Blow-up of solutions of critical elliptic equations in three dimensions

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-06-20 DOI:10.2140/apde.2024.17.1633

Rupert L. Frank, Tobias König, Hynek Kovařík

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引用次数: 0

Abstract

We describe the asymptotic behavior of positive solutions $u_{𝜀}$ of the equation $- Δ u + a u = 3 u^{5 - 𝜀}$ in $Ω \subset ℝ^{3}$ with a homogeneous Dirichlet boundary condition. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon, and the functions $u_{𝜀}$ are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Brezis and Peletier (1989). Similar results are also obtained for solutions of the equation $- Δ u + (a + 𝜀 V) u = 3 u^{5}$ in $Ω$ .

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三维临界椭圆方程解的膨胀

我们描述了方程 -Δu+au= 3u5-𝜀 在 Ω⊂ℝ3 中的正解 u𝜀 的渐近行为，该方程具有同质 Dirichlet 边界条件。假设函数 a 是 Hebey 和 Vaugon 意义上的临界值，并假设函数 u𝜀 是 Sobolev 不等式的优化序列。在一个自然非退化假设下，我们得出了炸毁的精确速率和集中点的位置，从而证明了 Brezis 和 Peletier（1989 年）的猜想。对于方程 -Δu+(a+𝜀V)u= 3u5 在 Ω 中的解，我们也得到了类似的结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.