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

In this article, we study nonlinear nonlocal equations with coercive gradient nonlinearity of the form $$(-{\mathrm{\Delta }}_p{)}^{s}u(x)+H(x,\mathrm{\nabla }u)=f,$$ where $f$ is Lipschitz continuous. We show that any viscosity solution $u$ is locally Lipschitz continuous, provided $$p\in (1,\frac{2}{1-s})\cup (1,m+1)\mathrm{.}$$ We also establish Hölder 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 $m,p\in (1,\mathrm{\infty })$.