论文标题
关键椭圆方程的积极解决方案
Positive solutions for a critical elliptic equation
论文作者
论文摘要
在本文中，我们关注以下椭圆方程\ begin {qore*} \ begin {cases}-Δu= q（x） ＆{\ text {in} 〜Ω}，\\ [1mm] u = 0，＆{\ text {on}〜\ partialω}，\ end {cases} \ end {eark {quesation {equation*}，其中$ n \ geq 3 $，$ n \ geq 3 $，$ s \ in [1,2^* - 1） $ \ varepsilon> 0 $，$ω$是$ \ mathbb {r}^n $中的平滑有限域。 在某些条件下，$ q（x）$，cao和en in nonlin。肛门。 TMA（第29卷，1997年，第461--483页）证明，如果$ n \ geq 4 $和$ s \ in（1,2^* - 1）$，则有一个小$ \ varepsilon $的单峰解决方案。他们在论文1.7中提出了 \ vskip 0.1cm \ begin {center} \ emph {````很有趣的是，知道对于小$ \ varepsilon $和$ s = 1 $''。} \ end} \ end {center {center} \ vskip 0.1cm vskip n。 \ noindent也在他们的论文1.8中解决了 \ vskip 0.1cm \ begin {center} \ emph {````同时集中在几个点的解决方案的问题仍然是打开的'。} \ end {center {center {center} \ vskip 0.1cm \ noindent 在这里，我们对以上两个问题给出了一些确认答案。此外，我们证明了多峰解决方案的局部唯一性。我们的结果表明，上述问题的浓度是微妙的，无论$ s = 1 $还是$ s> 1 $。
In this paper, we are concerned with the following elliptic equation \begin{equation*} \begin{cases} -Δu= Q(x)u^{2^*-1 }+\varepsilon u^{s},~ &{\text{in}~Ω},\\[1mm] u>0,~ &{\text{in}~Ω},\\[1mm] u=0, &{\text{on}~\partial Ω}, \end{cases} \end{equation*} where $N\geq 3$, $s\in [1,2^*-1)$ with $2^*=\frac{2N}{N-2}$, $\varepsilon>0$, $Ω$ is a smooth bounded domain in $\mathbb{R}^N$. Under some conditions on $Q(x)$, Cao and Zhong in Nonlin. Anal. TMA (Vol 29, 1997, 461--483) proved that there exists a single-peak solution for small $\varepsilon$ if $N\geq 4$ and $s\in (1,2^*-1)$. And they proposed in Remark 1.7 of their paper that \vskip 0.1cm\begin{center} \emph{``it is interesting to know the existence of single-peak solutions for small $\varepsilon$ and $s=1$''.} \end{center}\vskip 0.1cm \noindent Also it was addressed in Remark 1.8 of their paper that \vskip 0.1cm \begin{center} \emph{``the question of solutions concentrated at several points at the same time is still open''.} \end{center}\vskip 0.1cm \noindent Here we give some confirmative answers to the above two questions. Furthermore, we prove the local uniqueness of the multi-peak solutions. And our results show that the concentration of the solutions to above problem is delicate whether $s=1$ or $s>1$.
