关于二维光滑有界域上的Ambrosetti-Malchiodi-Ni猜想：聚类集中层
On Ambrosetti-Malchiodi-Ni conjecture on two-dimensional smooth bounded domains: clustering concentration layers
摘要：
我们考虑问题$$ \varepsilon^2 {\mathrm {div}}\big( \nabla_{{\mathfrak a}(y)} u\big)- V(y)u+u^p\, =\, 0, \quad u>0 \quad\mbox{in }Ω, \qquad \nabla_{{\mathfrak a}(y)} u\cdot ν\, =\, 0\quad\mbox{on } \partial Ω, $$的解在曲线上的聚类集中，其中$Ω$是$\mathbb R^2$中具有光滑边界的有界区域，指数$p$大于$1$，$\varepsilon>0$是小参数，$V$是$\barΩ$上的一致正光滑势，$ν$表示$\partial Ω$的外法线.对于$\barΩ$上的两个正光滑函数${\mathfrak a}_1(y), {\mathfrak a}_2(y)$，算子$\nabla_{{\mathfrak a}(y)}$由$$ \nabla_{{\mathfrak a}(y)} u=\Bigg({\mathfrak a}_1(y)\frac{\partial u}{\partial y_1}, \, {\mathfrak a}_2(y)\frac{\partial u}{\partial y_2}\Bigg). $$给出
英文摘要：
We consider the clustering concentration on curves for solutions to the problem $$ \varepsilon^2 {\mathrm {div}}\big( \nabla_{{\mathfrak a}(y)} u\big)- V(y)u+u^p\, =\, 0, \quad u>0 \quad\mbox{in }Ω, \qquad \nabla_{{\mathfrak a}(y)} u\cdot ν\, =\, 0\quad\mbox{on } \partial Ω, $$ where $Ω$ is a bounded domain in $\mathbb R^2$ with smooth boundary, the exponent $p$ is greater than $1$, $\varepsilon>0$ is a small parameter, $V$ is a uniformly positive smooth potential on $\barΩ$, and $ν$ denotes the outward normal of $\partial Ω$. For two positive smooth functions ${\mathfrak a}_1(y), {\mathfrak a}_2(y)$ on $\barΩ$, the operator $\nabla_{{\mathfrak a}(y)}$ is given by $$ \nabla_{{\mathfrak a}(y)} u=\Bigg({\mathfrak a}_1(y)\frac{\partial u}{\partial y_1}, \, {\mathfrak a}_2(y)\frac{\partial u}{\partial y_2}\Bigg). $$
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
