Maximum principles and monotonicity of solutions for fractional p-equations in unbounded domains

Abstract

In this paper, we consider the following non-linear equations in unbounded domains $\Omega$ with exterior Dirichlet condition: \begin{equation*}\begin{cases} (-\Delta)_p^s u(x)=f(u(x)), & x\in\Omega,\\ u(x)>0, &x\in\Omega,\\ u(x)\leq0, &x\in \mathbb{R}^n\setminus \Omega, \end{cases}\end{equation*} where $(-\Delta)_p^s$ is the fractional p-Laplacian defined as \begin{equation} (-\Delta)_p^s u(x)=C_{n,s,p}P.V.\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^{p-2}[u(x)-u(y)]}{|x-y|^{n+s p}}dy \label{0} \end{equation} with $0<s<1$ and $p\geq 2$. We first establish a maximum principle in unbounded domains involving the fractional p-Laplacian by estimating the singular integral in (\ref{0}) along a sequence of approximate maximum points. Then, we obtain the asymptotic behavior of solutions far away from the boundary. Finally, we develop a sliding method for the fractional p-Laplacians and apply it to derive the monotonicity and uniqueness of solutions. There have been similar results for the regular Laplacian \cite{BCN1} and for the fractional Laplacian \cite{DSV}, which are linear operators. Unfortunately, many approaches there no longer work for the fully non-linear fractional p-Laplacian here. To circumvent these difficulties, we introduce several new ideas, which enable us not only to deal with non-linear non-local equations, but also to remarkably weaken the conditions on $f(\cdot)$ and on the domain $\Omega$. We believe that the new methods developed in our paper can be widely applied to many problems in unbounded domains involving non-linear non-local operators.