Deterministic KPZ-type equations with nonlocal “gradient terms”

The main goal of this paper is to prove existence and non-existence results for deterministic Kardar–Parisi–Zhang type equations involving non-local “gradient terms”. More precisely, let Ω⊂ℝ^N , N ≥ 2 , be a bounded domain with boundary ∂Ω of class C^2 . For s ∈ (0,1) , we consider problems of the form { (-Δ )^s u = μ (x) |𝔻(u)|^q + λ f(x), in Ω , u = 0, in ℝ^N∖Ω , . (KPZ) where q > 1 and λ > 0 are real parameters, f belongs to a suitable Lebesgue space, μ∈ L^∞(Ω ) and 𝔻 represents a nonlocal “gradient term”. Depending on the size of λ > 0 , we derive existence and non-existence results. In particular, we solve several open problems posed in [Abdellaoui in Nonlinearity 31(4): 1260-1298 (2018), Section 6] and [Abdellaoui in Proc Roy Soc Edinburgh Sect A 150(5): 2682-2718 (2020), Section 7]