Towards a deterministic KPZ equation with fractional diffusion: The stationary problem

TL;DR

This paper investigates the existence of solutions to a fractional quasilinear PDE with gradient nonlinearity, analyzing different regimes based on the relation between p and 2s, and considering various conditions on the data.

Contribution

It provides a comprehensive existence analysis for the fractional quasilinear problem across subcritical, critical, and supercritical cases, extending previous results to fractional diffusion.

Findings

Existence results in subcritical case p<2s

Existence in the critical case p=2s

Analysis of the supercritical case p>2s

Abstract

In this work we analyze the existence of solution to the fractional quasilinear problem, \begin{equation*} \left\{ \begin{array}{rcll} (-\Delta)^s u &= & |\nabla u|^{p}+ \l f & \text{ in }\Omega , u &=& 0 &\hbox{ in } \mathbb{R}^N\setminus\Omega, u&>&0 &\hbox{ in }\Omega, \end{array}% \right. \end{equation*}% where $\Omega \subset \ren$ is a bounded regular domain ($\mathcal{C}^2$ is sufficient), $s\in (\frac 12, 1)$, $1<p$ and $f$ is a measurable nonnegative function with suitable hypotheses. The analysis is done separately in three cases, subcritical, $1<p<2s$, critical, $p=2s$, and supercritical, $p>2s$.