Some Qualitative Analysis For Solutions Of A Critical Nonlocal Equation

The aim of this paper is to classify the solutions of the weighted critical nonlocal equation(?) where N ? 3,0<?<N,?>0,0<2?+??N and 2-2?+?/N<p<2?,?*with 2?,?*=(2M-2?-?)/(N-2).The critical exponent 2?,?*is due to the weighted Hardy-Littlewood-Sobolev inequality and Sobolev imbedding.This thesis contains 7 parts.In chapter 1,we introduce the research background.In chapter 2,we use a nonlocal version of Concentration-Compactness principle to prove the existence of the positive ground state solutions.In chapter 3,we prove the integrability of solutions can be lifted frol L2*(RN)when the exponents ? and ? vary in suitable ranges by the regularity lifting lemma and the integrability of solutions can be lifited to L?(RN)and C?(RN-{0})under certain conditions.In chapter 4 and 5,we prove the symmetry and decaying property of positive solutions by the moving plane method in integral forms.Particularly,when a=0,we consider the equation-?u=(I?*u2?*)u2?*-1,x ? RN,where 2?*=2N-?/N-2 and I? is the Riesz potential defined by I?(x)=?(?/2)/?(N-?/2)?N/2 2N-?|x|?with ?(s)=?0-? xs-1 e-x dx,s>0.In chapter 6,we prove the positive solutions of equation assume the form c(t/t2+|x-x0|2)N-2/2 about some x0 with positive c and t.In chapter 7,for N equal to 3 or 4,we prove the unique solutions are non-degenerate when ?? N.