论文标题
布雷兹 - 尼伦贝格问题的积极解决方案数量
The number of positive solutions to the Brezis-Nirenberg problem
论文作者
论文摘要
在本文中，我们关注的是众所周知的brezis-nirenberg问题\ begin {equination*} \ begin {cases}-Δu= u^u^{\ frac {n+2} {n-2} {n-2}}+\ varepsilon U， ＆{\ text {in} 〜Ω}，\\ u = 0，＆{\ text {on}〜\ partialω}。 \ end {cases} \ end {equation*}在Musso和Pistoia获得了上述小问题的多峰解决方案中，对于小$ \ Varepsilon> 0 $。但是，上述问题的唯一性或正面解决方案的确切数量仍然未知。在这里，我们专注于多峰解决方案的局部唯一性，以及针对小$ \ varepsilon> 0 $的上述问题的确切数量。 通过使用各种局部Pohozaev身份和爆炸分析，我们首先检测到爆炸解决方案的轮廓与域$ω$的绿色功能之间的关系，然后获得爆炸解决方案的局部唯一性结果。最后，我们描述了小型$ \ varepsilon $的积极解决方案的数量，这也取决于绿色的功能。
In this paper we are concerned with the well-known Brezis-Nirenberg problem \begin{equation*} \begin{cases} -Δu= u^{\frac{N+2}{N-2}}+\varepsilon u, &{\text{in}~Ω},\\ u>0, &{\text{in}~Ω},\\ u=0, &{\text{on}~\partial Ω}. \end{cases} \end{equation*} The existence of multi-peak solutions to the above problem for small $\varepsilon>0$ was obtained by Musso and Pistoia. However, the uniqueness or the exact number of positive solutions to the above problem is still unknown. Here we focus on the local uniqueness of multi-peak solutions and the exact number of positive solutions to the above problem for small $\varepsilon>0$. By using various local Pohozaev identities and blow-up analysis, we first detect the relationship between the profile of the blow-up solutions and the Green's function of the domain $Ω$ and then obtain a type of local uniqueness results of blow-up solutions. At last we give a description of the number of positive solutions for small positive $\varepsilon$, which depends also on the Green's function.
