Qualitative Properties for Positive Solutions of Nonlocal Equations

Abstract

This thesis is devoted to the study of properties for nonnegative solutions to nonlocal problems and integral equations. The main tools we use are the method of moving planes and the method of moving spheres._____________ First, we focus on the nonlocal problems involving fractional p-Laplacian (p 2) in unbounded domains. Without assuming any asymptotic behavior of positive solutions near in nity, we develop narrow region principles in unbounded domains, then using the method of moving planes, we establish the monotonicity of positive solutions.__________ Second, we study the symmetry of positive solutions for nonlinear equations involving fractional Laplacian. In bounded domain, we prove that all positive solutions of fractional equations with Hardy Leray potential are radically symmetric about the origin. Then we consider a nonlocal problem in unbounded cylinders. By using the method of moving planes, we establish the symmetry and monotonicity of positive solutions. Furthermore, we obtain the nonexistence of nonnegative solutions for nonlocal problems in the whole space Rn.___________ Third, we establish a strong maximum principle and a Hopf type lemma for antisymmetric solutions of fractional parabolic equations in unbounded domains. These will become most commonly used basic techniques in the study of monotonicity and symmetry of solutions.________ Finally, we consider general integral systems on a half space and integral equations in bounded domains. Under natural integrability conditions, we obtain a classi cation of positive solutions for an integral system on half space by using a slight variant of the method of moving spheres. Here we removed the global integrability hypothesis on positive solutions by introducing some new ideas. In addition, we study the symmetry and monotonicity of positive solutions to over-determined problems and partially overdetermined problems. The main technique we use is the method of moving planes in an integral form.