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An analytic construction of singular solutions related to a critical Yamabe problem
Communications in Partial Differential Equations ( IF 1.9 ) Pub Date : 2020-07-06 , DOI: 10.1080/03605302.2020.1784209
Hardy Chan 1 , Azahara DelaTorre 2
Affiliation  

Abstract We answer affirmatively a question posed by Aviles in 1983, concerning the construction of singular solutions of semilinear equations without using phase-plane analysis. Fully exploiting the semilinearity and the stability of the linearized operator in any dimension, our techniques involve a careful gluing in weighted spaces that handles multiple occurrences of criticality, without the need of derivative estimates. The above solution constitutes an Ansatz for the Yamabe problem with a prescribed singular set of maximal dimension for which, using the same machinery, we provide an alternative construction to the one given by Pacard. His linear theory uses Lp-theory on manifolds, while our strategy relies solely on asymptotic analysis and is suitable for generalization to non-local problems. Indeed, in a forthcoming paper, we will prove analogous results in the fractional setting.

中文翻译:

与临界 Yamabe 问题相关的奇异解的解析构造

摘要 我们肯定地回答了 Aviles 在 1983 年提出的关于不使用相平面分析构造半线性方程的奇异解的问题。充分利用线性化算子在任何维度上的半线性和稳定性,我们的技术涉及在加权空间中仔细粘合,以处理多次出现的临界性,而不需要导数估计。上述解决方案构成了 Yamabe 问题的 Ansatz,具有指定的最大维数奇异集,为此,我们使用相同的机制提供了 Pacard 给出的构造的替代构造。他的线性理论在流形上使用 Lp 理论,而我们的策略仅依赖于渐近分析,适用于非局部问题的泛化。事实上,在即将发表的论文中,
更新日期:2020-07-06
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