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 \[ (-Δ_{p})^{s}u(x)+H(x,\nabla u)=f , \] where $f$ is Lipschitz continuous and the Hamiltonian $H$ satisfies the coercivity condition \[ H(x,ξ)\;\geq\;θ|ξ|^{m}-θ_{1} \qquad\text{for all } x,ξ\in\mathbb{R}^n, \] for some constants $\theta ,{\theta }_1>0$ and $m>1$. Under an additional local Lipschitz assumption on $H$, we show that every viscosity solution $u$ is locally Lipschitz continuous provided that \[ \text{either}\; sp>p-2\; \text{or}\; 1<p<m+1. \] We also establish Hölder and Lipschitz continuity of subsolutions, for which boundedness of $f$ alone suffices. Furthermore, when $f=0$ and $H$ is independent of $x$, we prove that the equation admits only the trivial solution within the class of bounded solutions, for all $m,p\in (1,\infty )$.