Qualitative Properties Of Solutions To Several Classes Of Hardy-Littlewood-Sobolev-type Integral Equations

This paper is concerned with some qualitative properties of the integral system related to Hardy-Littlewood-Sobolev inequality.In the first chapter,we mainly study the existence/nonexistence of positive solutions of a weighted Hardy-Littlewood-Sobolev type integral system.Such a syst,em is related to the extremal functions of the weighted Hardy-Littlewood-Sobolev inequality.The Serrin-type condition is critical for existence of positive solutions in L_loc~?(R~n\{0}).When the Serrin-type condition does not hold,we prove the nonexistence by an iteration process.In addition,we find three pairs of radial solutions when the Serrin-type condition holds.One is singular,and the other two are integrable in R~n and decaying fast and slowly respectively.Then we also study a weighted integral system which comes from the conformal properties of the reversed HardyLittlewood-Sobolev inequality.We consider the existence of positive entire solutions and present several sufficient conditions of the existence/nonexistence.In the second chapter,we study an integral system(?) Where u_i>0 and K_i(x)(i=1,2,…,m)are double bounded functions.When p_i>0(i=1,2,…,m),if ??(0,n),then the Serrintype condition is critical for existence of positive solutions.We will study optimal integrability intervals if the positive solutions have some other initial integrability integrability.Now,the Sobolev-type critical condition is not necessary,and we apply a weaker condition,the Serrin-type condition,to establish some important relations of exponents which come into play to lift the regularity.In addition,we also generalize this result to the case of m?Z~+.When p_i<0(i=1,2,…,m),if ??(0,n),then the system has no positive solution for any double bounded K_i(x)(i=1,2,…,m),if?>n,and maxi{-p_i}>?/?-n,then the system has positive solutions increasing with the rate a-n.