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

• Published in: arXiv.org 2020-07
• Main Authors: Guo, Helin, Huan-Song, Zhou
• Format: Article
• Language: English
• Subjects: Optimization; Sobolev space
• ISSN: 2331-8422

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.