On global minimizers for a mass constrained problem

Abstract

In any dimension $N \ge 1$, for given mass $m > 0$ and for the $C^1$ energy functional

$$\begin{aligned} I(u):=\frac{1}{2}\int _{{\mathbb {R}}^N}|\nabla u|^2dx-\int _{{\mathbb {R}}^N}F(u)dx, \end{aligned}$$

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

$$\begin{aligned} S_m:=\left\{ u\in H^1({\mathbb {R}}^N)~|~\Vert u\Vert ^2_{L^2({\mathbb {R}}^N)}=m\right\} . \end{aligned}$$

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 show that any energy ground state is a least action solution of the associated action functional. This last result answers positively, under general assumptions, a long standing issue.