Uniqueness and Radial Symmetry of Least Energy Solution for a Semilinear Neumann Problem  

作　　者：Zheng-ping Wang Huan-song Zhou 

机构地区：[1]Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O.Box 71010, Wuhan 430071, China

出　　处：《Acta Mathematicae Applicatae Sinica》2008年第3期473-482,共10页应用数学学报（英文版）

基　　金：the National Natural Science Foundation of China(No.10571174,10631030);Chinese Academy oF Sciences grant KJCX3-SYW-S03.

摘　　要：Consider the following Neumann problem d△u- u ＋ k（x）u^p = 0 and u 〉 0 in B1, δu/δv =0 on OB1,where d 〉 0, B1 is the unit ball in R^N, k（x） = k（｜x｜） ≠ 0 is nonnegative and in C（-↑B1）, 1 〈 p 〈 N＋2/N-2 with N≥ 3. It was shown in [2] that, for any d 〉 0, problem （＊） has no nonconstant radially symmetric least energy solution if k（x） ≡ 1. By an implicit function theorem we prove that there is d0 〉 0 such that （＊） has a unique radially symmetric least energy solution if d 〉 d0, this solution is constant if k（x） ≡ 1 and nonconstant if k（x） ≠ 1. In particular, for k（x） ≡ 1, do can be expressed explicitly.Consider the following Neumann problem d△u- u ＋ k（x）u^p = 0 and u 〉 0 in B1, δu/δv =0 on OB1,where d 〉 0, B1 is the unit ball in R^N, k（x） = k（｜x｜） ≠ 0 is nonnegative and in C（-↑B1）, 1 〈 p 〈 N＋2/N-2 with N≥ 3. It was shown in [2] that, for any d 〉 0, problem （＊） has no nonconstant radially symmetric least energy solution if k（x） ≡ 1. By an implicit function theorem we prove that there is d0 〉 0 such that （＊） has a unique radially symmetric least energy solution if d 〉 d0, this solution is constant if k（x） ≡ 1 and nonconstant if k（x） ≠ 1. In particular, for k（x） ≡ 1, do can be expressed explicitly.

关 键 词：Implicit function theorem least energy solution radial symmetry Neumann problem ELLIPTIC 

分 类 号：O18[理学—数学]