Zeitschrift für angewandte Mathematik und Physik ( IF 2 ) Pub Date : 2021-03-22 , DOI: 10.1007/s00033-021-01510-0 Huifang Jia , Xiao Luo
In this paper, we consider the following Gross–Pitaevskii equation describing Bose–Einstein condensation of trapped dipolar quantum gases:
$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta u+\lambda Q(x)u+\omega _{1}|u|^{2}u+\omega _{2}(K\star |u|^{2})u=0, &{} x\in {\mathbb {R}}^{3},\\ \displaystyle u>0,\quad u\in H^{1}({\mathbb {R}}^{3}),\\ \end{array}\right. } \end{aligned}$$(0.1)where \(\lambda >0\), \(Q\in C({\mathbb {R}}^{3},{\mathbb {R}})\) is a potential well, \(\star \) denotes the convolution, \(K(x)=\frac{1-3\cos ^{2}\theta }{|x|^{3}}\) and \(\theta =\theta (x)\) is the angle between the dipole axis determined by the vector x and the vector (0, 0, 1). When \((\omega _{1},\omega _{2})\in {\mathbb {R}}^{2}\) lies in the defined unstable regime, under some suitable conditions on Q, the existence and concentration of nontrivial solutions to problem (0.1) are proved by using variational methods. In particular, the trapping potential is allowed to be sign-changing. Moreover, when \((\omega _{1},\omega _{2})\in {\mathbb {R}}^{2}\) lies in the stable regime, we show that problem (0.1) with small bounded sign-changing potential has only trivial solutions.
中文翻译：
具有陡势阱的Gross–Pitaevskii方程解的存在和集中
在本文中，我们考虑以下Gross-Pitaevskii方程，该方程描述了俘获的偶极量子气体的Bose-Einstein凝聚：
$$ \ begin {aligned} {\ left \ {\ begin {array} {ll} \ displaystyle-\ Delta u + \ lambda Q（x）u + \ omega _ {1} | u | ^ {2} u + \ omega _ {2}（K \ star | u | ^ {2}）u = 0，＆{} x \ in {\ mathbb {R}} ^ {3}，\\ \ displaystyle u> 0，\ quad u \ in H ^ {1}（{\ mathbb {R}} ^ {3}），\\ \ end {array} \ right。} \ end {aligned} $$（0.1）其中\（\ lambda> 0 \），\（Q \ in C（{\ mathbb {R}} ^ {3}，{\ mathbb {R}}）\）是一口势阱，\（\ star \）表示卷积\（K（x）= \ frac {1-3 \ cos ^ {2} \ theta} {| x | ^ {3}} \）和\（\ theta = \ theta（x）\）是由向量x和向量（0，0，1）确定的偶极轴之间的角度。当{\ mathbb {R}} ^ {2} \中的\（（\ omega _ {1}，\ omega _ {2}）\位于定义的不稳定状态时，在Q的某些合适条件下，存在和通过变分方法证明了对问题（0.1）的非平凡解的集中性。特别地，允许俘获势能改变符号。而且，什么时候\（（\ omega _ {1}，\ omega _ {2}）\ {\ mathbb {R}} ^ {2} \中的）处于稳定状态，我们证明了问题（0.1）的小界符号-改变潜力只有微不足道的解决方案。
京公网安备 11010802027423号