@article{Abdellaoui2023, 
author = {Boumediene Abdellaoui and Kheireddine Biroud and Ana Primo and Fernando Soria and Abdelbadie Younes},
title = {Fractional KPZ equations with fractional gradient term and Hardy potential},
year = {2023},
journal = {Mathematics in Engineering},
volume = {5},
number = {2},
pages = {1-36},
keywords = {fractional elliptic equations, nonlocal gradient term, Hardy potential, stationary Kardar-Parisi-Zhang equations, existence and nonexistence results},
url = {https://www.sciopen.com/article/10.3934/mine.2023042},
doi = {10.3934/mine.2023042},
abstract = {In this work we address the question of existence and non existence of positive solutions to a class of fractional problems with non local gradient term. More precisely, we consider the problem   {(−Δ)su=λu|x|2s+(F(u)(x))p+ρfinΩ,u>0inΩ,u=0in(RN∖Ω),where  Ω⊂RN is a  C1,1 bounded domain,  N>2s,ρ>0,  0<s<1,  1<p<∞ and  0<λ<ΛN,s, the Hardy constant defined below. We assume that  f is a non-negative function with additional hypotheses. Here  F(u) is a nonlocal "gradient" term. In particular, if  F(u)(x)=|(−Δ)s2u(x)|, then we are able to show the existence of a critical exponents  p+(λ,s) such that: 1) if  p>p+(λ,s), there is no positive solution, 2) if  p<p+(λ,s), there exists, at least, a positive supersolution solution for suitable data and  ρ small. Moreover, under additional restriction on  p, there exists a solution for general datum  f.}
}