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, ( − Δ ) s u = λ u | x | 2 s + | ∇ u | p + μ f  in  Ω , u > 0  in  Ω , u = 0  in  ( R N ∖ Ω ) , where Ω is a C 1 , 1 bounded domain in R N , N > 2 s , μ > 0 , 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.