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一类质量约束问题的全局极小解

On global minimizers for a mass constrained problem

Louis Jeanjean, Sheng-Sen Lu

备注：
This version is the final one, corresponding to the paper now published in Calc. Var. Partial Differential Equations
发表信息：
Calculus of Variations and Partial Differential Equations, 61 (2022): 214
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摘要：

在任何维度$N \geq 1$中，对于给定的质量$m > 0$和对于$C^1$能量泛函\begin{equation*} I(u):=\frac{1}{2}\int_{\mathbb{R}^N}|\nabla u|^2dx-\int_{\mathbb{R}^N}F(u)dx, \end{equation*}，我们重新考虑寻找关于$F \in C^1(\mathbb{R},\mathbb{R})$的条件的经典问题，以确保$I$允许质量约束\begin{equation*} S_m:=\left\{u\in H^1(\mathbb{R}^N)~|~\|u\|^2_{L^2(\mathbb{R}^N)}=m\right\}. \end{equation*}上的全局最小值。在我们认为接近最优的假设下，特别是在不假设$F$是偶数的情况下，任何这样的全局最小值，称为能量基态，证明了具有常数符号，并关于$\mathbb{R}^N$中的某点是径向对称单调的。此外，我们还证明了任何能量基态都是相应作用泛函的最小作用量解。在一般假设下，最后一个结果肯定地回答了一个长期存在的问题。

英文摘要：

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.

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