Properties For Positive Solutions Of Several Elliptic Systems Involving Nonlocal Operators

Nonlocal elliptic equations have an important theoretical and practical significance in the fields of finance,physics,ecology and materials science and so on.In this thesis,we focus on the properties of positive solutions for elliptic equations involving nonlocal operators,including nonexistence of solutions for fractional elliptic equations;symmetry and monotonicity of positive solutions for weighted fractional elliptic equations on an unit ball and the whole space,as well as nonexistence of positive solutions on a half space;existence of positive solutions for weighted fractional elliptic equations with gradient terms on a bounded domain;symmetry and monotonicity of positive solutions for elliptic equations involving fully nonlinear nonlocal operators on the whole space,as well as nonexistence of positive solutions on a half space;symmetry and monotonicity of positive solutions to elliptic system involving fully nonlinear nonlocal operators with gradient terms in bounded and unbounded convex domains.This work is composed of the following five parts:The first part is concerned with the Liouville type theorems of semilinear fractional elliptic equations in the whole space.Applying the method of moving planes,we prove monotonicity of positive bounded solutions for the semilinear fractional elliptic equations without any decay assumption at infinity by constructing a suitable auxiliary function.And then the eigenvalue of the equations and maximum principle are used to obtain nonexistence of positive bounded solutions.In the second part,we concentrate on the analysis of solutions to weighted fractional elliptic equations.First of all,we establish a narrow region principle and a decay at infinity by the iteration method.Based on these results,we conclude the radial symmetry and monotonicity of positive solutions for weighted fractional equations on an unit ball and the whole space by the direct method of moving planes.Then we obtain nonexistence of positive solutions on a half space.In the third part,we investigate existence results of positive viscosity solutions for weighted fractional equations with gradient terms in a bounded domain.It is worth noting that weighted fractional equations with gradient terms is not variational,and consequently we will use the topological method to prove existence of positive viscosity solutions.The main difficulty when using the topological method is how to obtain a priori bounds for the solutions.A priori bounds for the solutions are derived via the direct blow-up and rescaling method.In the fourth part,we give the analysis of solutions to elliptic equations involving fully nonlinear nonlocal operators.Based on a narrow region principle and a decay at infinity,we conclude radial symmetry and monotonicity of positive solutions for equations on an unit ball and the whole space by the direct method of moving planes.Then we obtain nonexistence of positive solutions on a half space.In the last part we study equations involving fully nonlinear nonlocal operators with gradient terms in bounded and unbounded convex domains.Because any narrow region principle we used before fails in a general convex domain which is not strictly convex,we establish a delicate estimate of fully nonlinear nonlocal operators at the minimum point of antisymmetry functions,and prove symmetry and monotonicity of positive solutions for equations involving fully nonlinear nonlocal operators with gradient terms in bounded and unbounded convex domains by the method of moving planes.Furthermore,as applications we give some examples.