A priori estimates and existence of positive solutions of nonlinear cooperative elliptic systems

We study the existence of positive solutions for elliptic systems of the form (1) below, when $f$ has asymptotic behavior at $+\infty$ as $u^\sigma$ and $g$ satisfies some subcritical growth. Our result guarantees a nontrivial solution $(u,v)$ to the system $$ -\Delta u = u^\sigma + v^q,\;\;\; -\Delta v = u^p \quad \text {in} \quad \Omega; \quad u=v=0 \quad \text {on} \quad \partial \Omega, $$ where $1 <pq <\sigma < (N+2)/(N-2).$