This work studies positive minimizers of the Gross-Pitaevskii energy functional \[ E_a(u):=\int_{\mathbb{R}^2}\left(|\nabla u|^2+ V(x)|u|^2\right)dx-\frac{a}{2}\int_{\mathbb{R}^2}|u|^4\, dx \] with \(u\in H^1(\mathbb{R}^2)\), \(\int_{\mathbb{R}^2} V(x)|u|^2\,dx<\infty\), under the constraint \(\left\|u\right\|_2=1\).
The quantity \(a>0\) measures the strength of the attractive interactions, while the nonnegative potential \(V(x)\) is considered to be trapping, i.e. satisfying \(\lim_{|x|\to\infty}V(x)=\infty\), homogeneous of degree \(p\geq 2\), and such that the function \(H(y)=\int_{\mathbb{R}^2} V(x+y)w^2(x)\,dx\), where \(w\) is the unique positive solution of \(-\Delta w=-w+w^3\) in \(\mathbb{R}^2\), has a unique nondegenerate critical point.
It is known that minimizers exist if and only if \(a\) is smaller than the critical value \(a^*:=\left\|w\right\|_2^2\). This works proves, for the first time in the non-radial case, that they are unique for \(a\) close to \(a^*\). In addition, their refined spike profile is determined as \(a\uparrow a^*\).