Monotonicity and uniqueness of positive solutions to elliptic fractional p-equations

In this paper, we first obtain a maximum principle for parabolic fractional p -Laplacians. Then combining the maximum principle for parabolic fractional p -Laplacians with the parabolic inequalities, we give a new proof for monotonicity and uniqueness of solutions to elliptic fractional p -equations with gradient terms by the sliding method. In order to overcome the difficulties caused by the gradient term, we introduce some new tricks when applying the sliding method.We believe that parabolic inequalities and the maximum principle introduced here can be applied to many nonlinear nonlocal elliptic equations with gradient terms.