Fractional KPZ equations with critical growth in the gradient respect to Hardy potential

In this work we study the existence of positive solution to the fractional quasilinear problem, $$\{\begin{array}{rcll}(-\mathrm{\Delta }{)}^{s}u& =& \lambda {\displaystyle \frac{u}{\mathrm{\mid }x{\mathrm{\mid }}^{2s}}}+\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^p+\mu f& \text{inn}\mathrm{\Omega },\\ u& >& 0& \text{inn}\mathrm{\Omega },\\ u& =& 0& \text{inn}({\mathbb{R}}^N\setminus \mathrm{\Omega }),\end{array}$$ where $\mathrm{\Omega }$ is a $C^{1,1}$ bounded domain in ${\mathbb{R}}^N$, $N>2s,\mu >0$, $\frac{1}{2}<s<1$, and $0<\lambda <{\mathrm{\Lambda }}_{N,s}$ is defined in (3) . We assume that $f$ is a non-negative function with additional hypotheses. As we will see, there are deep differences with respect to the case $\lambda =0$. More precisely, If $\lambda >0$, there exists a critical exponent $p_{+}(\lambda ,s)$ such that for $p>p_{+}(\lambda ,s)$ there is no positive solution. Moreover, $p_{+}(\lambda ,s)$ is optimal in the sense that, if $p<p_{+}(\lambda ,s)$ there exists a positive solution for suitable data and $\mu$ sufficiently small.