Existence And Property For Nonlinear Elliptic Equation With Weight

In this paper, we will solve the following nonlinear elliptic equation with weight. First we are concerned with properties of positive solutions of the following fractional elliptic systemwhere N≥3,0≤β<α<N, p, q>1and p+q<N+α-β/N-α+β. We show that positive solutions of (1.1) are radially symmetric and belong to L∞(Rn). Moreover, if α=2,β=0,p≤q, we show that positive solution pair (u,v) of (1.1) is unique and u=v=U, where U is the unique positive solution of the problem-Δu+u=up+q in RN. Second, we are concerned with the existence of positive solution of the following nonlinear elliptic involving critical Hardy-Sobolev exponentwhere RN=Rk×RN-k, with2≤k<N,λ>0and x=(y, z)∈Rk x RN-k. For a given real numbers s, such that Suppose that Ω is a C1bounded domain in RN with0∈(?)Ω,(?)Ω is C2at0, and the principal curvature of (?)Ω at0is nonpositive,but do not all vanish. We may also suppose RN-k∩Ω=φ,RN-κ∩(?)Ω≠φ.We can also prove that the following equation has a positive solution if λ>0,1≤p<N/N-2, and the equation has a positive solution if λ>0,1<p<N+2/N-2.