On global minimizers for a mass constrained problem

In any dimension $N \geq 1$, for given mass $m > 0$ and for the $C^1$ energy functional \begin{equation*} I(u):=\frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^2dx-\int_{\mathbb{R}^N}F(u)dx, \end{equation*} we revisit the classical problem of finding conditions on $F \in C^1(\mathbb{R},\mathbb{R})$ insuring that $I$ admits global minimizers on the mass constraint \begin{equation*} S_m:=\left\{u\in H^1(\mathbb{R}^N)~|~\|u\|^2_{L^2(\mathbb{R}^N)}=m\right\}. \end{equation*} Under assumptions that we believe to be nearly optimal, in particular without assuming that $F$ is even, any such global minimizer, called energy ground state, proves to have constant sign and to be radially symmetric monotone with respect to some point in $\mathbb{R}^N$. Moreover, we manage to show that any energy ground state is a least action solution of the associated free functional. This last result settles, under general assumptions, a long standing open problem