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].
Categories: math.AP