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arXiv · 2404.13341

Nirenberg problem on high dimensional spheres: Blow up with residual mass phenomenon

Mohameden Ahmedou·Mohamed Ben Ayed·Khalil El Mehdi
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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.

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Mohameden Ahmedou, Mohamed Ben Ayed, Khalil El Mehdi. 2024-04-20. Nirenberg problem on high dimensional spheres: Blow up with residual mass phenomenon. https://arxiv.org/abs/2404.13341

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