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Finitely many solutions for the prescribed scalar curvature problem at the critical flatness order
Abstract
We study the following prescribed scalar curvature problem −Δu=K(y)u2*−1, u>0 in ℝN, where 2*=2N/(N−2). For N≥4, we construct a positive K(y) with maximum set {|y|=1} and critical order of flatness m=N−2 at |y|=1, such that the equation admits only finitely many positive solutions in D1,2(ℝN). To this end, we prove the compactness of the solution set for this equation by blow-up analysis and then apply Sard's theorem to choose K for which each solution is non-degenerate and hence isolated, so that the solution set is finite. Our results also give a definite answer to Wei--Yan's conjecture in [J. Wei and S. Yan, JFA, 2010].
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BibTeX
@misc{ji2026finitely,
title = {{Finitely many solutions for the prescribed scalar curvature problem at the critical flatness order}},
author = {Wei Ji and Yixing Pu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32707},
eprint = {2610.32707},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32707},
}
RIS
TY - UNPB
TI - Finitely many solutions for the prescribed scalar curvature problem at the critical flatness order
AU - Wei Ji
AU - Yixing Pu
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.32707
UR - https://arxiv.org/abs/2610.32707
AB - We study the following prescribed scalar curvature problem −Δu=K(y)u^2^*−1, u>0 in ℝ^N, where 2^*=2N/(N−2). For N≥4, we construct a positive K(y) with maximum set {|y|=1} and critical order of flatness m=N−2 at |y|=1, such that the equation admits only finitely many positive solutions in D^1,2(ℝ^N). To this end, we prove the compactness of the solution set for this equation by blow-up analysis and then apply Sard's theorem to choose K for which each solution is non-degenerate and hence isolated, so that the solution set is finite. Our results also give a definite answer to Wei--Yan's conjecture in [J. Wei and S. Yan, JFA, 2010].
ER -