The Properties Of Solutions Of Equations Related To Fractional Laplacian Operators

As a generalization of Laplacian operator,the fractional Laplacian operator and fully nonlinear operator are widely used in the research of partial differential equation.This dissertation mainly includes three aspects: firstly,the properties of solutions to Schr¨odinger systems with fractional Laplacian operators in different regions,we consider the symmetry and monotonicity of the solutions in the ball and the whole space,the monotonicity increasing of the solutions in the paraboloid and the nonexistence of positive solutions in upper half space;secondly,we consider the monotonicity and symmetry of solutions in equations with completely nonlinear operators and the nonlinear term with gradients in bounded convex region and unbounded convex region;finally,we consider the nonexistence of positive solutions for a class of mixed fully nonlinear equations in upper half space.