Fractional elliptic reaction-diffusion systems with coupled gradient terms and different diffusion

In this work, we study the existence and nonexistence of nonnegative solutions to a class of nonlocal elliptic systems set in a bounded open subset of ${\mathbb{R}}^N$. The diffusion operators are of type $u_i\mapsto d_i(-\mathrm{\Delta }{)}^{s_i}{u}_i$ where $0<s_1\mathrm{\ne }{s}_2<1$, and the gradients of the unknowns act as source terms. Existence results are obtained by proving some fine estimates when data belong to weighted Lebesgue spaces. Those estimates are new and interesting in themselves.