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In this work we study the existence of positive solution to the fractionalquasilinear problem, $$ \left\{ \begin{array}{rcll} (-\Delta )^s u &=&\lambda\dfrac{u}{|x|^{2s}}+ |\nabla u|^{p}+ \mu f &\inn \Omega,\\ u&>&0 &\inn\Omega,\\ u&=&0 & \inn(\mathbb{R}^N\setminus\Omega), \end{array}\right. $$where $\Omega$ is a $C^{1,1}$ bounded domain in $\mathbb{R}^N$, $N> 2s, \mu>0$,$\frac{1}{2}<s<1$, and $0<\lambda<\Lambda_{N,s}$ is defined in (3) . We assumethat $f$ 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 positivesolution. 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.

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