Infinitely many solutions for the prescribed scalar curvature problem on Sᴺ

Title
Infinitely many solutions for the prescribed scalar curvature problem on Sᴺ
Publication Date
2010
Author(s)
Wei, Juncheng
Yan, Shusen
Type of document
Journal Article
Language
en
Entity Type
Publication
Publisher
Elsevier Inc
Place of publication
United States of America
DOI
10.1016/j.jfa.2009.12.008
UNE publication id
une:7648
Abstract
In this paper, we consider the simplest case, i.e., K is rotationally symmetric, K = K(r), r = |y|. It follows from the Pohozaev identity (1.2) that (1.6) has no solution if K'(r) has fixed sign. Thus we assume that K is positive and not monotone. On the other hand, Bianchi [5] showed that any solution of (1.6) is radially symmetric if there is an rₒ > 0, such that K(r) is non-increasing in (0, rₒ], and non-decreasing in [rₒ,+∞). Moreover, in [6], it was proved that (1.6) has no solutions for some function K(r), which is non-increasing in (0, 1], and nondecreasing in [1,+∞). Therefore, we see that to obtain a solution for (1.6), it is natural to assume that K(r) has a local maximum at rₒ > 0. The purpose of this paper is to answer the following two questions: Q1. Does the existence of a local maximum of K guarantee the existence of a solution to (1.6)? Q2. Are there non-radially symmetric solutions to (1.6)?
Link
Citation
Journal of Functional Analysis, 258(9), p. 3048-3081
ISSN
1096-0783
0022-1236
Start page
3048
End page
3081

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