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 $\Omega \subset {\mathbb{R}}^N$, $N\ge 2$, be a bounded domain with boundary $\partial \Omega$ 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 $\lambda >0$ are real parameters, $f$ belongs to a suitable Lebesgue space, $\mu$ belongs to $L^{\infty }(\Omega )$ and $\mathbb{D}$ represents a nonlocal "gradient term". Depending on the size of $\lambda >0$, we derive existence and non-existence results. In particular, we solve several open problems posed in [4, Section 6] and [2, Section 7].

Minor changes have been made; to appear in "Annali di Matematica Pura ed Applicata"