Lipschitz regularity for fractional $p$-Laplacian with coercive gradients

Abstract

In this article, we study nonlinear nonlocal equations with coercive gradient nonlinearity of the form \[ (-\Delta_p)^s u(x) + H(x, \nabla u) = f, \] where $f$ is Lipschitz continuous. We show that any viscosity solution $u$ is locally Lipschitz continuous, provided \[ p \in \left(1, \frac{2}{1-s}\right) \cup (1, m+1). \] We also establish H\"older continuity of subsolutions. Furthermore, in the case $f=0$ and $H$ is independent of $x$, we prove that the equation admits only the trivial solution in the class of bounded solutions, for all $m, p \in (1,\infty)$.