Properties of the minimizers for a constrained minimization problem arising in Kirchhoff equation

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.

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