Results on entire solutions for a degenerate critical elliptic equation with anisotropic coefficients  

作　　者：CHEN ShaoWei LIN LiShan 

机构地区：[1]School of Mathematical Sciences, Capital Normal University, Beijing 100048, China [2]institute of Applied Mathematics, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100190, China

出　　处：《Science China Mathematics》2011年第2期221-242,共22页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant Nos. 10901112, 11001255);Beijing Natural Science Foundation (Grant No. 1102013);China Postdoctoral Science Foundation (Grant No. 20090460548)

摘　　要：In this paper, we study the following degenerate critical elliptic equation with anisotropic coefficients-div(|x N | 2α▽u) = K(x)|x N | α·2 * (s)-s |u| 2 * (s)-2 u in R N ,where x = (x 1 , . . . , x N ) ∈ R N , N≥3, α > 1/2, 0≤ s ≤2 and 2 * (s) = 2(N-s)/(N-2). Some basic properties of the degenerate elliptic operator -div(|x N |2α▽u) are investigated and some regularity, symmetry and uniqueness results for entire solutions of this equation are obtained. We also get some variational identities for solutions of this equation. As consequences, we obtain some nonexistence results for this equation.In this paper, we study the following degenerate critical elliptic equation with anisotropic coefficients-div（｜x N ｜ 2α▽u） = K（x）｜x N ｜ α·2 ＊ （s）-s ｜u｜ 2 ＊ （s）-2 u in R N ,where x = （x 1 , . . . , x N ） ∈ R N , N≥3, α 〉 1/2, 0≤ s ≤2 and 2 ＊ （s） = 2（N-s）/（N-2）. Some basic properties of the degenerate elliptic operator -div（｜x N ｜2α▽u） are investigated and some regularity, symmetry and uniqueness results for entire solutions of this equation are obtained. We also get some variational identities for solutions of this equation. As consequences, we obtain some nonexistence results for this equation.

关 键 词：weighted Sobolev inequalities Harnack inequality moving sphere method variational identities 

分 类 号：O175.25[理学—数学] TU43[理学—基础数学]