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.