论文标题
在Kirchhoff方程中产生的最小化最小化问题的最小化属性的属性
Properties of the minimizers for a constrained minimization problem arising in Kirchhoff equation
论文作者
论文摘要
令$ a> 0，b> 0 $和$ v（x）\ geq0 $是$ \ mathbb r^2 $中的强制功能。我们在适当加权的Sobolev空间$ \ MATHCAL {H} $：\ begin {equation*} e__ {a}（a}（b）：= \ feft \ left \ left \ left \ {e_ {a} a}^{a}^{b}（u \ in \ in \ intccal { \ int _ {\ Mathbb r^{2}} | u |^{2} dx = 1 \ right \}，\ end {equation*}其中$ e_ {a}^a}^{b}（b}（u）$是kirchhoff类型的能量，在$ \ nathcal { e_ {a}^{b}（u）= \ frac {1} {2} \ int _ {\ mathbb r^{2}}} [| \ nabla u |^{2}+v（x）+v（x） r^{2}} | \ nabla u |^{2} dx \ right）^{2} - \ frac {a} {4} {4} \ int _ {\ mathbb r^{2}} | \ end {qore*}众所周知，对于某些$ a^{\ ast}> 0 $，$ e_ {a}（b）$，如果$ b = 0 $和$ a \ geq a^{\ ast a^ast} $，但$ e_ {a}（a}（b）$始终为$ a $ a \ $ a \ geq 0。本文的目的是调查$ e_ {a}（b）$的最小化器的极限行为为$ b \ rightarrow0^{+} $。此外，还讨论了$ e_ {a}（b）$的最小化器的独特性，以$ b $接近0的内容进行了讨论。
Let $a>0,b>0$ and $V(x)\geq0$ be a coercive function in $\mathbb R^2$. We study the following constrained minimization problem on a suitable weighted Sobolev space $\mathcal{H}$: \begin{equation*} e_{a}(b):=\inf\left\{E_{a}^{b}(u):u\in\mathcal{H}\ \mbox{and}\ \int_{\mathbb R^{2}}|u|^{2}dx=1\right\}, \end{equation*} where $E_{a}^{b}(u)$ is a Kirchhoff type energy functional defined on $\mathcal{H}$ by \begin{equation*} E_{a}^{b}(u)=\frac{1}{2}\int_{\mathbb R^{2}}[|\nabla u|^{2}+V(x)u^{2}]dx+\frac{b}{4}\left(\int_{\mathbb R^{2}}|\nabla u|^{2}dx\right)^{2}-\frac{a}{4}\int_{\mathbb R^{2}}|u|^{4}dx. \end{equation*} It is known that, for some $a^{\ast}>0$, $e_{a}(b)$ has no minimizer if $b=0$ and $a\geq a^{\ast}$, but $e_{a}(b)$ has always a minimizer for any $a\geq0$ if $b>0$. The aim of this paper is to investigate the limit behaviors of the minimizers of $e_{a}(b)$ as $b\rightarrow0^{+}$. Moreover, the uniqueness of the minimizers of $e_{a}(b)$ is also discussed for $b$ close to 0.
