论文标题
$ l^{2} $的最小化器 - 在有限域中存在空间衰减的非线性的关键不均匀变分问题
Minimizers of $L^{2}$-critical inhomogeneous variational problems with a spatially decaying nonlinearity in bounded domains
论文作者
论文摘要
我们考虑$ l^{2} $的最小化器 - 关键的不均匀变分问题，在空间衰减的非线性术语中，在一个开放式界面域中$ \ mathbb {r}^{n} $包含$ 0 $。我们证明有一个阈值$ a^{*}> 0 $，因此以$ 0 <a <a <a^{*} $存在最小化的量，并且任何$ a> a> a^{*} $都不存在。与均匀的情况相反，我们表明，最小化器的存在和不存在的情况可能会在阈值$ a^*$处发生，具体取决于$ v（0）$的值，其中$ v（x）$表示陷阱的潜力。此外，根据$ v（x）$的一些合适假设，基于对最小化器作为$ a \ a \ nearRow a^*$的浓度行为的详细分析，当$ a $接近$ a^*$时，我们证明了局部的最小化量。
We consider the minimizers of $L^{2}$-critical inhomogeneous variational problems with a spatially decaying nonlinear term in an open bounded domain $Ω$ of $\mathbb{R}^{N}$ which contains $0$. We prove that there is a threshold $a^{*}>0$ such that minimizers exist for $0<a<a^{*}$ and the minimizer does not exist for any $a>a^{*}$. In contrast to the homogeneous case, we show that both the existence and nonexistence of minimizers may occur at the threshold $a^*$ depending on the value of $V(0)$, where $V(x)$ denotes the trapping potential. Moreover, under some suitable assumptions on $V(x)$, based on a detailed analysis on the concentration behavior of minimizers as $a\nearrow a^*$, we prove local uniqueness of minimizers when $a$ is close enough to $a^*$.
