The Existence And Uniqueness Of Positive Solutions For Semilinear Elliptic Equation With Neumann Boundary Value Condition

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).