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<μ<N,α>0,0<2α+μ≤N and 2-2α+μ/N<p<2α,μ*with 2α,μ*=(2M-2α-μ)/(N-2).The critical exponent 2α,μ*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 α and μ 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=(Iμ*u2μ*)u2μ*-1,x ∈ RN,where 2μ*=2N-μ/N-2 and Iμ is the Riesz potential defined by Iμ(x)=Γ(μ/2)/Γ(N-μ/2)πN/2 2N-μ|x|μwith Γ(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 μ→ N.