The following system of nonlinear elliptic PDE’s is studied: \(-\Delta u = g(v)\), \(-\Delta v = f(u)\) in an open subset of \({\mathbb R}^n\) with homogeneous boundary conditions. The functions \(f\) and \(g\) are assumed to be nonnegative, nondecreasing, with some Hölder regularity and superlinear or sublinear at \(0\) and infinity.