Uniqueness and monotonicity of solutions for fractional equations with a gradient term

Abstract

In this paper, we consider the following fractional equation with a gradient term su(x) = f(x, u(x), ∇u(x)), in a bounded domain and the upper half space. Firstly, we prove the monotonicity and uniqueness of solutions to the fractional equation in a bounded domain by the sliding method. In order to obtain maximum principle on unbounded domain, we need to estimate the singular integrals define the fractional Laplacians along a sequence of approximate maximum points by using a generalized average inequality. Then we prove monotonicity and uniqueness of solutions to fractional equation in Rn + by the sliding method. In order to solve the difficulties caused by the gradient term, some new techniques are developed. The paper may be considered as an extension of Berestycki and Nirenberg [J. Geom. Phys. 5(1988), 237–275].