Fractional KPZ equations with fractional gradient term and Hardy potential

In this work we address the question of existence and non existence of positive solutions to a class of fractional problems with non local gradient term. More precisely, we consider the problem

$$\{\begin{array}{rcll}(-\mathrm{\Delta }{)}^{s}u& =& \lambda {\displaystyle \frac{u}{|x{|}^{2s}}}+(\mathfrak{F}(u)(x){)}^p+\rho f& \text{in}\mathrm{\Omega },\\ u& >& 0& \text{in}\mathrm{\Omega },\\ u& =& 0& \text{in}({\mathbb{R}}^N\setminus \mathrm{\Omega }),\end{array}$$

where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a $C^{1,1}$ bounded domain, $N>2s,\rho >0$, $0<s<1$, $1<p<\mathrm{\infty }$ and $0<\lambda <{\mathrm{\Lambda }}_{N,s}$, the Hardy constant defined below. We assume that $f$ is a non-negative function with additional hypotheses. Here $\mathfrak{F}(u)$ is a nonlocal "gradient" term. In particular, if $\mathfrak{F}(u)(x)=|(-\mathrm{\Delta }{)}^{\frac{s}{2}}u(x)|$, then we are able to show the existence of a critical exponents $p_{+}(\lambda ,s)$ such that: 1) if $p>p_{+}(\lambda ,s)$, there is no positive solution, 2) if $p<p_{+}(\lambda ,s)$, there exists, at least, a positive supersolution solution for suitable data and $\rho$ small. Moreover, under additional restriction on $p$, there exists a solution for general datum $f$.