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arXiv 1007.4139math.AP
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关于平移不变约束极小化问题及其在薛定谔-泊松方程中的应用

On translation invariant constrained minimization problems with application to Schrödinger-Poisson equation

Jacopo Bellazzini, Gaetano Siciliano

备注：
This paper has been withdrawn by the author due to an error in the Application section
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摘要：

本文研究了一类平移不变的约束极小化问题极小解的存在性。我们称之为$$I_{ρ^{2}}:=\inf_{B_ρ}I(u) \ $$，其中$B_ρ=\{u\in H^{m}(\R^{N}):\|u\|_{2}=ρ\},$和$I(u)=1/2\|u\|^{2}_{D^{m,2}}+T(u)$，$T$满足一般假设。我们证明了函数$$(0,\infty)\ni s \mapsto I_{s^2}\,$$的正则性和$\frac{I_{s^2}}{s^2}$在零邻域的行为允许证明强次可加性不等式。作为副产品的强收敛性的最小化序列（翻译）证明。本文将能量泛函$I$应用于任意电荷的轨道稳定驻波中的Schrödinger-泊松方程：\label{SP} i_{t}+ Δ τ-（|x| ^{-1}*| ψ| ^{2}）+| ψ| ^{p-2} p =0 \text{in} \R^{3}，\text{with} \ 2<p<3。当$2<p<3$，特别是我们证明，$I$实现其最小的约束$\{u\in H^{1}(\R^{3}): \|u\|_{2}=ρ\}$为每个$ρ>0$，因此，一组极小是轨道稳定的。这涵盖了物理上相关的情况，$p=8/3$，所谓的薛定谔-泊松-斯莱特系统。

英文摘要：

In this paper we study the existence of minimizers for a class of constrained minimization problems that are invariant under translations. We call $$I_{ρ^{2}}:=\inf_{B_ρ}I(u) \ $$ where $B_ρ=\{u\in H^{m}(\R^{N}):\|u\|_{2}=ρ\},$ and $I(u)=1/2\|u\|^{2}_{D^{m,2}}+T(u)$, $T$ fulfilling general assumptions. We show that the regularity of the function $$(0,\infty)\ni s \mapsto I_{s^2}\,$$ and the behaviour of $\frac{I_{s^2}}{s^2}$ in the neighborhood of zero allows to prove the strong subadditivity inequality. As byproduct the strong convergence of the minimizing sequences (up to translations) is proved. We give an application to the energy functional $I$ associated to the Schrödinger-Poisson equation in $\R^{3}$ orbitally stable standing waves with arbitray charge for the following Schrödinger-Poisson type equation \label{SP} iψ_{t}+ Δψ- (|x|^{-1}*|ψ|^{2}) ψ+|ψ|^{p-2}ψ=0 \text{in} \R^{3}, \text{with} \ 2<p<3. When $2<p<3$, In particular we prove that $I$ achieves its minimum on the constraint $\{u\in H^{1}(\R^{3}): \|u\|_{2}=ρ\}$ for every $ρ>0$ and, as a consequence, the set of minimizers is orbitally stable. This covers the physically relevant case, $p=8/3$, the so called Schrödinger-Poisson-Slater system.

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