This paper is devoted to studying the mass concentration behavior of \(L^2\) -norm prescribed minimizers for the Gross-Pitaevskii energy functional with a mass subcritical perturbation \( E_a(u)=\int _{\mathbb {R}^2}|\nabla u|^2\textrm{d}x+\int _{\mathbb {R}^2}V(x)|u|^2\textrm{d}x -\frac{a}{2}\int _{\mathbb {R}^2}|u|^4\textrm{d}x-\frac{2\mu _a}{q+2}\int _{\mathbb {R}^2}|u|^{q+2}\textrm{d}x, \) where \(0<q<2\) , \(a>0\) , \(\mu _a\ge 0\) is a small parameter depending on a and V(x) is a trapping potential. It is shown that there is a constant \(a^*>0\) such that the minimizer exists if \(a<a^*\) and the minimizer does not exist if \(a=a^*\) . By establishing the delicate estimate on the minimal energy, we give a detailed description of limit behavior for minimizers as \(a\nearrow a^*\) , which implies that the minimizers must concentrate at the flattest global minimum of the trapping potential. Finally, we prove the local uniqueness of minimizers when \(a>0\) is sufficiently close to \(a^*\) . In particular, if V(x) is a radial function, the local uniqueness still holds when \(\mu _a=const>0\) .
