1. Introduction
In this paper, we study the following mass critical fractional Schrödinger equation
where $s\in (\frac {1}{2},\, 1)$, $V: \mathbb {R}^{2}\rightarrow \mathbb {R}$ is an external potential function, $\lambda \in \mathbb {R}$ and $a>0$. It's well known that the fractional Laplacian $(-\Delta )^{s} (s\in (0,\, 1))$ can be defined by
for $v\in S(\mathbb {R}^{2})$, where $P.V.$ denotes a Cauchy principal value, $S(\mathbb {R}^{2})$ is the Schwartz space of rapidly decaying $C^{\infty }$ function, $B_{\varepsilon }(x)$ denotes an open ball of radius $\varepsilon$ centred at $x$ and the normalization constant $C_{2, s}=(\int _{\mathbb {R}^{2}}\frac {1-\cos (\zeta _{1})}{|\zeta |^{2+2s}})^{-1}$(see e.g. [Reference Deren, Cerdik and Agarwal5, Reference Ragusa17, Reference Wu and Taarabti24] and reference therein). In fact, there are applications of operator $(-\Delta )^{s}$ in some areas such as fractional quantum mechanics, physics and chemistry, obstacle problems, optimization and finance, conformal geometry and minimal surfaces, please see [Reference Cont and Tankov1, Reference Chang and del Mar González2, Reference Laskin12, Reference Laskin13, Reference Metzler and Klafter15, Reference Silvestre18] and the references therein for more details.
For equation (1.1), a direct choice is to search for solutions $u\in H^{s}(\mathbb {R}^{2})$ by looking for critical points of the functional $I_{\lambda, a}: H^{s}(\mathbb {R}^{2})\rightarrow \mathbb {R}$ defined by
where $\lambda \in \mathbb {R}$ is fixed and the fractional Sobolev space $H^{s}(\mathbb {R}^{2})$ can be defined as follows
endowed with the norm
For this approach of finding solutions, we recommend the reader to see [Reference Felmer, Quaas and Tan8, Reference Liu and Ouyang14, Reference Secchi19, Reference Su, Chen, Liu and Fang20] for more details. Moreover, (1.1) is also the Euler–Lagrange equation of the following constrained minimization problem
where $J_{a}(u)$ is the mass critical fractional Schrödinger energy functional
Here we define
with the norm
and
Very recently, when $V(x)$ has satisfied the following assumption
Du, Tian, Wang and Zhang [Reference L.X. Tian, Wang and Zhang6] proved that for all $a\in [0,\, a^{\ast })$, $e(a)$ has at least one minimizer and has no minimizers if $a\geq a^{\ast }$, where $a^{\ast }=\|Q\|_{2}^{2s}$ and $Q$ is the unique positive radial solution of
Moreover, they also obtained that for $a\in [0,\, a^{\ast })$ small enough, $e(a)$ has a unique nonnegative minimizer.
When $s=1$ and replace $a$ by $b$, system (1.1) reduces to the following Schrödinger equation
For problem (1.8), we can also consider the following constrained minimization problem
where $\bar {J}_{b}(u)$ is defined by
Here we define
and
There are many works focusing on existence and nonexistence of minimizers for (1.9). For instance, in [Reference Guo and Seiringer9], Guo and Seiringer proved that there exists a critical value $b^{\ast }>0$ such that (1.9) has at least one minimizer if $0\leq b< b^{\ast }$, and (1.9) has no minimizers if $b\geq b^{\ast }$. Moreover, they also studied that the limit behaviour of minimizers for (1.9) as $b\nearrow b^{\ast }$. For more constrained minimization problem of (1.9), please see [Reference Guo, Wang, Zeng and Zhou10, Reference Guo, Zeng and Zhou11] and the references therein for more details.
The first purpose of this paper is to consider whether the minimizers of (1.3) are the ground states of (1.1) and whether the opposite is true? To solve these problems, we first give the definition of the ground states of (1.1). Let
and
We say $u\in E$ is a ground state of (1.1) if $u\in G_{\lambda, a}$. Moreover, for $a\in [0,\, a^{\ast })$, we define
If $u_{a}\in K_{a}$, we can assume that $u_{a}$ is nonnegative since $J_{a}(u)\geq J_{a}(|u|)$. As mentioned in the above introduction, $u_{a}$ satisfies (1.1) with a suitable $\lambda =\lambda _{a}$ and we can define
and $0< a^{\ast \ast }\leq a^{\ast }$. Now we state our first main result as follows.
Theorem 1.1 Suppose that $(V_{1})$ holds. Then, for all $a\in [0,\, a^{\ast \ast })$ and for a.e. $a\in [a^{\ast \ast },\, a^{\ast }),$ all minimizers of $e(a)$ satisfy (1.1) with the same Lagrange multiplier $\lambda =\lambda _{a}$ and $K_{a}=G_{\lambda _{a},\, a}$.
Next, we focus on the concentration behaviour of nonnegative minimizers of (1.3) as $a\nearrow a^{\ast }$. To our best knowledge, currently only in [Reference L.X. Tian, Wang and Zhang6], the authors have studied the concentration behaviour of nonnegative minimizers of (1.3) when $V(x)=h(x)\Pi _{i=1}^{n}|x-x_{i}|^{q_{i}}$, $q_{i}\in (0,\, 2\,s)$ and $C< h(x)<1/C$ for some $C>0$ and all $x\in \mathbb {R}^{2}$. Note that the method in [Reference L.X. Tian, Wang and Zhang6] relies heavily on the fact that $V(x)$ has a finite number of minima $\{x_{i}\in \mathbb {R}^{2},\, i=1,\,\cdot \cdot \cdot,\,n\}$.
So what happens if $V(x)$ has infinitely many minima. Therefore, another main purpose of this paper is to study the concentration behaviour of mass critical fractional Schrödinger energy functional with a potential $V(x)$ with infinitely many minima. Therefore, in order to study this problem, it is assumed that $V$ satisfies the following explicit expression
Obviously, $V$ is a ring-shaped trapping potential and all points in set $\{x\in \mathbb {R}^{2}: |x|=M\}$ are minima of $V(x)$. We state our main result as follows.
Theorem 1.2 Let $V(x)$ be given by (1.14) and let $u_{a}$ be a nonnegative minimizer of (1.3) for $a< a^{\ast }$. For any given a sequence $\{a_{k}\}$ with $a_{k}\nearrow a^{\ast }$ as $k\rightarrow \infty,$ there exists a subsequence, still denoted by $\{a_{k}\},$ such that each $u_{a_{k}}$ has a unique maximum point $x_{k}$ and $x_{k}\rightarrow y_{0}$ as $k\rightarrow \infty$ for some $y_{0}\in \mathbb {R}^{2}$ satisfying $|y_{0}|=M>0$. Moreover,
and
where $\mu _{0}>0$ satisfies
Remark 1.3 Now, we list some of the difficulties encountered in this study.
$(1)$ In order to prove theorem 1.2, we need to make a direct and accurate estimate of the mass critical fractional Schrödinger energy functional, however, similarly as the proof of lemma 5.1 in [Reference L.X. Tian, Wang and Zhang6], we can only obtain estimate of the following type
see lemma 4.1 in § 4. Therefore, we need to use the method in [Reference Guo, Zeng and Zhou11] to estimate the mass critical fractional Schrödinger energy functional so that the power $\frac {2}{2+s}$ on the left side of (1.18) decreases to $\frac {1}{1+s}$, see lemma 4.5 in § 4.
$(2)$ Compared with the Gross–Pitaevskii equations studied in [Reference Guo, Zeng and Zhou11], our minimization solution sequence does not satisfy the property of exponential decay, so we need to analyse the decay property of the sequence, see lemma 5.1 in § 5.
The next theorem shows that we can determine exactly the coefficients estimated in lemma 4.5.
Theorem 1.4 Let $V(x)$ be given by (1.14), then the mass critical fractional Schrödinger energy $e(a)$ satisfies
where $\mu _{0}$ is given in (1.17).
Ultimately, we consider the occurrence of a symmetry breaking for the minimizers of $e(a)$. We have
Theorem 1.5 Let $V(x)$ be given by (1.14). Then there exist two positive constant $a_{\ast }$ and $a_{\ast \ast }$ satisfying $a_{\ast \ast }\leq a_{\ast }< a^{\ast }$ such that
(i) $e(a)$ has a unique nonnegative minimizer which is radially symmetric about the origin if $a\in [0,\, a_{\ast \ast })$.
(ii) $e(a)$ has infinitely many different nonnegative minimizers, which are not radially symmetric if $a\in [a_{\ast },\, a^{\ast })$.
Throughout this paper, we shall make use of the following notations.
• For $\rho >0$ and $z\in \mathbb {R}^{2}$, $B_{\rho }(z)$ denotes the ball of radius $\rho$ centred at $z$.
• The symbol $\rightharpoonup$ denotes weak convergence and the symbol $\rightarrow$ denotes strong convergence.
• $L^{q}(\mathbb {R}^{2})$ denotes the usual Lebesgue space with norm $\|u\|_{q}:=(\int _{\mathbb {R}^{2}}|u|^{q}\,{\rm d}x)^{\frac {1}{q}}$, $1\leq q\leq \infty$.
• For any $x\in \mathbb {R}^{2}$, arg $x$ be the angle between $x$ and the positive $x$-axis, and $\langle x,\, y\rangle$ be the angle between the vectors $x$ and $y$.
• $C$, $C_{i}$ $(i=1,\, 2,\, 3\cdot \cdot \cdot )$ denotes various positive constants which may vary from one line to another and which is not important for the analysis of the problem.
The paper is organized as follows. In § 2, we present some preliminaries results. In § 3, we will prove theorem 1.1. In § 4, we will establish some preparatory energy estimates. Section 5 is devoted to proving theorems 1.2, 1.4 and 1.5.
2. Preliminaries
In this section, we give some lemmas which will be frequently used throughout the rest of the paper. First, we give the fractional Gagliardo–Nirenberg–Sobolev inequality. Taking $N=2$ and $p=2+2\,s$ in (1.13) in [Reference L.X. Tian, Wang and Zhang6], we have
Lemma 2.1 For $2< p<2_{s}^{\ast }=\frac {2}{1-s},$ the fractional Gagliardo–Nirenberg–Sobolev inequality
is attained at a function $Q(x)$ with the following properties:
(i) $Q(x)$ is radial, positive, and strictly decreasing in $|x|$.
(ii) $Q(x)$ is a solution of the fractional Schrödinger equation (1.7).
(iii) $Q(x)$ belongs to $H^{2s+1}(\mathbb {R}^{2})\cap C^{\infty }(\mathbb {R}^{2})$ and satisfies
(2.2)\begin{equation} \frac{C_{1}}{1+|x|^{2+2s}}\leq Q(x)\leq \frac{C_{2}}{1+|x|^{2+2s}}, \quad x\in \mathbb{R}^{2}. \end{equation}
From (1.7) and (2.1), it follows that
Let $\eta : \mathbb {R}^{2}\rightarrow \mathbb {R}$ be a smooth function such that $\eta (x)=1$ for $|x|\leq 1$, $\eta (x)=0$ for $|x|\geq 2$, $0\leq \eta \leq 1$ and $|\nabla \eta |\leq 2$. Define
for any $\tau >0$, where $Q(x)$ is given in lemma 2.1. According to lemma 3.2 in [Reference L.X. Tian, Wang and Zhang6], we have the following result.
Lemma 2.2 Let $s\in (0,\, 1)$. Then the following estimate holds true:
as $\tau \rightarrow \infty$.
From [Reference Cheng3], we can deduce the following compactness result.
Lemma 2.3 Suppose that $(V_{1})$ hold. Then the embedding $E\hookrightarrow L^{q}(\mathbb {R}^{2})$ is compact for all $q\in [2,\, 2_{s}^{\ast })$.
By [Reference Secchi19], we know that the following vanishing lemma for fractional Sobolev space.
Lemma 2.4 Assume that $\{u_{n}\}$ is bounded in $H^{s}(\mathbb {R}^{N})$ and it satisfies
where $\rho >0$. Then $u_{n}\rightarrow 0$ in $L^{r}(\mathbb {R}^{N})$ for $2< r<2_{s}^{\ast }$.
3. Normalized ground states
In this section, we give the proof of theorem 1.1. First, we give a smoothness result about $e(a)$.
Lemma 3.1 Suppose that $(V_{1})$ holds. Then, for $a\in (0,\, a^{\ast }),$ the left and right derivatives of $e(a)$ always exist in $[0,\, a^{\ast })$ and satisfy
where
and $K_{a}$ is given by (1.12).
Proof. By $(V_{1})$, lemma 2.1 and the definition of $e(a)$, we have
where $\delta _{1}$ is the first eigenvalue of $(-\Delta )^{s}+V(x)$ in $E$. Moreover, lemma 2.1 also shows that
From (3.1)–(3.2), it follows that
For any $a_{1},\, a_{2}\in [0,\, a^{\ast })$, it is easy to see that
and
which implies that
By using (3.4)–(3.5), for $u_{a_{1}}\in K_{a_{1}}$ and $u_{a_{2}}\in K_{a_{2}}$, we have
Without loss of generality, we set $0< a_{1}< a_{2}< a^{\ast }$, from (3.7), we can see that
for $\forall u_{a_{i}}\in K_{a_{i}}$, $i=1,\, 2$. This shows that
By (3.3) and lemma 2.3, we know that $\{u_{a_{2}}\}$ is bounded in $E$. Up to a subsequence, we may assume that there exists $u\in E$ such that
By (3.6), we deduce that
which implies that
Thus, (3.9) shows that
Moreover, by the definition of $\beta _{a}$, we get
In view of (3.10)–(3.11), we can obtain that
Similarly, we also have
Proof of theorem 1.1 By using (3.7), we can see that
which implies that $e(a)$ is locally Lipschitz continuous in $[0,\, a^{\ast })$. Thus, by using Rademacher's theorem, we know that $e(a)$ is differentiable for a.e. $a\in [0,\, a^{\ast })$. Moreover, lemma 3.1 shows that $e'(a)$ exists for all $a\in [0,\, a^{\ast \ast })$ and a.e. $a\in [a^{\ast \ast },\, a^{\ast })$ and
Thus, we know that all minimizers of $e(a)$ have the same $L^{2+2s}(\mathbb {R}^{2})$-norm. For $a\in [0,\, a^{\ast })$, taking each $u_{a}\in K_{a}$ such that $e'(a)$ satisfies (3.12), then $u_{a}$ satisfies (1.1) for some Lagrange multiplier $\lambda _{a}\in \mathbb {R}$. By (1.1) and (3.12), we have
which implies that all minimizers of $e(a)$ satisfy equation (1.1) with the same Lagrange multiplier $\lambda _{a}$, that is,
For $\forall \varphi \in E$, by (1.2) and (3.14), we deduce that
which implies that $u_{a}\in S_{\lambda _{a},\, a}$. Now, we show that $G_{\lambda _{a},\, a}$ is nonempty. In fact, according to the definition of $G_{\lambda _{a},\, a}$, we only need to show that for any $u\in S_{\lambda _{a},\, a}$ one has
Let $u_{a}\in K_{a}$ and $u\in S_{\lambda _{a},\, a}$, we have
which implies that
Similarly, we also have
Define
Clearly, $\int _{\mathbb {R}^{2}}|\hat {u}|^{2}\,{\rm d}x=1$. Thus, we get that $J_{a}(\hat {u})\geq J_{a}(u_{a})$. This shows that
Moreover, by using (3.16), we deduce that
Let $g(\sigma ):=\frac {1}{\sigma }(1+s-\frac {1}{\sigma ^{s}})$. It is easy to check that $g(\sigma )$ achieves the unique global maximum at $\sigma =1$. By (3.17), (3.19) and (3.20), we deduce that
which shows that (3.15) holds. Hence, $G_{\lambda _{a},\, a}$ is nonempty.
Next, we prove that $K_{a}=G_{\lambda _{a},\, a}$. For any given $a\in [0,\, a^{\ast })$, consider any $u_{a}\in K_{a}$ and $u\in G_{\lambda _{a},\, a}$. Clearly, $u\in S_{\lambda _{a},\, a}$. From (3.17)–(3.20), it follows that
which implies that
Define $h(\sigma ):=\sigma ^{1+s}-\frac {1+s}{s}\sigma ^{s}+\frac {1}{s}$. Taking the derivative of $h(\sigma )$, we have
By (3.22)–(3.23), we know that $\sigma =1$, that is, $\int _{\mathbb {R}^{2}}|u|^{2}\,{\rm d}x=1$. Therefore, (3.21) shows that
This implies that $u\in K_{a}$ and $u_{a}\in G_{\lambda _{a},\, a}$.
4. Estimates in the energy $e(a)$ as $a\nearrow a^{\ast }$
In this section, we mainly establish the following estimates on the energy $e(a)$ as $a\nearrow a^{\ast }$.
Lemma 4.1 Let $V(x)$ be given by (1.14). Then there exist two constants $C_{1},\, C_{2}>0$, independent of $a,\, s$, such that
Proof. For any $\kappa >0$ and $u\in E$ with $\|u\|_{2}^{2}=1$, by lemma 2.1, we have
where $[\cdot ]_+=\max \{0,\, \cdot \}$ denotes the positive part. Taking the variable $r=M+\sqrt {\kappa }\sin \theta$ with $-\frac {\pi }{2}\leq \theta \leq \frac {\pi }{2}$, by direct computation, we deduce that
for $\kappa >0$ small enough. Taking $\kappa =(\frac {a^{\ast }-a}{2C^{s}})^{\frac {2}{2+s}}$ in (4.2) and (4.3), we have
Clearly, we know that $1-\frac {s}{1+s}\frac {1}{2^{\frac {1}{s}}}>0$ since $s\in (\frac {1}{2},\, 1)$. Hence, (4.4) shows that
On the other hand, set a cut-off function $\eta \in C_{0}^{\infty }(\mathbb {R}^{2})$ such that $\eta (x)=1$ for $|x|\leq 1$, $\eta (x)=0$ for $|x|\geq 2$, $0\leq \eta \leq 1$ and $|\nabla \eta |\leq 2$. Define
where $x_{0}\in \mathbb {R}^{2}$, $R,\, \tau >0$ and $A_{R, \tau }>0$ is chosen so that $\int _{\mathbb {R}^{2}}|u|^{2}\,{\rm d}x=1$. First, we show that $\lim _{R\tau \rightarrow \infty }A_{R, \tau }=1$. In fact, by (4.5) and lemma 2.1, we can see that
Now, taking $R=1$ in (4.5), from lemma 2.1, lemma 2.2 and (4.6), it follows that
Moreover, from lemma 2.1, by direct computation, we get
In view of (4.7) and (4.8), we can deduce that
Taking $\tau =(a^{\ast }-a)^{-\frac {1}{2+2s}}$ in (4.9), we get
Similar to the proofs of lemma 2.2 and 2.3 in [Reference Guo, Zeng and Zhou11], we can obtain the following two main results, which are extensions of the classical local problem in [Reference Guo, Zeng and Zhou11] to the nonlocal problem.
Lemma 4.2 Let $V(x)$ be given by (1.14) and suppose $u_{a}$ is a nonnegative minimizer of (1.3) , then there exists a constant $K>0,$ independent of $a,\, s,$ such that
Proof. From (4.2), it follows that
which implies that the upper bounded of (4.10) since lemma 4.1.
Moreover, for any $0< b< a< a^{\ast }$, we have
Then, lemma 4.1 shows that
Taking $b=a-C_{3}(a^{\ast }-a)^{\frac {2+s}{2+2s}}$ in (4.11), where $C_{3}>0$ is large enough such that $C_{1}C_{3}^{\frac {2}{2+s}}>2C_{2}$. Then, we can see that
which implies that the lower bounded of (4.10).
Lemma 4.3 Let $V(x)$ be given by (1.14) and suppose $u_{a}$ is a nonnegative minimizer of (1.3) , and set
Then, we have
(i) $\epsilon _{a}\rightarrow 0$ as $a\nearrow a^{\ast }$.
(ii) There exist a sequence $\{y_{\epsilon _{a}}\}\subset \mathbb {R}^{2}$ and positive constants $R_{0},\, \eta$ such that the sequence
(4.13)\begin{equation} w_{a}(x):=\epsilon_{a}u_{a}(\epsilon_{a}x+\epsilon_{a}y_{\epsilon_{a}}) \end{equation}satisfies(4.14)\begin{equation} \liminf_{a\nearrow a^{{\ast}}}\int_{B_{R_{0}}(0)}|w_{a}|^{2}\,{\rm d}x\geq\eta>0. \end{equation}(iii) The sequence $\{\epsilon _{a}y_{\epsilon _{a}}\}$ is bounded uniformly for $\epsilon _{a}\rightarrow 0$. Moreover, for any sequence $\{a_{k}\}$ with $a_{k}\nearrow a^{\ast },$ there exists a convergent subsequence, still denoted by $\{a_{k}\},$ such that
(4.15)\begin{equation} \bar{x}:=\epsilon_{a_{k}}y_{\epsilon_{a_{k}}}\rightarrow x_{0}, \text{as }a_{k}\nearrow a^{{\ast}}, \end{equation}for some $x_{0}\in \mathbb {R}^{2}$ being a global minimum point of $V(x)$, i.e., $|x_{0}|=M>0$. Furthermore, we also have(4.16)\begin{equation} w_{a_{k}}\stackrel{k}{\longrightarrow}\frac{\beta_{1}}{s^{\frac{1}{2s}}\|Q\|_{2}}Q\Bigg(\frac{\beta_{1}}{s^{\frac{1}{2s}}}|x-\bar{y_{0}}|\Bigg) \end{equation}in $H^{s}(\mathbb {R}^{2})$ for some $\bar {y_{0}}\in \mathbb {R}^{2}$ and $\beta _{1}>0$.
Proof. $(i)$ By lemma 2.1 and lemma 4.1, we deduce that
and
Lemma 4.2 implies that
that is,
which implies that
where $m=\max \{\frac {2}{a^{\ast }},\, a^{\ast }\}$. Thus, from (4.20) and lemma 4.2, there exist $C_{3},\, C_{4}>0$ such that
which implies that $\epsilon _{a}\rightarrow 0$ as $a\nearrow a^{\ast }$.
$(ii)$ Set
Next, we show that there exist a sequence $\{y_{\epsilon _{a}}\}\subset \mathbb {R}^{2}$ and $R_{0},\, \eta >0$ such that
Suppose by contradiction, for any $R>0$, there exists a sequence $\{\tilde {w}_{a_{k}}\}$ with $a\nearrow a^{\ast }$ such that
From lemma 2.4, we get that $\tilde {w}_{a_{k}}\stackrel {k}{\longrightarrow }0$ in $L^{r}(\mathbb {R}^{2})$ for $2< r<2_{s}^{\ast }$. Hence, $\tilde {w}_{a_{k}}\stackrel {k}{\longrightarrow }0$ in $L^{2+2s}(\mathbb {R}^{2})$, which contradicts with (4.23). Therefore, from (4.22) and (4.24), we have
$(iii)$ From (4.13) and (4.17), it follows that
Now, we prove that
Indeed, assume by contradiction that there exist a constant $\alpha >0$ and a subsequence $\{a_{n}\}$ with $a_{n}\nearrow a^{\ast }$ as $n\rightarrow \infty$ such that
By (4.14) and Fatou's lemma, we can see that
which gives a contradiction by (4.25). Thus, (4.26) shows that $\{\epsilon _{a}y_{\epsilon _{a}}\}$ is bounded uniformly as $\epsilon _{a}\rightarrow 0$ and (4.15) holds true.
Next, we prove that (4.16) holds. Since $u_{a}$ is a nonnegative minimizer of (1.3), we have
where $\lambda _{a}\in \mathbb {R}$ is a Lagrange multiplier. Moreover, we also have
From (4.20), (4.28) and lemma 4.1, we can see that there exist $C_{5},\, C_{6}>0$, independent of $a,\, s$, such that
By (4.13) and (4.27), we deduce that
Passing if necessary to a subsequence of $\{a_{k}\}$, still denoted by $\{a_{k}\}$, we may assume that
From the boundedness of $\{\epsilon _{a}y_{\epsilon _{a}}\}$, by passing to the weak limit of (4.29), we get
Clearly, (4.14) implies that $w_{0}\not \equiv 0$. Similar argument to the proof of proposition 4.4 in [Reference Teng and Agarwal21], we know that $w_{0}\in C^{1, \alpha }$ for some $\alpha \in (0,\, 1)$. Then, by lemma 3.2 in [Reference Di Nezza, Palatucci and Valdinoci7], we have
Next, we show that $w_{0}>0$. Assume by contradiction that there exists $x_{0}\in \mathbb {R}^{2}$ such that $w_{0}(x_{0})=0$, then we can see that
since $w_{0}\geq 0$ and $w_{0}\not \equiv 0$. However, it is easy to see that
which gives a contradiction. Hence $w_{0}>0$ for all $x\in \mathbb {R}^{2}$. Now, by (4.30) and $Q$ is the unique positive radial solution of (1.7), we can deduce that
By simple computation, we know that $\|w_{0}\|_{2}^{2}=1$. From the norm preservation, we get that $w_{a_{k}}\stackrel {k}{\longrightarrow }w_{0}$ in $L^{2}(\mathbb {R}^{2})$. Hence, by the boundedness of $\{w_{a_{k}}\}$ in $H^{s}(\mathbb {R}^{2})$, we have
Therefore, in view of (4.29) and (4.30), we know that $w_{a_{k}}\stackrel {k}{\longrightarrow }w_{0}$ in $H^{s}(\mathbb {R}^{2})$, and thus (4.16) holds.
Lemma 4.4 Under the assumptions of lemma 4.3, and let $\{a_{k}\}$ be given by lemma 4.3-$(iii)$. Then, for any $R>0,$ there exists $C(R)>0,$ independent of $a_{k},\, s,$ such that
Proof. The proof is parallel to lemma 2.4 in [Reference Guo, Zeng and Zhou11], for the reader's convenience, we give a brief proof. From lemma 4.3, we know that $\epsilon _{a_{k}}y_{\epsilon _{a_{k}}}\stackrel {k}{\longrightarrow }x_{0}$ with $|x_{0}|=M>0$. Hence. we get
which implies that
Without loss of generality, we may assume that $x_{0}=(M,\, 0)$. Then, arg $y_{\epsilon _{a_{k}}}\stackrel {k}{\longrightarrow }0$. By setting $0<\theta <\frac {\pi }{16}$ small enough, we get that
Let
and
Clearly, $B_{R}(0)=\Delta ^{1}_{\epsilon _{a_{k}}}\cup \Delta ^{2}_{\epsilon _{a_{k}}}$ and $\Delta ^{1}_{\epsilon _{a_{k}}}\cap \Delta ^{2}_{\epsilon _{a_{k}}}=\emptyset$. Next, we consider the following two cases.
Case 1: $|\Delta ^{1}_{\epsilon _{a_{k}}}|\geq \frac {\pi R^{2}}{2}$. It is easy to check that $B_{\frac {R}{\sqrt {2}}}(0)\subset \Delta ^{1}_{\epsilon _{a_{k}}}$. Let
By simple computation, we get
By (4.34), we have
and
Moreover, by (4.33) and the Taylor expansion, we have
From (4.35), (4.38), (4.39) and (4.40), it follows that
Hence, by (4.32), (4.39) and (4.40), we have
which implies that
where $\theta =\frac {\pi }{20}$.
Case 2: $|\Delta ^{2}_{\epsilon _{a_{k}}}|\geq \frac {\pi R^{2}}{2}$. Clearly, $B_{0}(R)\backslash B_{\frac {R}{\sqrt {2}}}(0)\subset \Delta ^{2}_{\epsilon _{a_{k}}}$. Set
The rest of the proof is very similar to the case 1, we omit it.
Lemma 4.5 There exist two positive constants $C_{7}$ and $C_{8}$, independent of $a,\, s,$ such that
Proof. By lemma 4.1, it suffices to prove that there exists a $C>0,$ independent of $a,\, s,$ such that
From lemma 4.3, we know that for any sequence $\{a_{k}\}$ with $a_{k}\nearrow a^{\ast }$, there exists a convergent subsequence, still denoted by $\{a_{k}\}$, such that $w_{a_{k}}\rightarrow w_{0}>0$ in $L^{2+2s}(\mathbb {R}^{2})$, where $w_{0}$ satisfies (4.31). Thus, there exists $M_{1}>0$, independent of $a_{k},\, s$, such that
Lemma 4.4 shows that there exists $M_{2}>0$, independent of $a_{k},\, s$, such that
In view of (4.43)–(4.44), we deduce that
as $a_{k}\nearrow a^{\ast }$ and here the last equality is achieved at
Thus, (4.42) holds for the subsequence $\{a_{k}\}$. Actually, the above argument can be carried out for any subsequence $\{a_{k}\}$ satisfying $a_{k}\nearrow a^{\ast }$, which then implies that (4.42) holds for all $a\nearrow a^{\ast }$.
Now, by using lemma 4.5, instead of using lemma 4.1 in the proof of lemma 4.2, and taking $b=a-C_{3}(a^{\ast }-a)$, we have
Corollary 4.6 Let $V(x)$ be given by (1.14) and suppose $u_{a}$ is a nonnegative minimizer of (1.3) , then there exists a constant $M>0,$ independent of $a,\, s,$ such that
5. Concentration behaviour
In this last section we study the concentration behaviour of normalized ground states and give the proofs of theorems 1.2, 1.4 and 1.5. Let $u_{a}$ is a nonnegative minimizer of (1.3), we define
By lemma 2.1, we deduce that
Hence, from lemma 4.5, it follows that
Similar to the proof of (4.14), for $\varepsilon _{a}$ given by (5.1), we get that there exist a sequence $\{y_{\varepsilon _{a}}\}\subset \mathbb {R}^{2}$ and $R_{0},\, \eta >0$ such that
where
Moreover, by (5.2) and corollary 4.6, we have
Lemma 5.1 For any given sequence $\{a_{k}\}$ with $a_{k}\nearrow a^{\ast },$ let $\varepsilon _{k}:=\varepsilon _{a_{k}}=(a^{\ast }-a_{k})^{\frac {1}{2+2s}}>0,$ $u_{k}(x)=u_{a_{k}}(x)$ be a nonnegative minimizer of (1.3) , and $w_{k}:=w_{a_{k}}\geq 0$ be defined by (5.4). Then, there is a subsequence, still denoted by $\{a_{k}\},$ such that
Moreover, for any $\rho >0$ small enough, we have
Proof. We divide the proof into four steps. Step 1. By (4.27) and (5.4), we get
where $\lambda _{k}\in \mathbb {R}^{2}$ is a Lagrange multiplier. Similar to the proof of lemma 4.3-$(iii)$, we know that (5.6) holds.
Step 2. For all $k$, we assume that $v_{k}\geq 0$ satisfies
Next, we prove that $\|v_{k}\|_{\infty }\leq C$, for all $k$. In fact, (5.8) shows that
From (5.9) and (5.10), it is easy to see that
For $\beta \geq 1$ and $T>0$, let
Clearly, $\varphi$ is convex and Lipschitz continuous, we get
in the weak sense. By using Sobolev inequality, (5.11), (5.12), the fact $\lambda _{k}<0$, $\varphi '(v_{k})\varphi (v_{k})\leq \beta v_{k}^{2\beta -1}$, $v_{k}\varphi '(v_{k})\leq \beta \varphi (v_{k})$ and integrating by parts, we deduce that
where $C>0$ independent of $k$ and $\beta$.
Note that $\beta \geq 1$ and that $\varphi (v_{k})$ is linear when $v_{k}\geq T$, then we have
which implies that $\int _{\mathbb {R}^{2}}(\varphi (v_{k}))^{2}v_{k}^{2_{s}^{\ast }-2}\,{\rm d}x$ is well defined for every $T$.
Now, we let $\beta$ in (5.13) such that $2\beta -1=2_{s}^{\ast }$ and define $\beta _{1}=\frac {2_{s}^{\ast }+1}{2}$. Let $R>0$ be fixed later, by Hölder's inequality, we have
Similar to the proof of lemma 4.3-$(iii)$, we know that $\{v_{k}\}$ converges strongly in $H^{s}(\mathbb {R}^{2})$, then $\{v_{k}\}$ converges strongly in $L^{2_{s}^{\ast }}(\mathbb {R}^{2})$, so we can choose $R$ sufficiently large such that
From (5.13)–(5.15), it follows that
Thus, by applying $\varphi (v_{k})\leq v_{k}^{\beta _{1}}$ and letting $T\rightarrow \infty$, we have
which implies that
Assume that $\beta >\beta _{1}$. By taking $T\rightarrow \infty$ in (5.13), we can see that
Let
where $l=\frac {2_{s}^{\ast }(2_{s}^{\ast }-1)}{2(\beta -1)}$ and $m=2\beta -1-l$. Moreover, $\beta >\beta _{1}$ implies that $0< l,\, m<2_{s}^{\ast }$, by using Young's inequality, we get
In view of (5.18) and (5.19), we have
which shows that
Iterating this argument, we obtain
where
Setting $C_{i+1}=C\beta _{i+1}$ and
We can see that there exists a constant $C>0$ independent of $i$ , such that
Hence, we have
Step 3. We prove that $w_{k}(x)\rightarrow 0$ as $|x|\rightarrow \infty$ uniformly in $k$.
In fact, we rewrite problem (5.9) as follows
where $h_{k}(x)=v_{k}+\varepsilon _{k}^{2s}\lambda _{k}v_{k}+a_{k}w^{2s+1}_{k}$. Thus, step 2 shows that $h_{k}\in L^{\infty }(\mathbb {R}^{2})$ and is uniformly bounded. From interpolation inequality and $\{v_{k}\}$ converges strongly in $H^{s}(\mathbb {R}^{2})$, we know that $h_{k}\rightarrow h$ in $L^{q}(\mathbb {R}^{2})$ for $q\in [2,\, +\infty )$. Thus, by [Reference Felmer, Quaas and Tan8], we deduce that
where $\mathcal {K}$ is a Bessel potential and it satisfies
$(\mathcal {K}_{1})$ $\mathcal {K}$ is positive, radially symmetric and smooth in $\mathbb {R}^{2}\backslash \{0\}$.
$(\mathcal {K}_{2})$ There exists a $C>0$ such that $\mathcal {K}(x)\leq \frac {C}{|x|^{2+2s}}$ for $x\in \mathbb {R}^{2}\backslash \{0\}$.
$(\mathcal {K}_{3})$ $\mathcal {K}\in L^{r}(\mathbb {R}^{2})$ for $r\in [1,\, \frac {1}{1-s})$.
Now, for any $\zeta >0$, we have
By step 1 and $(\mathcal {K}_{2})$, we can see that
Moreover, by using Hölder's inequality and $(\mathcal {K}_{3})$, we deduce that
which implies that there exist $K_{0}\in \mathbb {N}$ and $R_{0}>0$ independent of $\zeta >0$ such that
where we have used the fact $s>\frac {1}{2}$ so that $2<\frac {1}{1-s}$ and $(\int _{\{|x-y|<\frac {1}{\zeta }\}}|h|^{2}{\rm d}y)^{\frac {1}{2}}\rightarrow 0$ as $|x|\rightarrow \infty$. Thus, by (5.23) and (5.24), we know that
On the other hand, for all $k\in \{1,\, 2,\,\cdot \cdot \cdot,\, K_{0}-1\}$, there exists $R_{k}>0$ such that
which implies that
Thus, setting $R=\max \{R_{0},\, R_{1},\,\cdot \cdot \cdot R_{K_{0}-1}\}$, we conclude that
which implies
From (5.11) and (5.25), it follows that
Step 4. Combining step 2, step 3 and the proof of theorem 1.1 in [Reference Teng and Cheng22], we can get that
For any $x\in B_{\rho }^{c}(y_{0})$, (5.6) shows that
From (5.27) and (5.28), it follows that
Inspired by [Reference Wang25], we now prove theorem 1.2.
Proof of theorem 1.2 Set $\varepsilon _{k}:=(a^{\ast }-a_{k})^{\frac {1}{2+2s}}>0$, where $a_{k}\nearrow a^{\ast }$. Define $u_{k}(x):=u_{a_{k}}(x)$ is a nonnegative minimizer of (1.3). Moreover, we set $\bar {z}_{k}$ be any local maximum point of $u_{k}$. Clearly, we have
Hence, from (5.29) and lemma 5.1, it follows that
Let
By (5.8), we deduce that
Next, we prove that $\{\frac {\bar {z}_{k}-z_{k}}{\varepsilon _{k}}\}\subset \mathbb {R}^{2}$ is bounded uniformly in $k$. Assume by contradiction that $|\frac {\bar {z}_{k}-z_{k}}{\varepsilon _{k}}|\rightarrow \infty$ as $k\rightarrow \infty$. (5.27) shows that
which implies a contradiction by (5.29). Thus, there exists $R_{1}>0$, independent of $k$, such that, $|\frac {\bar {z}_{k}-z_{k}}{\varepsilon _{k}}|<\frac {R_{1}}{2}$. By (5.3), we can see that
where we have used the fact $\bar {w}_{k}(x)=w_{k}(x+\frac {\bar {z}_{k}-z_{k}}{\varepsilon _{k}})$. Similar to the argument of lemma 4.3-$(iii)$, we know that there exists a subsequence, still denoted by $\{\bar {w}_{k}\}$, of $\{\bar {w}_{k}\}$ such that
where $\bar {w}_{0}$ satisfies
Note from (5.33) that $\bar {w}_{0}\not \equiv 0$. Thus, similar to the proof of lemma 4.3-$(iii)$, we know that $\bar {w}_{0}>0$ in $\mathbb {R}^{2}$. Since the origin is a critical point of $\bar {w}_{k}$, we get that the origin is also a critical point of $\bar {w}_{0}$. By (5.35) and $Q$ is the unique positive radial solution of (1.7), for the above $\beta >0$, we can deduce that
Clearly, we know that $\bar {w}_{k}\geq (\frac {\beta ^{2}}{2a^{\ast }})^{\frac {1}{2s}}$ at each local maximum point. Hence, lemma 5.1 implies that all the local maximum points of $\bar {w}_{k}$ stay in a finite ball in $\mathbb {R}^{2}$. By (5.26) and the definition of $\bar {w}_{k}$, we can get that
Next, we prove that $\{\bar {w}_{k}\}$ is bounded uniformly in $C^{2, \alpha }_{loc}(\mathbb {R}^{2})$ for some $0<\alpha <1$. In fact, we rewrite (5.32) as follows
where $f_{k}(x)=-\varepsilon _{k}^{2s}(|\varepsilon _{k}x+\bar {z}_{k}|-M)^{2}\bar {w}_{k}+\varepsilon _{k}^{2s}\lambda _{k}\bar {w}_{k}+a_{k}\bar {w}^{2s+1}_{k}$. Since $\varepsilon _{k}^{2s}(|\varepsilon _{k}x+\bar {z}_{k}|-M)^{2}$ is locally Lipschitz continuous in $\mathbb {R}^{2}$ and (5.37), we have
From (5.37), (5.38), (5.39) and lemma 2.3 in [Reference Teng and Wu23], we know that
and
From (5.40) and (5.41), it follows that
Thus, by (5.37), (5.38), (5.42) and lemma 4.4 in [Reference Cabré and Sire4], we know that $\{\bar {w}_{k}\}$ is bounded uniformly in $C^{2, \alpha }_{loc}(\mathbb {R}^{2})$ for some $0<\alpha <1$. Thus, we may assume that there exists $\hat {w}_{0}\in C^{2, \alpha }_{loc}(\mathbb {R}^{2})$ such that $\bar {w}_{k}\rightarrow \hat {w}_{0}$ in $C^{2}_{loc}(\mathbb {R}^{2})$ as $k\rightarrow \infty$. Moreover, (5.34) shows that $\hat {w}_{0}=\bar {w}_{0}$.
Since the origin is the only critical point of $\bar {w}_{0}$, then the content of the appeal discussion shows that all local maximum points of $\{\bar {w}_{k}\}$ must approach the origin and hence stay in a small ball $B_{\varrho }(0)$ as $k\rightarrow \infty$. Letting $\varrho >0$ small enough such that $\bar {w}''_{0}(\tau )<0$ for $0<\tau <\varrho$. By lemma 4.2 in [Reference Ni and Takagi16], we know that $\{\bar {w}_{k}\}$ has no critical points other than the origin. Therefore, we get that there exists a subsequence of $\{u_{k}\}$ concentrating at a unique global minimum point of potential $V(x)=(|x|-M)^{2}$ in $\mathbb {R}^{2}$.
Now, we turn to proving (1.15)–(1.17). In fact, by (5.31), we have
where $\bar {z}_{k}$ is the unique global maximum point of $u_{k}$, and $\bar {z}_{k}\rightarrow y_{0}\in \mathbb {R}^{2}$ as $k\rightarrow \infty$ for some $|y_{0}|=M>0$.
Next, we prove that $\{\frac {|\bar {z}_{k}|-|y_{0}|}{\varepsilon _{k}}\}\subset \mathbb {R}$ is bounded uniformly for $k\rightarrow \infty$. Assume by contradiction that there exists a subsequence $\{a_{k}\}$, still denoted by $\{a_{k}\}$ such that $|\frac {|\bar {z}_{k}|-|y_{0}|}{\varepsilon _{k}}|\rightarrow \infty$ as $k\rightarrow \infty$, by (5.33), for any $C>0$, we have
From (5.43)–(5.44), it follows that
holds for any $C>0$, which implies a contradiction by lemma 4.5. Thus, there exists a subsequence $\{a_{k}\}$, still denoted by $\{a_{k}\}$ such that
for some constant $C_{0}$. Since $Q$ a radially symmetric function and polynomial decay as $|x|\rightarrow \infty$, we then deduce from (5.36) that
where the equality holds if and only if $C_{0}=0$. By (5.43) and (5.46), we have
where the equality is achieved at
We take
as a trial functional for $J_{a}$, and minimizes over $\beta >0$. (5.47) shows that
Therefore, from (5.48), we get the following several conclusions.
$(I)$ $\beta$ is unique, which is independent of the choice of the subsequence, and takes the value of $\mu _{0}$ as above.
$(II)$ $C_{0}=0$, that is, (1.15) holds.
Finally, by (5.30), (5.34) and (5.36), we have
that is, (1.16) holds.
Proof of theorem 1.4 In fact, (5.48) shows that (1.19) holds for the subsequence $\{a_{k}\}$. Moreover, the proof of theorem 1.2 shows that (5.48) is correct for all $\{a_{k}\}$ with $a_{k}\nearrow a^{\ast }$. Therefore, (1.19) holds for all $a\nearrow a^{\ast }$.
Proof of theorem 1.5 It then follows from theorem 1.2 that all nonnegative minimizers of $e(a)$ concentrate at any point on the ring $\{x\in \mathbb {R}^{2}: |x|=M\}$. This further implies that there exists a $a_{\ast }$ satisfying $0< a_{\ast }< a^{\ast }$ such that for any $a\in [a_{\ast },\, a^{\ast })$, $e(a)$ has infinitely many different nonnegative minimizers, each of which concentrates at a specific global minimum point of potential $V(x)=(|x|-M)^{2}$. However, recall from theorem 1.3-(ii) in [Reference L.X. Tian, Wang and Zhang6] that $e(a)$ admits a unique nonnegative minimizer $u_{a}$ for all $a>0$ being small enough $(a< a^{\ast })$, and noting that the trapping potential $V(x)=(|x|-M)^{2}$ $(M>0)$ is radially symmetric. Then similar to the argument of corollary 1.7 in [Reference L.X. Tian, Wang and Zhang6], by rotation $u_{a}$ must be rotational symmetry with respect to the origin.
Data availability
All data, models, and code generated or used during the study appear in the submitted article.
Acknowledgments
L.T. Liu is supported by the Fundamental Research Funds for the Central Universities of Central South University (No. 2021zzts0048). J. Yang is supported by the Research Foundation of Education Bureau of Hunan Province, China (No. 20B457, 20A387). H.B. Chen is supported by the National Natural Science Foundation of China (No. 12071486).