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 \({E_p}(u) = \int_{{\mathbb{R}^N}} | \nabla u{|^2} - c{{|u{|^2}} \over {|x{|^2}}} + V(x)|u{|^2}{\rm{d}}x - {2 \over {p + 2}}\int_{{\mathbb{R}^N}} | u{|^{p + 2}}{\rm{d}}x.\) When \(p = {p^*}: = {4 \over N}\) , the precise concentration behavior of minimizers is analyzed as \(M_{p^{*}}\nearrow \|Q_{p^{*}}\|_{L^{2}}\) , where Qp* is the unique radially positive solution of \( - \Delta \varphi - c{\varphi \over {|x{|^2}}} - |\varphi {|^{{p^*} + 1}}\varphi = 0\) When 0 < p < p*, we prove that all minimizers must blow up if \(\mathop {\lim}\limits_{p \to {p*}} {M_p} \ge {\|}Q_{p*}{\|}_{L^2}\) . On this argument, the detailed concentration behavior of minimizers is established as p ↗ p*.
