The Nonexistence of Vortices for Rotating Bose–Einstein Condensates with Attractive Interactions

Abstract

This article is devoted to studying the model of two-dimensional attractive Bose–Einstein condensates in a trap V(x) rotating at the velocity $\Omega$. This model can be described by the complex-valued Gross–Pitaevskii energy functional. It is shown that there exists a critical rotational velocity $0<\Omega ^*:=\Omega ^*(V)\le \infty$, depending on the general trap V(x), such that for any rotational velocity $0\le \Omega <\Omega ^*$, minimizers (i.e., ground states) exist if and only if $a<a^*=\Vert w\Vert ^2_2$, where $a>0$ denotes the absolute product for the number of particles times the scattering length, and $w>0$ is the unique positive solution of $\Delta w-w+w^3=0$ in ${\mathbb {R}}^2$. If $V(x)=|x|^2$ and $0<\Omega <\Omega ^*(=2)$ is fixed, we prove that, up to a constant phase, all minimizers must be real-valued, unique and free of vortices as $a \nearrow a^*$, by analyzing the refined limit behavior of minimizers and employing the non-degenerancy of w.

Appendix

In this appendix, we shall establish Lemmas A.1 and A.2 which are used in the proof of Theorem 1.3.

Lemma A.1

For $i=1,2$, suppose that $f_i(x)\in C^1({\mathbb {R}}^2)$ satisfies

$$\begin{aligned} |f_i(x)|,\,|\nabla f_i(x)|\le Pe^{-Q|x|}\,\ \hbox {in} \,\ {\mathbb {R}}^2 \end{aligned}$$

(A.1)

for some constants $P>0$ and $Q>0$, and assume that there exists $b_i\in {\mathbb {R}}$ such that $V_i(x)\in C^1({\mathbb {R}}^2)$ satisfies

$$\begin{aligned} V_i(x)-|\nabla V_i|^2- \frac{b_i^2|x|^2}{4}-\big (|b_1|+|b_2|\big )\ge 2Q^2+\delta \,\ \hbox {in}\,\ {\mathbb {R}}^2 \end{aligned}$$

(A.2)

for some constant $\delta >0$. Let $(w_1,w_2)\in C^3({\mathbb {R}}^2)\times C^3({\mathbb {R}}^2)$ be a real solution of

$$\begin{aligned} \left\{ \begin{aligned} -\Delta w_1 +V_1(x)w_1&=b_1\big (x^{\perp }\cdot \nabla w_2\big ) +f_1(x)\,\ \hbox {in} \,\ {\mathbb {R}}^2,\\ -\Delta w_2+V_2(x)w_2&=b_2\big (x^{\perp }\cdot \nabla w_1\big )+f_2(x)\,\ \hbox {in} \,\ {\mathbb {R}}^2 \end{aligned} \right. \end{aligned}$$

(A.3)

satisfying

$$\begin{aligned} |w_i(x)|,\ |\nabla w_i(x)|\rightarrow 0\,\ \hbox {as}\,\ |x|\rightarrow \infty ,\ \ i=1,\, 2. \end{aligned}$$

(A.4)

Then for $i=1,\, 2$, $w_i(x)$ and $\nabla w_i(x)$ satisfy

$$\begin{aligned} |w_i(x)|,\ |\nabla w_i(x)|\le C(\delta )Pe^{-Q|x|}\,\ \hbox {in} \,\ {\mathbb {R}}^2, \end{aligned}$$

(A.5)

where the constant $C(\delta )>0$ depends only on $\delta >0$.

Proof

We first prove that (A.5) holds for $w_i(x)$ with $i=1$ and 2. Actually, we obtain from (A.3) that

$$\begin{aligned} \left\{ \begin{aligned} \Big [-\frac{1}{2}\Delta +V_1(x)\Big ]w^2_1+|\nabla w_1|^2&=b_1\big (x^{\perp } \cdot \nabla w_2\big )w_1+f_1w_1\,\ \hbox {in} \,\ {\mathbb {R}}^2,\\ \Big [-\frac{1}{2}\Delta +V_2(x)\Big ]w^2_2+|\nabla w_2|^2&=b_2\big (x^{\perp }\cdot \nabla w_1\big )w_2+f_2w_2\,\ \hbox {in} \,\ {\mathbb {R}}^2. \end{aligned} \right. \end{aligned}$$

(A.6)

For $i,j=1,2$, we get that

$$\begin{aligned} b_i\big (x^{\perp }\cdot \nabla w_j\big )w_i(x)\le \frac{b_i^2|x|^2|w_i|^2}{4}+|\nabla w_j|^2,\,\ f_i(x)w_i(x)\le \frac{\delta }{2} w_i^2+C_1(\delta )f_i^2, \end{aligned}$$

where $\delta >0$ is as in (A.2). Applying (A.1) and (A.2), it then follows from (A.6) that

$$\begin{aligned} (-\Delta +4Q^2+\delta )(w^2_1+w^2_2)\le C_1(\delta )P^2e^{-2Q|x|}\,\ \hbox {in} \,\ {\mathbb {R}}^2. \end{aligned}$$

(A.7)

Applying the comparison principle to (A.7), we thus deduce from (A.4) that

$$\begin{aligned} w^2_1+w^2_2\le C_2(\delta )P^2e^{-2Q|x|}\,\ \hbox {in} \,\ {\mathbb {R}}^2, \end{aligned}$$

which implies that (A.5) holds for $w_i(x)$ with $i=1$ and 2.

We next prove that (A.5) holds for $|\nabla w_i|$ with $i=1$ and 2. For $i,\,j=1,\, 2,$ denote $\partial _iw_j(x)=\frac{\partial w_j(x)}{\partial x_i}$, and let $\delta _{ij}$ be the Kronecker function satisfying $\delta _{ij}=1$ for $i=j$ and $\delta _{ij}=0$ for $i\ne j$. We then derive from (A.3) that $(\partial _iw_1,\partial _iw_2)$ satisfies

$$\begin{aligned} \left\{ \begin{aligned}&\big [-\Delta +V_1(x)\big ]\partial _iw_1+\partial _i V_1(x)w_1\\&\quad =b_1 \big [\big (x^{\perp }\cdot \nabla \partial _iw_2\big )-\delta _{i,2} \partial _1w_2+\delta _{i1}\partial _2w_2\big ]+\partial _if_1(x),\\&\big [-\Delta +V_2(x)\big ]\partial _iw_2+\partial _i V_2(x)w_2\\&\quad =b_2 \big [\big (x^{\perp }\cdot \nabla \partial _iw_1\big )-\delta _{i,2}\partial _1w_1 +\delta _{i1}\partial _2w_1\big ]+\partial _if_2(x).\\ \end{aligned} \right. \end{aligned}$$

(A.8)

The above system further yields that

$$\begin{aligned} \left\{ \begin{aligned}&\Big [-\frac{1}{2}\Delta +V_1(x)\Big ]|\nabla w_1|^2+\sum _{i=1}^{2}| \nabla \partial _iw_1|^2+\sum _{i=1}^{2}\partial _i V_1(x)w_1\partial _iw_1\\&=b_1\Big [\sum _{i=1}^{2}\big (x^{\perp }\cdot \nabla \partial _iw_2\big )\partial _iw_1+\partial _2w_2\partial _1w_1-\partial _1w_2 \partial _2w_1\Big ]+\sum _{i=1}^{2} \partial _if_1(x)\partial _iw_1\,\ \hbox {in} \,\ {\mathbb {R}}^2,\\&\Big [-\frac{1}{2}\Delta +V_2(x)\Big ]|\nabla w_2|^2+\sum _{i=1}^{2}|\nabla \partial _iw_2|^2+\sum _{i=1}^{2}\partial _i V_2(x)w_2\partial _iw_2 \\&=b_2\Big [\sum _{i=1}^{2}\big (x^{\perp }\cdot \nabla \partial _iw_1\big )\partial _iw_2+\partial _2w_1\partial _1w_2-\partial _1w_1 \partial _2w_2\Big ]+\sum _{i=1}^{2} \partial _if_2(x)\partial _iw_2\,\ \hbox {in} \,\ {\mathbb {R}}^2.\\ \end{aligned} \right. \end{aligned}$$

(A.9)

Note that for $i,j,l=1,2$ with $j\ne l$,

$$\begin{aligned}&\partial _i V_j(x)w_j\partial _iw_j\le |\partial _i V_j(x)|^2|\partial _iw_j|^2 +\frac{1}{4}w^2_j,\quad \partial _if_j(x)\partial _iw_j\le \frac{\delta }{2} (\partial _iw_j)^2+C_1(\delta )|\partial _if_j(x)|^2,\\&b_l(x^{\perp }\cdot \nabla \partial _iw_j\big )\partial _iw_l\le \frac{b_l^2|x|^2|\partial _i w_l|^2}{4}+|\nabla \partial _iw_j|^2,\quad b_l\partial _1w_j\partial _2w_l\le \frac{1}{2}|b_l|(|\nabla w_1|^2+|\nabla w_2|^2), \end{aligned}$$

where $\delta >0$ is again as in (A.2). Following above estimates, we infer from (A.2) that

$$\begin{aligned} \begin{aligned} \Big (-\Delta +4Q^2+\delta \Big )(|\nabla w_1|^2+|\nabla w_2|^2)\le (|w_1|^2+|w_2|^2)+2C_1(\delta )(|\nabla f_1|^2+|\nabla f_2|^2)\ \ \hbox {in} \,\ {\mathbb {R}}^2. \end{aligned} \end{aligned}$$

Applying (A.1) and (A.2), since (A.5) holds for $w_i(x)$ with $i=1$ and 2, we deduce from the above equation that

$$\begin{aligned} \Big [-\Delta +4Q^2+\delta \Big ](|\nabla w_1|^2+|\nabla w_2|^2)\le C_3(\delta )P^2e^{-2Q|x|}\ \ \hbox {in} \ \ {\mathbb {R}}^2. \end{aligned}$$

(A.10)

Applying the comparison principle to (A.10), we derive from (A.4) that

$$\begin{aligned} |\nabla w_1|^2+|\nabla w_2|^2\le C_4(\delta )P^2e^{-2Q|x|}\ \ \hbox {in} \ \ {\mathbb {R}}^2. \end{aligned}$$

We therefore conclude from above that (A.5) holds for $|\nabla w_i|$ with $i=1$ and 2, and we are done. $\square$

The rest part of this appendix is to derive some estimates used in the proof of Proposition 5.1, for which we consider the following minimization problem

$$\begin{aligned} e_a=\inf _{\{u\in \mathbb {H}, \, \Vert u\Vert ^2_2=1 \} }E_a(u), \end{aligned}$$

(A.11)

where $\mathbb {H} := \big \{u\in H^1({\mathbb {R}}^2, {\mathbb {R}}):\ \int _{{\mathbb {R}}^2} |x|^2u^2 \mathrm{d}x<\infty \big \}$, and $E_a(u)$ is defined by

$$\begin{aligned} E_a(u)=\int _{\mathbb {R}^2}\big (|\nabla u|^2+|x|^2u^2\big )\mathrm{d}x-\frac{a}{2}\int _{\mathbb {R}^2}u^4\mathrm{d}x,\ \ a>0. \end{aligned}$$

(A.12)

As stated in Theorem 2.1, $e_a$ admits minimizers if and only if $0<a <a^*=\Vert w\Vert ^2_{2}$. Let ${\hat{v}}_a>0$ be a real minimizer of $e_{a}$ as $a\nearrow a^*$. Then ${\hat{v}}_a>0$ satisfies the following Euler-Lagrange equation

$$\begin{aligned} -\Delta {\hat{v}}_a+|x|^2{\hat{v}}_a=\beta _a{\hat{v}}_a+a{\hat{v}}_a^3\ \ \text{ in }\,\ {\mathbb {R}}^2, \end{aligned}$$

(A.13)

where $\beta _a\in {\mathbb {R}}$ is the Lagrange multiplier and satisfies

$$\begin{aligned} \beta _a=e_{a}-\frac{a}{2}\int _{\mathbb {R}^2}{\hat{v}}_a^4 <0\ \ \text{ as }\,\ a\nearrow a^*. \end{aligned}$$

We also denote

$$\begin{aligned} \alpha _a=\frac{(a^*-a)^{\frac{1}{4}}}{\lambda }>0. \end{aligned}$$

The following lemma gives the estimates of $\beta _a$ and $\hat{v}_a$ as $a\nearrow a^*$:

Lemma A.2

Let ${\hat{v}}_a>0$ be a real minimizer of $e_{a}$. Then, as $a\nearrow a^*$,

• (i) $1+\alpha _a^2\beta _a =O(\alpha _a^4)$;

• (ii) $\big |\alpha _a{\hat{v}}_a(\alpha _ax)\big |\le Ce^{-\frac{3}{4}|x|},\,\Big |\nabla \big (\alpha _a{\hat{v}}_a(\alpha _ax)\big )\Big |\le Ce^{-\frac{2}{3}|x|}$ in ${\mathbb {R}}^2$;

• (iii) $\max _{i,j=1,2}\Big |\partial _i\partial _j\Big (\alpha _a{\hat{v}}_a(\alpha _ax)\Big )\Big |\le Ce^{-\frac{7}{12}|x|}$  in ${\mathbb {R}}^2$;

• (iv) $\Big |\alpha _a{\hat{v}}_a(\alpha _ax)-\frac{w(x)}{\sqrt{a^*}}\Big |\le C\alpha _a^{4}e^{-\frac{2}{3}|x|}$,   $\Big |\nabla \Big (\alpha _a{\hat{v}}_a(\alpha _ax)-\frac{w(x)}{\sqrt{a^*}}\Big ) \Big |\le C\alpha _a^{4}e^{-\frac{1}{2}|x|}$  in ${\mathbb {R}}^2$;

All above constants $C>0$ are independent of $0<a<a^*$.

Proof

• 1. The estimate (i) follows directly from (3.1) and (3.37) in [29].

• 2. Denote $\bar{v}_a(x)=\alpha _a{{\hat{v}}}(\alpha _ax)$. It then follows from (A.13) that $\bar{v}_a$ satisfies

  $$\begin{aligned} -\Delta {\bar{v}}_a+\alpha _a^4|x|^2{\bar{v}}_a=\alpha _a^2\beta _a{\bar{v}}_a +a{\bar{v}}_a^3\ \ \text{ in }\,\ {\mathbb {R}}^2. \end{aligned}$$

  (A.14)

Similarly to (3.13) in [31], one can obtain from (A.14) that $\Vert {\bar{v}}_a\Vert _{L^{\infty }({\mathbb {R}}^2)}\le C$ and ${\bar{v}}_a(x)\rightarrow 0$ as $|x|\rightarrow \infty$ for all $0<a<a^*$. Since the estimate (i) gives that $\alpha _a^2\beta _a\rightarrow -1$ as $a\nearrow a^*$, we derive from (A.14) that as $a\nearrow a^*$,

$$\begin{aligned} -\Delta {\bar{v}}_a+\frac{9}{16}{\bar{v}}_a\le 0 \ \ \text{ in }\, \ {\mathbb {R}}^2/B_R, \end{aligned}$$

(A.15)

where $R>0$ is large enough. By the comparison principle, we then deduce from (A.15) that as $a\nearrow a^*$,

$$\begin{aligned} |{\bar{v}}_a(x)|\le Ce^{-\frac{3}{4}|x|} \ \ \text{ in }\, \ {\mathbb {R}}^2/B_R, \end{aligned}$$

which implies that there exists a sufficiently large constant $C>0$ such that as $a\nearrow a^*$,

$$\begin{aligned} |{\bar{v}}_a(x)|\le Ce^{-\frac{3}{4}|x|} \ \ \text{ in }\, \ {\mathbb {R}}^2. \end{aligned}$$

(A.16)

Next, we give the estimate of $\nabla {\bar{v}}_a(x)$ as follows: denoting $\bar{v}_{a,j}:=\frac{\partial \bar{v}_a}{\partial x_j},$ $j=1, 2$, we then infer from (A.14) that $\bar{v}_{a,j}$ satisfies

$$\begin{aligned} -\Delta \bar{v}_{a,j}+\big (\alpha _a^4|x|^2-\alpha _a^2\beta _a-3a\bar{v}^2_{a}\big )\bar{v}_{a,j} =-2\alpha _a^4x_j\bar{v}_a\quad \hbox {in} \ {\mathbb {R}}^2. \end{aligned}$$

(A.17)

Applying (i) and (A.16), we have, for $a\nearrow a^*$, that

$$\begin{aligned} -\alpha _a^2\beta _a\rightarrow 1,\ |x_j\bar{v}_a|\le Ce^{-\frac{2}{3}|x|}\ \ \hbox {in}\,\ {\mathbb {R}}^2/B_R, \end{aligned}$$

where $R>0$ is large enough. Therefore, by the comparison principle, we deduce from (A.17) that as $a\nearrow a^*$,

$$\begin{aligned} |\bar{v}_{a,j}(x)|\le Ce^{-\frac{2}{3}|x|}\ \ \hbox {in}\ \ {\mathbb {R}}^2/B_R. \end{aligned}$$

On the other hand, similar to (3.13) in [31] again, one can get from (A.17) that $\bar{v}_{a,j}(x)$ is bounded uniformly in ${\mathbb {R}}^2$ for k, which further implies that as $a\nearrow a^*$,

$$\begin{aligned} |\bar{v}_{a,j}(x)|\le Ce^{-\frac{2}{3}|x|}\ \ \hbox {in}\,\ {\mathbb {R}}^2,\ \ j=1,\,2, \end{aligned}$$

(A.18)

from which we then obtain the desired estimate of $\nabla \bar{v}_{a}$ as $a\nearrow a^*$. This gives the estimate of (ii).

3. By the exponential estimate (A.18), applying gradient estimates [25, (3.15)] to (A.17) gives that as $a\nearrow a^*$,

$$\begin{aligned} |\nabla {\bar{v}}_{a,j}(x)|\le Ce^{-\frac{7}{12}|x|} \ \ \text{ in }\, {\mathbb {R}}^2, \end{aligned}$$

where $\bar{v}_{a,j}:=\frac{\partial \bar{v}_a}{\partial x_j}$ for $j=1, 2$. Taking $j=1,\,2$, we then obtain that for any $i,\,j=1,\,2$,

$$\begin{aligned} |\partial _i\partial _j{\bar{v}}_{a}(x)|\le Ce^{-\frac{7}{12}|x|} \ \ \hbox { in}\,\ {\mathbb {R}}^2 \end{aligned}$$

as $a\nearrow a^*$, which therefore gives the estimate of (iii).

4. Denote ${\tilde{v}}_a(x):={\bar{v}}_a(x)-\frac{w}{\sqrt{a^*}}=\alpha _a\hat{v}_a(\alpha _ax)-\frac{w}{\sqrt{a^*}}$. We then infer from [29, Theorem 1.4] that as, $a\nearrow a^*,$

$$\begin{aligned} \Vert {\tilde{v}}_a(x)\Vert _{L^{\infty }({\mathbb {R}}^2)}\le C\alpha _a^{4}. \end{aligned}$$

(A.19)

Moreover, because ${\tilde{v}}_a(x)$ satisfies the equation

$$\begin{aligned} \begin{aligned}&\quad -\Delta {\tilde{v}}_a-\alpha _a^2\beta _a{\tilde{v}}_a-a\big (\bar{v}^2_a +\frac{w\bar{v}_a}{\sqrt{a^*}}+\frac{w^2}{a^*}\big ){\tilde{v}}_a\\&=-\alpha _a^4|x|^2\bar{v}_a+\big (1+\alpha _a^2\beta _a\big )\frac{w}{\sqrt{a^*}} +\frac{(a-a^*)w^3}{(a^*)^{\frac{3}{2}}}\ \ \hbox {in}\,\ {\mathbb {R}}^2, \end{aligned} \end{aligned}$$

by the comparison principle, we obtain from (A.19) and (i) that, as $a\nearrow a^*$,

$$\begin{aligned} \big |{\tilde{v}}_a(x)\big |\le C\alpha _a^{4}e^{-\frac{2}{3}|x|}\ \ \hbox {in}\,\ {\mathbb {R}}^2/B_R, \end{aligned}$$

where $R>0$ is large enough. This further implies that, as $a\nearrow a^*$,

$$\begin{aligned} \big |{\tilde{v}}_a(x)\big |\le C\alpha _a^{4}e^{-\frac{2}{3}|x|}\ \ \hbox {in}\,\ {\mathbb {R}}^2. \end{aligned}$$

Applying the gradient estimate (3.15) in [25] and the above exponential decay of ${\hat{v}}_a(x)$, we finally obtain that

$$\begin{aligned} |\nabla {\tilde{v}}_a(x)|\le C\alpha _a^{4}e^{-\frac{|x|}{2}}\ \ \hbox {in}\,\ {\mathbb {R}}^2 \end{aligned}$$

as $a\nearrow a^*$, which therefore completes the proof of Lemma A.2. $\square$