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{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.

This version is the final one, corresponding to the paper now published in Calc. Var. Partial Differential Equations

References in corpus (4)

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• On the symmetry of minimizers
• Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation
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• Normalized solutions to Schrödinger equations in the strongly sublinear regime
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• On ground states for the 2D Schrodinger equation with combined nonlinearities and harmonic potential
• Two dimensional NLS ground states with attractive Coulomb potential and point interaction
• Existence and dynamics of normalized solutions to Schrödinger equations with generic double-behaviour nonlinearities
• On action ground states of defocusing nonlinear Schrödinger equations
• Non-uniqueness of normalized NLS ground states on polygons with homogeneous Neumann boundary conditions
• Normalized NLS ground states on a double plane hybrid