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 where f2 is a bounded regular (C2) domain of RN and p, q > 1. f, g are nonnegative measurable functions satisfying some additional hypotheses and rho, tau > 0.Here Ds 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.