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, $$ \left\{ \begin{array}{rcll} (-Δ)^s u &=&λ\dfrac{u}{|x|^{2s}}+ |\nabla u|^{p}+ μf &\inn Ω,\\ u&>&0 & \innΩ,\\ u&=&0 & \inn(\mathbb{R}^N\setminusΩ), \end{array}\right. $$ where $Ω$ is a $C^{1,1}$ bounded domain in $\mathbb{R}^N$, $N> 2s, μ>0$, $\frac{1}{2}<s<1$, and $0<λ<Λ_{N,s}$ 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 $λ=0$. More precisely, If $λ>0$, there exists a critical exponent $p_{+}(λ, s)$ such that for $p> p_{+}(λ,s)$ there is no positive solution. Moreover, $p_{+}(λ,s)$ is optimal in the sense that, if $p<p_{+}(λ,s)$ there exists a positive solution for suitable data and $μ$ sufficiently small.

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