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A Pohožaev type identity and its application to uniqueness of positive radial solutions of Brezis-Nirenberg problem on an annulus
Journal of Mathematical Analysis and Applications ( IF 1.3 ) Pub Date : 2020-12-30 , DOI: 10.1016/j.jmaa.2020.124901
Naoki Shioji , Kohtaro Watanabe

We study the uniqueness of positive radial solutions of the Brezis-Nirenberg problem{Δu(x)+λu(x)+u(x)p=0,xAa,b,u(x)=0,xAa,b, where n3, b>a>0, 0<λ<λ1, p>(n+2)/(n2) and λ1 is the first eigenvalue of −Δ under the Dirichlet boundary condition on Aa,b={xRn|a<|x|<b}. In particular, in the case n=3, we completely solve the problem without any additional assumption, and in the case n4, we show the uniqueness result under 0<λ1λ1. These results are obtained through a kind of Pohožaev identity.



中文翻译:

Pohožaev型恒等式及其在环上Brezis-Nirenberg问题正径向解的唯一性中

我们研究Brezis-Nirenberg问题的正径向解的唯一性{ΔüX+λüX+üXp=0X一种一种büX=0X一种一种b 哪里 ñ3b>一种>00<λ<λ1个p>ñ+2/ñ-2λ1个 是Dirichlet边界条件下-Δ的第一个特征值 一种一种b={X[Rñ|一种<|X|<b}。特别是ñ=3,我们将完全解决问题,而无需任何其他假设,并且在这种情况下 ñ4,我们在下显示唯一性结果 0<λ1个-λ1个。这些结果是通过一种Pohožaev身份获得的。

更新日期:2021-01-07
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