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 \(\Omega \subset \mathbb {R}^N\), \(N \ge 2\), be a bounded domain with boundary \(\partial \Omega \) class \(C^2\). For \(s \in (0,1)\), we consider problems the form $$\begin{aligned} \left\{ \begin{aligned} (-\Delta )^s u&= \mu (x)\, |{\mathbb {D}}(u)|^q + \lambda f(x), \quad{} & {} \text { in } ,\\ 0,{} {R}^N\setminus , \end{aligned} \right. \quad (\hbox {KPZ}) \end{aligned}$$where \(q > 1\) \(\lambda 0\) are real parameters, f belongs suitable Lebesgue space, \(\mu L^{\infty }(\Omega )\) \({\mathbb {D}}\) represents nonlocal term”. Depending on size 0\), derive results. In particular, solve several open posed [Abdellaoui Nonlinearity 31(4): 1260-1298 (2018), Section 6] Proc Roy Soc Edinburgh Sect A 150(5): 2682-2718 (2020), 7]