关键词: minimization problem, minimizers, concentration behavior

Abstract: We study the following minimization problem: $$d_{p}(M_{p}):=\inf\{E_{p}(u): \|u\|_{L^{2}}=M_{p}\},$$ where the Gross-Pitaevskii energy functional $$E_{p}(u)=\int_{\mathbb{R}^{N}}|\nabla u|^{2}-c\frac{|u|^{2}}{|x|^{2}}+V(x)|u|^{2}{\rm d}x-\frac{2}{p+2}\int_{\mathbb{R}^{N}}|u|^{p+2}{\rm d}x.$$ When $p=p^{*}:=\frac{4}{N}$, the precise concentration behavior of minimizers is analyzed as $M_{p^{*}}\nearrow \|Q_{p^{*}}\|_{L^{2}}$, where $Q_{p^{*}}$ is the unique radially positive solution of $-\Delta \varphi-c\frac{\varphi}{|x|^{2}}-|\varphi|^{p^{*}+1}\varphi=0$. When $0<p<p^{*}$, we prove that all minimizers must blow up if $\lim\limits_{p\to p^{*}}M_{p}\geq \|Q_{p^{*}}\|_{L^{2}}$. On this argument, the detailed concentration behavior of minimizers is established as $p\nearrow p^{*}$.

Key words: minimization problem, minimizers, concentration behavior