In this article we study existence and uniqueness of positive solutions for elliptic systems of the form $$ \begin{align} -\Delta v = &f(x,u) \quad \hbox{in} \quad \Omega \\ -\Delta u = & v^\beta \quad \hbox{in}\quad \Omega, \end{align} $$ with Dirichlet boundary condition on a bounded smooth domain in $\Bbb R^N$. The nonlinearity $f$ is assumed to have a sub-$\beta$ growth with $\beta>0$, that in case $f(x,u)=u^\alpha, \alpha>0$, corresponds to $\alpha\beta<1$. The results are also valid for a larger class of systems, including some infinite dimensional Hamiltonian Systems.

In this paper we generalize and improve some well-known solvability conditions for semilinear elliptic boundary value problems at resonance within the context of Besov and Triebel-Lizorkin spaces. In particular we will show that many such solvability conditions can be viewed as special cases of a single generalized Landesman-Lazer condition. Our methods apply an adaptation of Leray-Schauder degree ideas to quasi-Banach spaces.