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

Abstract

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 $\Omega$ 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 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.