Fractional KPZ equations with critical growth in the gradient respect to Hardy potential. (December 2020) - Abdellaoui, Boumediene; Peral, Ireneo; Primo, Ana; Soria, Fernando
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.