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, $$ \left\{ \begin{array}{rcll} (-\Delta )^s u &=&\lambda \dfrac{u}{|x|^{2s}}+ |\nabla u|^{p}+ \mu f &\inn \Omega,\\ u&>&0 & \inn\Omega,\\ u&=&0 & \inn(\mathbb{R}^N\setminus\Omega), \end{array}\right. $$ 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 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.