On global minimizers for a mass constrained problem

TL;DR

This paper investigates conditions under which energy minimizers with fixed mass exist for a class of nonlinear functionals, showing they are radially symmetric, monotone, and relate to least action solutions, solving a long-standing problem.

Contribution

It establishes nearly optimal conditions for the existence of global minimizers on mass constraints without requiring evenness of the nonlinearity, and proves their symmetry and sign properties.

Findings

Global minimizers are radially symmetric and monotone.

Minimizers have constant sign.

Minimizers are least action solutions.

Abstract

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 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.