Sign-changing blowing-up solutions for the Brezis--Nirenberg problem in dimensions four and five
Sign-changing blowing-up solutions for the Brezis--Nirenberg problem in dimensions four and five
复制标题
第四维和第五维 Brezis-Nirenberg 问题的变号爆炸解决方案
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Giusi Vaira
中科院分区：
文献类型：
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作者：
A. Iacopetti;Giusi Vaira
我们考虑Brezis-Nirenberg问题：$$-Delta u =lambda u +| u| ^{p-1}uqquad mbox{in}，，Omega，quad u=0，，mbox{on}，，partialOmega，$$其中$Omega$是$mathbb R^N$，$Ngeq 3$中的光滑有界域，$p=frac{N+2}{N-2}$且$lambda>0$。本文证明了，若Ω是对称的，且N= 4，5，则存在一个变号解，其正部分在区域的对称中心集中并爆破，而负部分消失，如λ
其中$Delta$在$Omega$上的第一特征值为$Delta$，Dirichlet边界条件为零。
We consider the Brezis-Nirenberg problem: $$-Delta u =lambda u + |u|^{p-1}uqquad mbox{in},, Omega,quad u=0,, mbox{on},, partialOmega,$$ where $Omega$ is a smooth bounded domain in $mathbb R^N$, $Ngeq 3$, $p=frac{N+2}{N-2}$ and $lambda>0$. In this paper we prove that, if $Omega$ is symmetric and $N=4,5$, there exists a sign-changing solution whose positive part concentrates and blows-up at the center of symmetry of the domain, while the negative part vanishes, as $lambda
ightarrow lambda_1$, where $lambda_1=lambda_1(Omega)$ denotes the first eigenvalue of $-Delta$ on $Omega$, with zero Dirichlet boundary condition.