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 $$\begin{aligned} \left\{ \begin{aligned} (-\Delta )^s u&= \mu (x)\, |{\mathbb {D}}(u)|^q + \lambda f(x), \quad{} & {} \text { in } \Omega ,\\ u&= 0,{} & {} \text { in } \mathbb {R}^N\setminus \Omega , \end{aligned} \right. \quad \quad \quad \quad \quad \quad (\hbox {KPZ}) \end{aligned}$$where \(q > 1\) and \(\lambda > 0\) are real parameters, f belongs to a suitable Lebesgue space, \(\mu \in L^{\infty }(\Omega )\) 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 [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]