elliptic systems with Dirichlet boundary value are considered \begin{align} \left{\begin{array}{ll} -\Delta v=f(u) & $\mathrm{i}\mathrm{n}$\ \Omega, -\Delta u=g(v) & $\mathrm{i}\mathrm{n}$\ \Omega, u=0, \ v=0 & $\mathrm{o}\mathrm{n}$
\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].