MAXIMUM PRINCIPLES FOR NONLINEAR INTEGRO-DIFFERENTIAL EQUATIONS AND SYMMETRY OF SOLUTIONS

In this paper, we study the semilinear integro-differential equations LKu(x)equivalent to CnP.V.integral(Rn)(u(x)-u(y))K(x-y)dy=f(x,u), and the full nonlinear integro-differential equations F(G,K)u(x)equivalent to CnP.V.integral(Rn)G(u(x)-u(y))K(x-y)dy=f(x,u),, where K(<middle dot>) isa symmetric jumping kernel and K(<middle dot>) > C| <middle dot>|(-n-alpha), G(<middle dot>) is some nonlinear function without non-degenerate condition. We adopt the direct method of moving planes to study the symmetry and monotonicity of solutions for integro-differential equations, and investigate the limit of some non-lo cal operators GK as alpha -+ 2. Our results extend some results obtained in [12] and [14].