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 $\Omega \subset \mathbb{R}^N$, $N \geq 2$, be a bounded domain with boundary $\partial \Omega$ of class $C^2$. For $s \in (0,1)$, we consider problems of the form \[ \tag{KPZ} \left\{ \begin{aligned} (-\Delta)^s u & = \mu(x) |\mathbb{D}(u)|^q + \lambda f(x), \quad && \mbox{ in } \Omega,\\ u & = 0, && \mbox{ in } \mathbb{R}^N \setminus \Omega, \end{aligned} \right. \] 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].