Study On Properties Of Solutions To Several Types Of Nonlocal Operator Problems

In recent years,due to various applications in physics,biology,finance and other disciplines with anomalous diffusion phenomena,nonlocal problems have attracted more and more attention.The study of the qualitative properties of solutions such as symmetry and monotonicity is an important subject in nonlocal problems.In this thesis,we mainly study the qualitative properties of solutions to some fractional p-Laplacian equations and systems.In addition,we also study the properties of solutions and Hopf lemma of fully nonlinear parabolic problems involving with the fractional Laplacian.This thesis is organized as follows:Chapter 1 briefly introduces the research background and research status of problems and the main work of this paper.In Chapter 2,we study the symmetry and monotonicity of solutions to Hardy type equation and Henon type equation involving the fractional p-Laplacian.The key of the proof is to apply the boundary estimate lemma in the process of moving planes.Furthermore,based on the proved monotonicity,we obtain the nonexistence result of positive solutions for a Henon type problem by a comparison with the first eigenfunction-For the case of general fractional p-Laplacian system,we first establish a boundary estimate lemma and a decay at infinity principle for system then,we use them to prove the symmetry and monotonicity results of positive solutions to several types of fractional p-Laplacian systems.Chapter 3 concerns the properties of solutions of nonlocal operator problems with negative power nonlinearity in Rn.Different from the study of fractional order problems on bounded domains or with Lipschitz nonlinearities,here we need to overcome the difficulties caused by negative power nonlinearity.We first prove the symmetry and monotonicity of the fractional Laplacian problem,then we generalize the results to the nonlinear fractional p-Laplacian problem.Chapter 4 is devoted to the study of fully nonlinear parabolic problems involving with the fractional Laplacian.We prove the radial symmetry and monotonicity results of positive solutions on a ball domain and the whole space.In addition,we also obtain Hopf lemma for both on half space and bounded domain with smooth boundary.