fractional elliptic systems with nonlinearities of arbitrary growth

In this article we discuss the existence, uniqueness and regularity of solutions of the following system of coupled semilinear Poisson equations on a smooth bounded domain $\Omega$ in $\mathbb{R}^n$: $$\displaylines{ \mathcal{A}^s u= v^p \quad\text{in }\Omega\cr \mathcal{A}^s v = f(u) \quad\text{in }\Omega\cr u= v=0 \quad\text{on }\partial\Omega }$$ where $s\in (0, 1)$ and $\mathcal{A}^s$ denote spectral fractional Laplace operators. We assume that $1< p<\frac{2s}{n-2s}$, and the function f is superlinear and with no growth restriction (for example $f(r)=re^r$); thus the system has a nontrivial solution. Another important example is given by $f(r)=r^q$. In this case, we prove that such a system admits at least one positive solution for a certain set of the couple (p,q) below the critical hyperbola