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

Abstract

Let \begin{document}$ a>0,b>0 $\end{document} and \begin{document}$ V(x)\geq0 $\end{document} be a coercive function in \begin{document}$ \mathbb R^2 $\end{document} . We study the following constrained minimization problem on a suitable weighted Sobolev space \begin{document}$ \mathcal{H} $\end{document} : \begin{document}$ \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*} $\end{document} where \begin{document}$ E_{a}^{b}(u) $\end{document} is a Kirchhoff type energy functional defined on \begin{document}$ \mathcal{H} $\end{document} by \begin{document}$ \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*} $\end{document} It is known that, for some \begin{document}$ a^{\ast}>0 $\end{document} , \begin{document}$ e_{a}(b) $\end{document} has no minimizer if \begin{document}$ b = 0 $\end{document} and \begin{document}$ a\geq a^{\ast} $\end{document} , but \begin{document}$ e_{a}(b) $\end{document} has always a minimizer for any \begin{document}$ a\geq0 $\end{document} if \begin{document}$ b>0 $\end{document} . The aim of this paper is to investigate the limit behaviors of the minimizers of \begin{document}$ e_{a}(b) $\end{document} as \begin{document}$ b\rightarrow0^{+} $\end{document} . Moreover, the uniqueness of the minimizers of \begin{document}$ e_{a}(b) $\end{document} is also discussed for \begin{document}$ b $\end{document} close to 0.