The Lieb-Yau Conjecture for Ground States of Pseudo-Relativistic Boson Stars

TL;DR

This paper proves that for small enough stellar mass, the ground states of pseudo-relativistic Boson stars are unique, confirming a longstanding conjecture by Lieb and Yau in a specific case.

Contribution

It establishes the validity of the Lieb-Yau conjecture on the uniqueness of ground states for small stellar masses in pseudo-relativistic Boson stars.

Findings

Ground states exist if and only if stellar mass N is positive and less than a critical value N*.

The Lieb-Yau conjecture on uniqueness is confirmed for sufficiently small N.

The proof applies to the specific case where N is small enough.

Abstract

It is known that ground states of the pseudo-relativistic Boson stars exist if and only if the stellar mass $N>0$ satisfies $N<N^*$, where the finite constant $N^*$ is called the critical stellar mass. Lieb and Yau conjecture in [Comm. Math. Phys., 1987] that ground states of the pseudo-relativistic Boson stars are unique for each $N<N^*$. In this paper, we prove that the above uniqueness conjecture holds for the particular case where $N>0$ is small enough.

Equations

$$e(N):=\inf\Big{\{}\mathcal{E}(u):\,u\in H^{\frac{1}{2}}({\mathbb{R}}^{3})\,\ \mbox{and}\ \int_{{\mathbb{R}}^{3}}|u(x)|^{2}dx=N\Big{\}},$$

$$\mathcal{E}(u):=\int_{{\mathbb{R}}^{3}}\bar{u}\big{(}\sqrt{-\Delta+m^{2}}-m\big{)}udx-\frac{1}{2}\int_{{\mathbb{R}}^{3}}\big{(}|x|^{-1}\ast|u|^{2}\big{)}|u|^{2}dx,\ \ m>0.$$

$$\int_{\mathbb{R}^{3}}\big{(}\frac{1}{|x|}\ast|u|^{2}\big{)}|u|^{2}dx\leq\frac{2}{\|w\|^{2}_{2}}\|(-\Delta)^{1/4}u\|^{2}_{2}\,\|u\|^{2}_{2},\ \ u\in H^{\frac{1}{2}}({\mathbb{R}}^{3}),$$

$$\sqrt{-\Delta}\,u+u-\Big{(}\frac{1}{|x|}\ast|u|^{2}\Big{)}u=0\ \mbox{ in }\ {\mathbb{R}}^{3},\ \mbox{ where }\ u\in H^{\frac{1}{2}}({\mathbb{R}}^{3}).$$

$$\big{(}\sqrt{-\Delta+m^{2}}-m\big{)}u-\Big{(}\frac{1}{|x|}\ast|u|^{2}\Big{)}u=-\mu u\ \ \text{in}\,\ \mathbb{R}^{3},$$

$$(-\Delta {)}^{s}u+u-u^{\alpha +1}=0\text{mbox}in{\mathbb{R}}^N,u\in H^s({\mathbb{R}}^N),$$

$$\tilde{u}(x)=c^{2}u(cx),\text{mbox}wherec>0,$$

$$\mathcal{E}(u)=c^{-3}{\mathcal{E}}_c(\tilde{u})\text{ and }{\int }_{{\mathbb{R}}^3}|\tilde{u}{|}^{2}dx=c{\int }_{{\mathbb{R}}^3}|u{|}^{2}dx,$$

$$\mathcal{E}_{c}(u):=\int_{{\mathbb{R}}^{3}}\bar{u}\big{(}\sqrt{-c^{2}\Delta+c^{4}m^{2}}-c^{2}m\big{)}udx-\frac{1}{2}\int_{{\mathbb{R}}^{3}}\big{(}|x|^{-1}\ast|u|^{2}\big{)}|u|^{2}dx,\ \ m>0.$$

$$\bar{e}(c):=\inf\Big{\{}\mathcal{E}_{c}(u):\,u\in H^{\frac{1}{2}}({\mathbb{R}}^{3})\,\ \mbox{and}\ \int_{{\mathbb{R}}^{3}}|u(x)|^{2}dx=1\Big{\}}.$$

$$\big{(}\sqrt{-c^{2}\Delta+c^{4}m^{2}}-c^{2}m\big{)}Q_{c}-\Big{(}\frac{1}{|x|}\ast|Q_{c}|^{2}\Big{)}Q_{c}=-\mu_{c}Q_{c}\ \ \text{in}\,\ \mathbb{R}^{3}.$$

$${\mu }_c\to -\lambda <0\text{mbox}asc\to \infty$$

$$\begin{split}&\big{(}\sqrt{-c^{2}\Delta+m^{2}c^{4}}-mc^{2}\big{)}\phi_{c}-\Big{(}\frac{1}{|x|}\ast Q_{c}^{2}\Big{)}\phi_{c}-2k_{1}\Big{\{}\frac{1}{|x|}\ast\big{(}Q_{c}\phi_{c}\big{)}\Big{\}}Q_{c}\\ &-k_{2}(c)Q_{c}=-\mu_{c}\phi_{c}\ \text{ in}\ \ {\mathbb{R}}^{3}.\end{split}$$

$$|{\varphi }_c(x)|\le C{e}^{-\delta |x|}\text{mbox}in{\mathbb{R}}^3$$

$${\bar{H}}_c:=\sqrt{-c^2\Delta +m^2{c}^4}-m{c}^2.$$

$$Q_c\to Q_{\infty }\text{mbox}uniformlyin{\mathbb{R}}^3\text{ as }c\to \infty ,$$

$${e}_{m}:=\inf\Big{\{}{E}_{m}(u):\,u\in H^{1}({\mathbb{R}}^{3})\,\ \mbox{and}\ \int_{{\mathbb{R}}^{3}}|u(x)|^{2}dx=1\Big{\}},$$

$${E}_{m}(u):=\frac{1}{2m}\int_{{\mathbb{R}}^{3}}|\nabla u|^{2}dx-\frac{1}{2}\int_{{\mathbb{R}}^{3}}\big{(}|x|^{-1}\ast|u|^{2}\big{)}|u|^{2}dx,\ \forall u\in H^{1}({\mathbb{R}}^{3}),\ \ m>0.$$

$$-\frac{1}{2m}\Delta Q_{\infty }+\lambda Q_{\infty }=(|x{|}^{-1}\ast |Q_{\infty }{|}^2)Q_{\infty }\text{mbox}in{\mathbb{R}}^3,Q_{\infty }\in H^1({\mathbb{R}}^3),$$

$$L_{+}\xi:=\Big{(}-\frac{1}{2m}\Delta+\lambda-|x|^{-1}\ast|Q_{\infty}|^{2}\Big{)}\xi-2\big{(}|x|^{-1}\ast(Q_{\infty}\xi)\big{)}Q_{\infty},$$

$$\ker L_{+}=\text{\rm span}\Big{\{}\frac{\partial Q_{\infty}}{\partial x_{1}},\frac{\partial Q_{\infty}}{\partial x_{2}},\frac{\partial Q_{\infty}}{\partial x_{3}}\Big{\}}.$$

$$\big{(}\sqrt{-c^{2}\Delta+m^{2}c^{4}}-mc^{2}\big{)}Q_{c}-(|x|^{-1}\ast Q_{c}^{2})Q_{c}=\mu_{c}Q_{c}\ \text{ in}\,\ {\mathbb{R}}^{3},$$

$${\mu }_c\to -\lambda <0\text{ as }c\to \infty ,$$

$$\begin{split}\mathcal{L}_{k_{1},k_{2}}\xi:=&\big{(}\sqrt{-c^{2}\Delta+m^{2}c^{4}}-mc^{2}\big{)}\xi-\Big{(}\frac{1}{|x|}\ast Q_{c}^{2}\Big{)}\xi\\ &-2k_{1}\Big{\{}\frac{1}{|x|}\ast\big{(}Q_{c}\xi\big{)}\Big{\}}Q_{c}-k_{2}(c)Q_{c}\ \text{ in}\ \ {\mathbb{R}}^{3},\end{split}$$

$${\mathcal{L}}_{k_1,k_2}{\phi }_c={\mu }_c{\phi }_c\text{ in}{\mathbb{R}}^3,$$

$$|{\phi }_c(x)|\le C{e}^{-\delta |x|}\text{ in}{\mathbb{R}}^3$$

$$|Q_c(x)|,|\nabla Q_c(x)|\le C{e}^{-\delta |x|}\text{mbox}in{\mathbb{R}}^3$$

$$Q_c\to Q_{\infty }\text{mbox}uniformlyin{\mathbb{R}}^3\text{ as }c\to \infty ,$$

$$Q_c\to Q_{\infty }\text{ in }{H}^1({\mathbb{R}}^3)\text{ as }c\to \infty ,$$

$$\begin{split}\Big{(}-\frac{1}{2m}\Delta-\mu_{c}\Big{)}Q_{c}&=\big{(}|x|^{-1}\ast Q_{c}^{2}\big{)}Q_{c}-\Big{(}\sqrt{-c^{2}\Delta+m^{2}c^{4}}-mc^{2}+\frac{1}{2m}\Delta\Big{)}Q_{c}\\ &:=\big{(}|x|^{-1}\ast Q_{c}^{2}\big{)}Q_{c}-F_{c}(\nabla)Q_{c}\ \text{ in}\,\ {\mathbb{R}}^{3},\end{split}$$

Full text

Keywords: Uniqueness; Ground states; Boson stars; Pohozaev identity

1 Introduction

Various models of pseudo-relativistic boson stars have attracted a lot of attention in theoretical and numerical astrophysics over the past few decades, see [29, 30] and references therein. In this paper, we are interested in ground states of pseudo-relativistic Boson stars in the mean field limit (cf. [10, 14, 29]), which can be described by constraint minimizers of the following variational problem

[TABLE]

where $N>0$ denotes the stellar mass of Boson stars, and the pseudo-relativistic Hartree energy functional $\mathcal{E}(u)$ is of the form

[TABLE]

Here the operator $\sqrt{-\Delta+m^{2}}$ is defined via multiplication in the Fourier space with the symbol $\sqrt{|\xi|^{2}+m^{2}}$ for $\xi\in{\mathbb{R}}^{3}$, which describes the kinetic and rest energy of many self-gravitating and relativistic bosons with rest mass $m>0$, and the symbol $\ast$ stands for the convolution on ${\mathbb{R}}^{3}$. Because of the physical relevance, without special notations we always focus on the case $m>0$ throughout the whole paper. The main purpose of this paper is to prove the uniqueness of minimizers for $e(N)$, provided that $N>0$ is small enough.

The variational problem $e(N)$ is essentially in the class of $L^{2}-$critical constraint minimization problems, which were studied recently in the nonrelativistic cases, e.g. [5, 19, 20, 22] and references therein. Comparing with these mentioned works, it however deserves to remark that the analysis of $e(N)$ is more complicated in a substantial way, which is mainly due to the nonlocal nature of the pseudo-differential operator $\sqrt{-\Delta+m^{2}}$, and the convolution-type nonlinearity as well. Starting from the pioneering papers [29, 30], many works were devoted to the mathematical analysis of the variational problem $e(N)$ over the past few years, see [10, 12, 13, 14, 15, 23, 26, 33, 37] and references therein. The existing results show that the analysis of $e(N)$ is connected well to the following Gagliardo-Nirenberg inequality of fractional type

[TABLE]

where $w(x)=w(|x|)>0$ is a ground state, up to translations and suitable rescaling (cf. [14, 29]), of the fractional equation

[TABLE]

By making full use of (1.3), Lenzmann in [26] established the following interesting existence and analytical characters of minimizers for $e(N)$:

Theorem A ([26, Theorem 1]) Under the assumption $m>0$, the following results hold for $e(N)$:

$e(N)$* has minimizers if and only if $0<N<N^{*}$, where the finite constant $N^{*}$ is independent of $m$.* 2. 2.

Any minimizer $u$ of $e(N)$ satisfies $u\in H^{s}({\mathbb{R}}^{3})$ for all $s\geq\frac{1}{2}$. 3. 3.

Any nonnegative minimizer of $e(N)$ must be strictly positive and radially symmetric-decreasing, up to phase and translation.

We remark that the existence of the critical constant $N^{*}>0$ stated in Theorem A, which is called the critical stellar mass of boson stars, was proved earlier in [14, 29]. Further, the dynamics and some other analytic properties of minimizers for $e(N)$ were also investigated by Lenzmann and his collaborators in [12, 13, 14, 15, 25, 26] and references therein. Stimulated by [20, 22], the related limit behavior of minimizers for $e(N)$ as $N\nearrow N^{*}$ were studied more recently in [23, 33, 37], where the Gagliardo-Nirenberg type inequality (1.3) also played an important role. Note also from the variational theory that any minimizer $u$ of $e(N)$ satisfies the Euler-Lagrange equation

[TABLE]

where $\mu\in{\mathbb{R}}$ is the associated Lagrange multiplier. Following (1.5) and Theorem A, one can deduce that any minimizer of $e(N)$ must be either positive or negative, see [23] for details. Therefore, it is enough to consider positive minimizers of $e(N)$, which are called ground states of $e(N)$ throughout the rest part of this paper.

Whether a physics system admits a unique ground state or not is an interesting and fundamental problem. Lieb and Yau [29] conjectured in 1987 that for each $N<N^{*}$, there exists a unique ground state (minimizer) of $e(N)$. As expected by Lieb and Yau there, the analysis of this uniqueness conjecture is however challenging extremely. Actually, it is generally difficult to prove whether any two different ground states of $e(N)$ satisfy the equation (1.5) with the same Lagrange multiplier $\mu\in{\mathbb{R}}$. On the other hand, more difficultly, it seems very challenging to address the uniqueness of ground states for (1.5). Essentially, even though the uniqueness of ground states for the following fractional equation

[TABLE]

where $0<s<1$, $\alpha>0$ and $N\geq 1$, was already proved in the celebrated works [11, 12], the uniqueness of ground states for (1.4) or (1.5) is still open, due to the nonlocal nonlinearity of Hartree type. Therefore, whether the above Lieb-Yau conjecture is true for all $0<N<N^{*}$ remains mainly open after three decades, except Lenzmann’s recent work [26] in 2009.

As an important first step towards the Lieb-Yau conjecture, Lenzmann proved in [26] that for each $0<N\ll N^{*}$ and except for at most countably many $N$, the uniqueness of minimizers for $e(N)$ holds true. We emphasize that the additional assumption “except for at most countably many $N$” seems essential in Lenzmann’s proof, since the smoothness of the GP energy $e(N)$ with respect to $N$ was employed there. In this paper we intend to remove the above additional assumption and prove the Lieb-Yau conjecture in the particular case where $0<N<N^{*}$ is small enough. More precisely, our main result of this paper is the following uniqueness of minimizers for $e(N)$.

Theorem 1.1**.**

If $N>0$ is small enough, then the problem $e(N)$ admits a unique positive minimizer, up to phase and translation.

The similar uniqueness results of Theorem 1.1 were established recently in [2, Theorem 2] and [31, Theorem 1.1] (see also [21, Corollary 1.1]) for the nonrelativistic Hartree minimization problems with trapping potentials, under the additional assumption that the associated nonrelativistic operator admits the first eigenvalue. We however emphasize that the arguments of [2, 31, 21] are not applicable for proving Theorem 1.1, since the associated pseudo-relativistic operator $H:=\sqrt{-\Delta+m^{2}}-m$ does not admit any eigenvalue in our problem $e(N)$. Therefore, a different approach is needed for proving Theorem 1.1. Towards this purpose, since positive minimizers $u$ of $e(N)$ vanish uniformly as $N\searrow 0$, motivated by [26] we define

[TABLE]

so that

[TABLE]

where the energy functional $\mathcal{E}_{c}(\cdot)$ is given by

[TABLE]

Consider the minimization problem

[TABLE]

Note from Theorem A and (1.7) that if $c>0$ is large enough, then $\bar{e}(c)$ in (1.9) admits positive minimizers. More importantly, setting $c=N^{-1}>0$, studying positive minimizers of $e(N)$ as $N\searrow 0$ is then equivalent to investigating positive minimizers of the minimization problem (1.9) as $c\nearrow\infty$. Therefore, to establish the uniqueness of Theorem 1.1, it suffices to prove the following uniqueness theorem.

Theorem 1.2**.**

If $c>0$ is large enough, then $\bar{e}(c)$ in (1.9) admits a unique positive minimizer, up to phase and translation.

Suppose now that $Q_{c}>0$ is a minimizer of $\bar{e}(c)$ defined in (1.9). Then there exists a Lagrange multiplier $\mu_{c}\in{\mathbb{R}}$ such that $Q_{c}>0$ solves

[TABLE]

Recall from [26, Proposition 1] that up to a subsequence if necessary, $\mu_{c}\in{\mathbb{R}}$ satisfies

[TABLE]

for some constant $\lambda>0$. In order to prove Theorem 1.2, associated to $Q_{c}$, we need to study the uniformly exponential decay of $\phi_{c}\in H^{\frac{1}{2}}({\mathbb{R}}^{3})$ as $c\to\infty$, where $\phi_{c}$ satisfies

[TABLE]

Here $\mu_{c}\in{\mathbb{R}}$ satisfies (1.11), $k_{1}\geq 0$ and $k_{2}(c)\in{\mathbb{R}}$ is bounded uniformly in $c>0$. As proved in Lemma 2.1, we shall derive the following uniformly exponential decay of $\phi_{c}$ as $c\to\infty$:

[TABLE]

holds uniformly as $c\to\infty$, where the constants $\delta>0$ and $C=C(\delta)>0$ are independent of $c$. Since $\mu_{c}\in{\mathbb{R}}$ satisfies (1.11), stimulated by [1, 14, 24, 35], the proof of (1.13) is based on the uniformly exponential decay (2.25) of the Green’s function $G_{c}(\cdot)$ for \big{(}\bar{H}_{c}+\mu_{c}\big{)}^{-1} as $c\to\infty$, where the operator $\bar{H}_{c}$ is defined by

[TABLE]

As shown in Lemma 2.3, it however deserves to remark that because $G_{c}(\cdot)$ depends on $c>0$, one needs to carry out more delicate analysis, together with some tricks, for addressing the uniformly exponential decay (2.25) of $G_{c}(\cdot)$ as $c\to\infty$. On the other hand, as a byproduct, the exponential decay (1.13) can be useful in analyzing the limiting procedure of solutions for Schrodinger equations involving the above fractional operator $\bar{H}_{c}$, which were investigated widely in [6, 7, 8] and references therein.

Following (1.13) and the regularity of $Q_{c}$, in Section 2 we shall finally prove the following limit behavior

[TABLE]

where $Q_{\infty}>0$ is the unique positive minimizer of (2.1) described below. Based on the refined estimates of Section 2, motivated by [9, 18, 19], we shall employ the nondegenerancy and uniqueness of $Q_{\infty}$ to complete the proof of Theorem 1.2 by establishing Pohozaev identities.

This paper is organized as follows. In Section 2 we shall address some refined estimates of $Q_{c}$ as $c\to\infty$, where Lemma 2.1 is proved in Subsection 2.1. Following those estimates of Section 2, Section 3 is devoted to the proof of Theorems 1.2 on the uniqueness of minimizers for $\bar{e}(c)$ as $c\to\infty$. Theorem 1.1 then follows immediately from Theorem 1.2 in view of the relation (1.7).

2 Analytical Properties of $Q_{c}$ as $c\to\infty$

The main purpose of this section is to give some refined analytical estimates of $Q_{c}$ as $c\to\infty$, where $Q_{c}>0$ is a positive minimizer of $\bar{e}(c)$ defined in (1.9). Note also from Theorem A and (1.7) that $Q_{c}$ is radially symmetric in $|x|$.

We first introduce the following limit problem associated to $\bar{e}(c)$:

[TABLE]

where the energy functional $E_{m}(\cdot)$ satisfies

[TABLE]

For any given $m>0$, it is well-known that, up to translations, problem (2.1) has a unique positive minimizer denoted by $Q_{\infty}(|x|)$, which must be radially symmetric, see [26, 27] and references therein. Further, $Q_{\infty}>0$ solves the following equation

[TABLE]

where the Lagrange multiplier $\lambda>0$ depends only on $m$ and is determined uniquely by the constraint condition $\int_{{\mathbb{R}}^{3}}Q_{\infty}^{2}dx=1$. Note from [32, Theorem 3] that $Q_{\infty}>0$ is a unique positive solution of (2.3). Moreover, recall from [26, Theorem 4] that $Q_{\infty}$ is non-degenerate, in the sense that the linearized operator $L_{+}:\,H^{2}({\mathbb{R}}^{3})\mapsto L^{2}({\mathbb{R}}^{3})$, which is defined by

[TABLE]

satisfies

[TABLE]

As a positive minimizer of $\bar{e}(c)$, $Q_{c}>0$ satisfies the following equation

[TABLE]

where $\mu_{c}\in{\mathbb{R}}$ is a suitable Lagrange multiplier. Recall from [26, Proposition 1] that up to a subsequence if necessary, the Lagrange multiplier $\mu_{c}$ of (2.6) satisfies

[TABLE]

where $\lambda>0$ is the same as that of (2.3). Associated to the positive minimizer $Q_{c}>0$ of (2.6), we next define the linearized operator

[TABLE]

where the constants $m>0$ and $k_{1}\geq 0$, and $k_{2}(c)\in{\mathbb{R}}$ is bounded uniformly in $c>0$.

Lemma 2.1**.**

Suppose $\varphi_{c}=\varphi_{c}(x)\in H^{\frac{1}{2}}({\mathbb{R}}^{3})$ is a solution of

[TABLE]

where $\mu_{c}$ satisfies (2.7), and the operator $\mathcal{L}_{k_{1},k_{2}}$ is defined by (2.8) for some constants $m>0$ and $k_{1}\geq 0$, and $k_{2}(c)\in{\mathbb{R}}$ being bounded uniformly in $c>0$. Then there exist $\delta>0$ and $C=C(\delta)>0$, which are independent of $c>0$, such that

[TABLE]

uniformly for all sufficiently large $c>0$.

Since the proof of Lemma 2.1 is a little involved, we leave it to Subsection 2.1. Applying Lemma 2.1, we next address the following estimates of $Q_{c}>0$ as $c\to\infty$, which are crucial for the proof of Theorem 1.2.

Proposition 2.2**.**

Let $Q_{c}>0$ be a positive minimizer of $\bar{e}(c)$ defined in (1.9) as $c\to\infty$. Then we have

There exist $\delta>0$ and $C=C(\delta)>0$, which are independent of $c>0$, such that

[TABLE]

uniformly as $c\to\infty$. 2. 2.

$Q_{c}$* satisfies*

[TABLE]

where $Q_{\infty}>0$ is the unique positive minimizer of (2.1).

Proof. 1. Since $Q_{c}>0$ solves (2.6), the uniformly exponential decay (2.11) for $Q_{c}$ as $c\to\infty$ follows directly from (2.45) below. Because $\frac{\partial Q_{c}}{\partial x_{i}}$ satisfies (2.9) for $k_{1}=1$ and $k_{2}(c)\equiv 0$, where $i=1,2,3$, the exponential decay (2.11) holds for $\nabla Q_{c}$ by applying Lemma 2.1.

2. Following [31, Lemma 4.9] and references therein, we first recall from [26, Proposition 1] that $Q_{c}$ satisfies

[TABLE]

where the convergence holds for the whole sequence of $\{c\}$, due to the uniqueness of $Q_{\infty}>0$. Rewrite (2.6) as

[TABLE]

where we denote the pseudo-differential operator

[TABLE]

with the symbol

[TABLE]

Recall from (2.11) that $Q_{c}$ decays exponentially as $|x|\to\infty$ for all sufficiently large $c>0$. Moreover, since the operator $\bar{H}_{c}:=\sqrt{-c^{2}\Delta+m^{2}c^{4}}-mc^{2}$ is uniformly bounded from below for all $c>0$, the similar argument of [14, Theorem 4.1(i)] or [7, Proposition 4.2] applied to (2.6) and (2.7) yields that

[TABLE]

which further implies the uniform smoothness of $Q_{c}$ in $c>0$, and

[TABLE]

Applying the Taylor expansion, we obtain from (2.15) that for $|\xi|\geq\frac{mc}{2}$,

[TABLE]

and for $|\xi|\leq\frac{mc}{2}$,

[TABLE]

due to the fact that $|\sqrt{1+t}-1-\frac{1}{2}t|\leq\frac{1}{8}t^{2}$ holds for all $0<t\leq\frac{1}{2}$. Following above estimates, we then derive from (2.15) and (2.16) that for sufficiently large $c>0$,

[TABLE]

where $M_{1}>0$ and $M_{2}>0$ are independent of $c>0$. Also, since

[TABLE]

we derive from (2.16) that for any $p\geq 2$,

[TABLE]

where $K_{p}>0$ is independent of $c>0$. Employing (2.17) and (2.19), together with Sobolev imbedding theorem, the bootstrap argument applied to (2.14) yields that

[TABLE]

On the other hand, one can easily deduce from (2.3) that $Q_{\infty}$ decays exponentially as $|x|\to\infty$. Together with (2.11), this indicates that for any $\varepsilon>0$, there exists a constant $R_{\varepsilon}>0$, independent of $c>0$, such that

[TABLE]

and hence,

[TABLE]

Moreover, it follows from (2.20) that for sufficiently large $c>0$,

[TABLE]

The above two estimates thus yield that for sufficiently large $c>0$,

[TABLE]

which implies that (2.12) holds true. The lemma is therefore proved. ∎

2.1 Uniformly exponential decay as $c\to\infty$

In this Subsection, we address the proof of Lemma 2.1 on the uniformly exponential decay as $c\to\infty$. We remark that even though the proof of Lemma 2.1 is stimulated from [1, 14, 24, 35], as shown in proving Lemma 2.3 below, we need to carry out more delicate analysis together with some tricks.

We first suppose that $\varphi_{c}\in H^{\frac{1}{2}}({\mathbb{R}}^{3})$ is a solution of

[TABLE]

where the constant $m>0$ and

[TABLE]

for some positive constant $\lambda$. Define

[TABLE]

Therefore, $\varphi_{c}$ can be thought of as an eigenfunction of the Schrodinger operator $H:=\bar{H}_{c}+V(x)$. Moreover, the argument of [14, Theorem 4.1] or [7, Proposition 4.2] gives that $\varphi_{c}\in H^{s}({\mathbb{R}}^{3})$ for all $s\geq\frac{1}{2}$, which implies the smoothness of $\varphi_{c}$. Further, the spectrum of $\bar{H}_{c}$ satisfies

[TABLE]

for all $c>0$. Under the assumption (2.22), then \big{(}\bar{H}_{c}+\lambda_{c}\big{)}^{-1} exists for all sufficiently large $c>0$, and (2.21) can be rewritten as

[TABLE]

Note also from (2.21) and (2.23) that

[TABLE]

where $G_{c}(x-y)$ is the Green’s function of \big{(}\bar{H}_{c}+\lambda_{c}\big{)}^{-1} defined in (2.23). The following lemma gives the uniformly exponential decay of $G_{c}(\cdot)$ as $c\to\infty$.

Lemma 2.3**.**

Suppose $\lambda_{c}\in{\mathbb{R}}$ satisfies (2.22) for some $\lambda>0$. Then for each $0<\delta<\min\{\frac{m}{2},\sqrt{\lambda m}\}$, there exists a constant $M:=M(\delta)>0$, independent of $c>0$, such that the Green’s function $G_{c}(x-y)$ of \big{(}\bar{H}_{c}+\lambda_{c}\big{)}^{-1} satisfies

[TABLE]

uniformly for all sufficiently large $c>0$.

Proof. Under the assumption (2.22), since $G_{c}(\cdot)$ is the Green’s function of \big{(}\bar{H}_{c}+\lambda_{c}\big{)}^{-1} defined in (2.23) for all sufficiently large $c>0$, we have

[TABLE]

and $f_{c}^{-1}:\mathcal{S^{\prime}}\to\mathcal{S}^{\prime}$ denotes the inverse Fourier transform of $f_{c}$. We obtain from (2.26) that for all sufficiently large $c>0$,

[TABLE]

where

[TABLE]

In view of (2.27), we next define

[TABLE]

so that

[TABLE]

Note from pp. 183 of [28] that

[TABLE]

where $K_{2}(\cdot)>0$ denotes the modified Bessel function of the third kind. We then derive from above that

[TABLE]

Recall from [24] that there exist positive constants $\bar{M}_{1}$ and $\bar{M}_{2}$, independent of $c>0$, such that

[TABLE]

where $w>0$ is a real number. We next follow (2.29) and (2.30) to complete the proof by discussing separately the following two cases, which involve very complicated estimates together with some tricks:

(1). Case of $|z|\geq\frac{1}{mc}$. In this case, we have $\sqrt{t^{2}+|z|^{2}}\geq\frac{1}{mc}$ for all $t\geq 0$. We then obtain from (2.22), (2.29) and (2.30) that for all sufficiently large $c>0$,

[TABLE]

where $A(t,z)$ satisfies

[TABLE]

For $I_{A}(z)+I_{D}(z)$, we note that if $t\geq 0$ satisfies

[TABLE]

then one can check that

[TABLE]

for sufficiently large $c>0$. We thus obtain from (2.32) and (2.33) that for sufficiently large $c>0$,

[TABLE]

where $\sigma>0$ is arbitrary, and the constants $C_{1}>0$ and $C_{2}>0$ are independent of $c>0$.

For $I_{B}(z)$, we observe that if $t>0$ satisfies

[TABLE]

then we have

[TABLE]

for sufficiently large $c>0$. We thus obtain from (2.32) and (2.35) that for sufficiently large $c>0$,

[TABLE]

where the constants $C_{3}>0$ and $C_{4}>0$ are independent of $c>0$.

As for $I_{C}(z)$, we get that if $t>0$ satisfies

[TABLE]

then we have

[TABLE]

for sufficiently large $c>0$, where $0<\varepsilon<\frac{1}{4}$ is arbitrary. We thus obtain from (2.32) and (2.37) that for sufficiently large $c>0$,

[TABLE]

Note that if $\frac{1}{\sqrt{c}}\leq s$, then $2\sqrt{c}ds\leq cds^{2}$. We thus derive from (2.38) that for $\tau:=s^{2}>0$,

[TABLE]

where the constant $C_{5}>0$ is also independent of $c>0$, and $0<\varepsilon<\frac{1}{4}$ is arbitrary as before.

Following (2.31), we now conclude from above that for 0<\delta_{0}:=\min\{\big{(}\frac{1}{2}-\varepsilon\big{)}m,\sqrt{\lambda m}\}, where $0<\varepsilon<\frac{1}{4}$ is arbitrary, there exists a constant $M_{0}:=M_{0}(\delta_{0})>0$ such that for all sufficiently large $c>0$,

[TABLE]

This further implies that for each $0<\delta_{1}<\min\{\frac{m}{2},\sqrt{\lambda m}\}$, there exists a constant $M_{1}:=M_{1}(\delta_{1})>0$ such that for $|z|\geq\frac{1}{mc}$,

[TABLE]

uniformly for all sufficiently large $c>0$.

(2). Case of $|z|\leq\frac{1}{mc}$. In this case, we deduce from (2.22), (2.29) and (2.30) that for all sufficiently large $c>0$,

[TABLE]

Similar to (2.31) and (2.40), one can obtain that for all sufficiently large $c>0$,

[TABLE]

where the constants $C_{6}>0$ and $M_{2}>0$ are independent of $c>0$. As for $I_{2}(z)$, we infer that

[TABLE]

where the constant $C_{7}>0$ is independent of $c>0$. We therefore derive from (2.42) and above that for all sufficiently large $c>0$,

[TABLE]

where the constant $M_{3}>0$ is also independent of $c>0$.

We finally conclude from (2.41) and (2.44) that (2.25) holds true, and we are done. ∎

Proof of Lemma 2.1. We first prove that the positive solution $Q_{c}>0$ of (2.6), where the Lagrange multiplier $\mu_{c}$ is as in (2.7), satisfies the following exponential decay

[TABLE]

uniformly for all sufficiently large $c>0$, where the constants $\delta>0$ and $C=C(\delta)>0$ are independent of $c>0$. Actually, recall from (2.6) that $Q_{c}$ can be rewritten as

[TABLE]

where $G_{c}(x-y)$ is the Green’s function of \big{(}\bar{H}_{c}+(-\mu_{c})\big{)}^{-1} defined by (2.23), and the potential $V(x):=-(\frac{1}{|x|}\ast Q_{c}^{2})(x)$ satisfies $V(x)\in C^{0}({\mathbb{R}}^{3})$ and $\lim_{|x|\to\infty}V(x)=0$. Since $-\mu_{c}\to\lambda>0$ as $c\to\infty$ in view of (2.7), $G_{c}(x-y)$ satisfies the exponential decay of Lemma 2.3. Following the above properties, the uniformly exponential decay (2.45) as $c\to\infty$ can be proved in a similar way of [14, Appendix C] and [24, Theorem 2.1], where the Slaggie-Wichmann method (e.g. [24]) is employed.

To finish the proof of Lemma 2.1, we next rewrite the solution $\varphi_{c}\in H^{\frac{1}{2}}({\mathbb{R}}^{3})$ of (2.9) as

[TABLE]

where the operator $\bar{H}_{c}$ satisfies (2.23) as before and

[TABLE]

Here $\mu_{c}\in{\mathbb{R}}$ satisfies (2.7), $k_{1}\geq 0$, and $k_{2}(c)\in{\mathbb{R}}$ is bounded uniformly in $c>0$. Note from (2.47) that $\varphi_{c}\in H^{\frac{1}{2}}({\mathbb{R}}^{3})$ solves

[TABLE]

where the Green’s function $G_{c}(\cdot)$ of \big{(}\bar{H}_{c}+(-\mu_{c})\big{)}^{-1} satisfies as before the uniformly exponential decay of Lemma 2.3 in view of (2.7). Since $Q_{c}$ satisfies the uniformly exponential decay (2.45) as $c\to\infty$, the uniformly exponential decay (2.10) of $\varphi_{c}$ as $c\to\infty$ can be further proved in a similar way of [13, Lemma 4.9]. We omit the detailed proof for simplicity. This completes the proof of Lemma 2.1. ∎

3 Proof of Theorem 1.2

Following the refined estimates of previous section, in this section we shall complete the proof of Theorem 1.2. We begin with the following two lemmas.

Lemma 3.1**.**

Suppose $Q_{\infty}$ is the unique radially symmetric positive solution of (2.3) and let the operator $L_{+}$ be defined by (2.4). Then we have

[TABLE]

where $\lambda>0$ is as in (2.3).

Proof. Direct calculations give that

[TABLE]

and

[TABLE]

We then have

[TABLE]

Taking the action $x\cdot\nabla$ on (2.3), we deduce that

[TABLE]

Together with (3.2), this indicates that

[TABLE]

Since

[TABLE]

it follows from (3.3) that

[TABLE]

Moreover, recall from (2.3) that

[TABLE]

Combining (3.5) with (3.6) thus yields that

[TABLE]

and the proof of this lemma is therefore complete. ∎

Lemma 3.2**.**

Let $Q_{c}$ be a radially symmetric positive minimizer of $\bar{e}(c)$ defined in (1.9). Then we have the following Pohozaev identity

[TABLE]

Proof. In the proof of this lemma, we denote $Q_{c}$ by $Q$ for convenience. We first note that

[TABLE]

Multiplying $x\cdot\nabla Q(x)$ on both sides of (2.6) and integrating over ${\mathbb{R}}^{3}$, we have

[TABLE]

Moreover, we derive from the exponential decay (2.11) that

[TABLE]

where the argument of deriving (3.4) is used in the last equality. Since

[TABLE]

we obtain from (3.9) that

[TABLE]

One can easily check that

[TABLE]

Thus, it follows from (2.6), (3.8) and (3.10) that

[TABLE]

By (3.11), multiplying $Q$ on both sides of (2.6) and integrating over ${\mathbb{R}}^{3}$ yield that

[TABLE]

We also derive from (2.6) that

[TABLE]

which therefore implies that (3.7) holds true by applying (3.12). ∎

Following previous estimates, we are now ready to finish the proof of Theorem 1.2.

Proof of Theorem 1.2. Up to the phase and translation, it suffices to prove that $\bar{e}(c)$ in (1.9) admits a unique positive minimizer for sufficiently large $c>0$. On the contrary, suppose that $Q_{1c}$ and $Q_{2c}$ are two different radially symmetric (about the origin) positive minimizers of problem (1.9), where $m>0$ is fixed.

Then $Q_{ic}\in H^{s}({\mathbb{R}}^{3})$, where $s\geq\frac{1}{2}$, satisfies the following equation

[TABLE]

where $\mu_{ic}\in{\mathbb{R}}$ is the Lagrange multiplier associated to $Q_{ic}$ for $i=1,2$. Since $Q_{1c}\not\equiv Q_{2c}$ in ${\mathbb{R}}^{3}$, we define

[TABLE]

It then follows from (3.13) that

[TABLE]

Recall from (2.7) that

[TABLE]

We also note from (3.13) that

[TABLE]

which implies that

[TABLE]

where $k_{2}(c)$ is bounded uniformly in $c>0$ by (2.11) and (2.16). Applying Lemma 2.1 to the equation (3.15), we then deduce from (3.16) and (3.17) that there exist $\delta>0$ and $C=C(\delta)>0$, which are independent of $c>0$, such that

[TABLE]

uniformly as $c\to\infty$. Similar to the proof of Lemma 2.1, following (3.15) to consider the equation of $\frac{\partial w_{c}}{\partial x_{i}}$ ($i=1,2,3$), we further derive from (3.16)–(3.18) that there exist $\delta_{1}>0$ and $C_{1}=C_{1}(\delta_{1})>0$, which are independent of $c>0$, such that

[TABLE]

uniformly as $c\to\infty$.

Rewrite the equation (3.15) as

[TABLE]

for any $\mu\in{\mathbb{R}}$, where the uniformly bounded function $k_{2}(c)\in{\mathbb{R}}$ is as in (3.17), and the operator $\bar{H}_{c}$ satisfies

[TABLE]

Since $\|w_{c}\|_{L^{\infty}({\mathbb{R}}^{3})}\leq 1$ for all $c>0,$ we obtain from (2.11), (2.16) and (2.18) that for sufficiently large $c>0$,

[TABLE]

where $M>0$ is independent of $c>0$. In view of (2.16) and (3.21), the similar argument of [14, Theorem 4.1(i)] or [7, Proposition 4.2] applied to (3.20) and (3.16) yields that

[TABLE]

which further implies the uniform smoothness of $w_{c}$ in $c>0$.

We next rewrite the equation (3.15) as

[TABLE]

where the uniformly bounded function $k_{2}(c)\in{\mathbb{R}}$ is again as in (3.17), and the pseudo-differential operator $F_{c}(\nabla)$ is the same as (2.15) with the symbol

[TABLE]

Note that $F_{c}(\nabla)$ satisfies the estimate (2.17). We then derive from (2.17) and (3.22) that

[TABLE]

where $M_{1}>0$ and $M_{2}>0$ are independent of $c>0$.

Using the uniformly exponential decays (2.11) and (3.18), by the standard elliptic regularity we derive from (2.16), (3.23) and (3.24) that $\|w_{c}\|_{C^{\alpha}_{loc}({\mathbb{R}}^{3})}\leq M_{3}$ for some $\alpha\in(0,1)$, where the constant $M_{3}>0$ is independent of $c$. Therefore, there exists a function $w_{0}=w_{0}(x)$ such that up to a subsequence if necessary, we have

[TABLE]

Moreover, applying the estimates (3.16), (3.17) and (3.24), we deduce from (3.23) that $w_{0}$ satisfies

[TABLE]

where the uniformly exponential decay (2.11) is also used. Applying (2.5) and Lemma 3.1, we now derive from (3.25) that there exist constants $b_{0}$ and $c_{i}$ ($i=1,2,3$) such that

[TABLE]

Further, since $Q_{1c}$ and $Q_{2c}$ are both radially symmetric in $|x|$ for all $c>0$, the definition of $w_{c}$ implies that $w_{0}$ is also radially symmetric in $|x|$, i.e., $w_{0}\in L^{2}_{rad}({\mathbb{R}}^{3})$. Applying [26, Proposition 2], it then follows from the above expression that

[TABLE]

We next prove $b_{0}=0$ in (3.26) so that $w_{0}\equiv 0$ in ${\mathbb{R}}^{3}$. Indeed, multiplying (3.7) by $Q_{1c}$ and $Q_{2c}$ respectively, and integrating over ${\mathbb{R}}^{3}$, we obtain that

[TABLE]

Moreover, since $\frac{1}{\sqrt{1+t}}=1-\frac{t}{2}+O(t^{2})$ as $t\to 0$ and $w_{c}$ is smooth for all $c>0$, we have

[TABLE]

Putting it into (3.27) yields that

[TABLE]

where the exponential decays (2.11), (3.18) and (3.19) are used again. Applying (2.11), (2.12) and (3.22), it then follows from above that

[TABLE]

We thus conclude from (3.26) and (3.28) that

[TABLE]

which therefore implies that $b_{0}=0$ in (3.26) and thus $w_{0}\equiv 0$ in ${\mathbb{R}}^{3}$.

We are now ready to derive a contradiction. In fact, let $y_{c}\in{\mathbb{R}}^{3}$ be a point satisfying $|w_{c}(y_{c})|=\|w_{c}\|_{L^{\infty}({\mathbb{R}}^{3})}=1$. Since it follows from (3.18) that $w_{c}$ admits the exponential decay uniformly for all $c>0$, we have $|y_{c}|\leq M$ uniformly in $c$ for some constant $M>0$. Therefore, we obtain that $w_{c}\to w_{0}\not\equiv 0$ uniformly on ${\mathbb{R}}^{3}$ as $c\to\infty$, which however contradicts to the fact that $w_{0}\equiv 0$ on ${\mathbb{R}}^{3}$. This completes the proof of Theorem 1.2. ∎

Acknowledgements: The authors thank Professor Enno Lenzmann very much for his helpful discussions on the subject of the present work.