Deterministic KPZ-type equations with nonlocal "gradient terms"

Abstract

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 $Ω\subset \mathbb{R}^N$, $N \geq 2$, be a bounded domain with boundary $\partial Ω$ of class $C^2$. For $s \in (0,1)$, we consider problems of the form \[ \tag{KPZ} \left\{ \begin{aligned} (-Δ)^s u & = μ(x) |\mathbb{D}(u)|^q + λf(x), \quad && \mbox{ in } Ω,\\ u & = 0, && \mbox{ in } \mathbb{R}^N \setminus Ω, \end{aligned} \right. \] where $q > 1$ and $λ> 0$ are real parameters, $f$ belongs to a suitable Lebesgue space, $μ$ belongs to $L^{\infty}(Ω)$ and $\mathbb{D}$ 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 [4, Section 6] and [2, Section 7].