Maximum principles involving the uniformly elliptic nonlocal operator

In this paper, we consider equations involving a uniformly elliptic nonlocal operator A(beta)u(x) - CN,beta P.V. integral(RN)a(x - y)(u(x) - u(y)/vertical bar x - y vertical bar(N+beta) dy, where the function a : R-N bar right arrow R is uniformly bounded and radial decreasing. We establish some maximum principles for Ap in bounded and unbounded domains. Since there is no decay condition in the unbounded domain, we make use of an approximate method to estimate the singular integral to get the maximum principle. As applications of these principles, by carrying out the method of moving planes, we give the monotonicity of solutions to the semilinear equation in the coercive epigraph, which extends the result of Dipierro-Soave-Valdinoci [Math. Ann.2017, 369(3-4): 1283-1326]. Moreover, we obtain the radial symmetry and monotonicity of solutions to the generalized Schrodinger equation in a weaker condition, which is the improvement of the result of Tang [Math. Methods Appl. Sci. 2017, 40(7): 2596-2609]. In addition, the maximum principle also plays an important role in acquiring monotonicity of solutions and a Liouville theorem.