具有吸引三体相互作用的三维偶极玻色-爱因斯坦凝聚体的归一化基态
Normalized ground states for 3D dipolar Bose-Einstein condensate with attractive three-body interactions
摘要：
本文研究了具有吸引三体相互作用的三维偶极玻色-爱因斯坦凝聚方程：\begin{align}\label{1} -Δu+βu+λ_1|u|^2 u+λ_2 (K*|u|^2)u-|u|^4u=0.\tag{DBEC} \end{align}当$λ_2=0$或$u$是径向的时，（\ref{1}）约化为Soave最近在[31]中研究的五次NLS \begin{align}\label{2} -Δu+βu+λ_1|u|^2 u-|u|^4u=0\tag{CQNLS}, \end{align}.特别是，它表明，对于任何$λ_1<0$和$c>0$，（\ref{2}）拥有一个径向对称的基态解与质量$c$和$λ_1\geq 0$，（\ref{2}）没有非平凡的解决方案。我们发现，通过向（\ref{2}）添加偶极-偶极相互作用，（\ref{2}）的几何性质发生了显著变化，并且不能再使用[31]中的技术来获得类似的结果。更确切地说，由于偶极-偶极相互作用势的轴对称性质，对应于（\ref{1}）的能量在对称重排下是不稳定的，因此基于解的径向对称性的传统论点是不适用的。我们将克服这一困难，呼吁微妙的变分和微扰方法，并证明了以下几点：（i）如果对$(λ_1,λ_2)$是不稳定的和$λ_1<0$，那么对于任何$c>0$，（\ref{1}）有一个基态解与质量$c$。(ii)如果$(λ_1,λ_2)$和$λ_1\geq 0$对是不稳定的，则存在一些$c^*=c^*(λ_1,λ_2)\geq 0$，使得对于所有$c>c^*$，（\ref{1}）具有质量为$c$的基态解。此外，在这种情况下，（\ref{1}）的任何非平凡解必须是非径向的。(iii)如果$(λ_1,λ_2)$对是\textit{stable}，则（\ref{1}）没有非平凡解。
英文摘要：
We study the existence of normalized ground states for the 3D dipolar Bose-Einstein condensate equation with attractive three-body interactions: \begin{align}\label{1} -Δu+βu+λ_1|u|^2 u+λ_2 (K*|u|^2)u-|u|^4u=0.\tag{DBEC} \end{align} When $λ_2=0$ or $u$ is radial, (\ref{1}) reduces to the cubic-quintic NLS \begin{align}\label{2} -Δu+βu+λ_1|u|^2 u-|u|^4u=0\tag{CQNLS}, \end{align} which has been recently studied by Soave in [31]. In particular, it was shown that for any $λ_1<0$ and $c>0$, (\ref{2}) possesses a radially symmetric ground state solution with mass $c$ and for $λ_1\geq 0$, (\ref{2}) has no non-trivial solution. We show that by adding a dipole-dipole interaction to (\ref{2}), the geometric nature of (\ref{2}) changes dramatically and techniques as the ones from [31] cannot be used anymore to obtain similar results. More precisely, due to the axisymmetric nature of the dipole-dipole interaction potential, the energy corresponding to (\ref{1}) is not stable under symmetric rearrangements, hence conventional arguments based on the radial symmetry of solutions are inapplicable. We will overcome this difficulty by appealing to subtle variational and perturbative methods and prove the following: (i) If the pair $(λ_1,λ_2)$ is unstable and $λ_1<0$, then for any $c>0$, (\ref{1}) has a ground state solution with mass $c$. (ii) If the pair $(λ_1,λ_2)$ is unstable and $λ_1\geq 0$, then there exists some $c^*=c^*(λ_1,λ_2)\geq 0$ such that for all $c>c^*$, (\ref{1}) has a ground state solution with mass $c$. Moreover, any non-trivial solution of (\ref{1}) in this case must be non-radial. (iii) If the pair $(λ_1,λ_2)$ is \textit{stable}, then (\ref{1}) has no non-trivial solutions.
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