Fractional KPZ system with nonlocal "gradient terms"

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