Concentration of Solutions for the Scalar Curvature Equation on RN
Concentration of Solutions for the Scalar Curvature Equation on RN
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DOI：
10.1006/jdeq.1999.3718
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发表时间：
2000-05
影响因子：
2.4
通讯作者：
Shusen Yan
中科院分区：
文献类型：
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作者：
Shusen Yan
&2u= K（y）u 2* &1 y# Rn {u> 0，y# RN（1.1）u <$0，as| y|其中2*= 2N（N&2），且N = 3。问题（1.1）来源于SN上的规定数量曲率问题。实际上，在球面投影到SN的赤道平面之后，SN上的规定标量曲率问题变为问题（1.1）。关于标量曲率Eq.（1.1）。例如，见[4，6，7，12 18]。除[4]外，[6，7，12，18]中关于（1.1）的存在性结果都是在对系数K（y）的一些对称性假设下得到的.本文的目的是在K（y）有一列向无穷大移动的严格局部极大值点的条件下，构造（1.1）的无穷多个解。本文构造的解基本上都集中在K（y）的两个极大点上，这两个极大点之间的距离很大。在我们陈述本文的主要结果之前，我们引入一些符号。对于任何x# RN，*# R+，我们设置
&2u= K (y) u 2* &1 y# Rn {u> 0, y# RN(1.1) u Ä0, as| y| Ä+, where 2*= 2NĀ (N&2), and N 3. Problem (1.1) comes from the prescribed scalar curvature problem on SN. Indeed, after stereographic projection to the equatorial plane of SN, the prescribed scalar curvature problem on SN changes to problem (1.1). There are many works on the scalar curvature Eq.(1.1). See for example [4, 6, 7, 12 18]. Except [4], the existence results in [6, 7, 12 18] for (1.1) were obtained under some symmetry assumptions on the coefficient K (y). The aim of this paper is to construct infinitely many solutions for (1.1) under the condition that K (y) has a sequence of strictly local maximum points moving to infinity. Basically speaking, the solutions we construct in this paper concentrate at two local maximum points of K (y), whose distance is very large. Before we state the main results of this paper, we introduce some notation. For any x# RN,*# R+ we set