Fractional KPZ system with nonlocal "gradient terms"

In the present work we study the existence and non-existence of nonnegative solutions to a class of deterministic KPZ system with nonlocal gradient term. More precisely we will consider the system$ \begin{equation*} \left\{ \begin{array}{rcll} (-\Delta)^{s} u & = &|\mathbb{D}_{s} v|^q + \rho f\,, & \quad {\rm{in }}\; \Omega,\\ (-\Delta)^{s} v & = & |\mathbb{D}_{s} u|^p + \tau g\,, & \quad {\rm{in }}\; \Omega,\\ u = v& = & 0 &\quad {\text{in }} \mathbb{R}^N \setminus \Omega \end{array} \right. \end{equation*} $where $ \Omega $ is a bounded regular ($ C^2 $) domain of $ \mathbb{R}^N $ and $ p, q\ge 1 $. $ f,g $ are nonnegative measurable functions satisfying some additional hypotheses and $ \rho, \tau \ge 0 $.Here $ \mathbb{D}_{s} $ represents a nonlocal 'gradient term' that will be specified below. In some particular cases, we are able to show the optimality of the condition imposed on the data $ f,g $ and $ \rho,\tau $.