Nirenberg problem on high dimensional spheres: Blow up with residual mass phenomenon
Abstract
In this paper, we extend the analysis of the subcritical approximation of the Nirenberg problem on spheres recently conducted in \cite{MM19, MM}. Specifically, we delve into the scenario where the sequence of blowing up solutions exhibits a non-zero weak limit, which necessarily constitutes a solution of the Nirenberg problem itself. Our focus lies in providing a comprehensive description of such blowing up solutions, including precise determinations of blow-up points and blow-up rates. Additionally, we compute the topological contribution of these solutions to the difference in topology between the level sets of the associated Euler-Lagrange functional. Such an analysis is intricate due to the potential degeneracy of the involved solutions. We also provide a partial converse, wherein we construct blowing up solutions when the weak limit is non-degenerate.
Keywords
Cite
@article{arxiv.2404.13341,
title = {Nirenberg problem on high dimensional spheres: Blow up with residual mass phenomenon},
author = {Mohameden Ahmedou and Mohamed Ben Ayed and Khalil El Mehdi},
journal= {arXiv preprint arXiv:2404.13341},
year = {2024}
}