Structure And Properties Of Positive Solutions For An Integral Equation On Half Space

In this paper,we consider positive solutions of an integral equation on half space whereτ= is the reflection of the point x about the plane Rn-1.We prove that every positive solution u(x)of(0-1)is rotationally symmetric about some line parallel to xn-axis and the regularity of positive solutions. Our main results are the following theorems. Theorem 3.1 Every positive solution u(x)of(0-1)is rotationally symmetric aboutsome line parallel to xn-axis. Theorem 3.2 Let u(x)be a solution of(0-1). Assume that i(x)∈L2n/n-a(R+n).Then u(x)is in Lp(R+n)∩L∞(R+n),for any 1<p<∞.Hence u is continuous. Theorem 3.3 Let u(x)be a smooth solution ofThen u(x)satisfiesIn previous works,people pay attention to symmetric results on higher order equations and integral equations on whole space.We use a new method of moving planes introduced by Chen-Li-Ou to study the symmetric properties of solutions of(0-1)in a domain with a curvesurface boundary.This result is new.Moreover,the relation between smooth solutions to a higher order PDE on half space and an integral equation on half space is also studied.The integral form of method of moving planes is introduced by Chen,Li and Ou[3], which is different from the traditional method of moving planes for partial differential equations. The method of moving planes and its variant-the method of moving spheres are powerful tools in establishing symmetries and monotonicity of solutions of partial dif-ferential equations. They can also be used to obtain a priori estimates, to derive useful inequalities, and to prove non-existence of solutions. The traditional method of moving planes for partial differential equations relies on some local properties of the solutions, the integral form of moving planes does not rely on the local properties of the solutions. Our proofs depend on regularity of the solutions, the extremum principle of integral inequalities, the Hardy-Littlewood-Sobolev inequality.