Fractional Elliptic Systems with Nonlinearities of Arbitrary Growth

In this paper we discuss the existence, uniqueness and regularity of solutions of the following system of coupled semilinear Poisson equations on a smooth bounded domain $\mathrm{\Omega }$ in ${\mathbb{R}}^n$: \left\{{llll} \mathcal{A}^s u= v^p & {\rm in} \ \ \Omega \mathcal{A}^s v = f(u) & {\rm in} \ \ \Omega u= v=0 & {\rm on} \ \ \partial\Omega \right. 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 $f(r)=r{e}^r$); 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 $$\frac{1}{p+1}+\frac{1}{q+1}=\frac{n-2s}{n}$$ whenever $n>2s$. For such weak solutions, we prove an $L^{\mathrm{\infty }}$ estimate of Brezis-Kato type and derive the regularity property of the weak solutions.