Monotonicity of Polynomial Growing Solutions for Uniformly Elliptic Fractional Equations

ABSTRACT In this paper, we consider a nonlinear equation involving uniformly elliptic nonlocal operator in the upper half space. We first develop a narrow region principle for antisymmetric functions in unbounded domains, in which we assume that has polynomial growth instead of the usual decay condition at infinity. Then we obtain the monotonicity of positive solutions for uniformly elliptic nonlocal equations in the upper half space by a direct method of moving planes, which extends the monotonic result of the nonlinear fractional elliptic equation. Based on the monotonicity results, we prove the existence of positive bounded solutions for uniformly elliptic nonlocal equations on the whole space. We believe that these methods will also be helpful for equations involving other nonlocal operators.