In this paper, we consider the existence and uniqueness of positive solutions for the following problemwhenΩis a smooth bounded domain in R~N with 0∈Ω, d is a positive constant,γis the unit outer normal to (?)Ω, and a(x)∈C~0(Ω|-), a(x)≥v > 0, a(0)=1 , 1 < p≤(N+2)/(N-2), the Laplacian of u is .In this paper, we have proved the uniqueness of (1.1) by the proposition and corresponding scale and Pohozaev identity.The organization of this paper is as follows:In section 1, as the introduction, we list the research situation and major results in this paper forΩ= B, a(x) = 1.In section 2, when 1 < p≤(N+2)/(N-2),Ωis a bounded domain in R~N, we obtain a positive solution of (1.1).In section 3, we prove the proposition which u_d has at most one local maximum inΩ|- and it is attained exactly at one point which must lie on the boundary, provided that d is sufficiently small.In section 4, when 1 < p < (N+2)/(N-2),Ω= B, we obtain the uniqueness of (1.1).In section 5, when p = (N+2)/(N-2),Ω= B, we obtain the uniqueness of one-bubbling solutions for (1.1).
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