1 Introduction and main results
We study the pseudorelativistic Hartree equation
$$ \begin{align} i\partial_t\psi=(\sqrt{-c^2\Delta +m^2c^4}-mc^2)\psi-(|x|^{-1}*|\psi|^2)\psi \quad\text{with } (t,x)\in\mathbb{R}\times\mathbb{R}^3. \end{align} $$
In the physical context, the parameter
$m>0$
is the mass of a particle and the symbol
$*$
stands for the convolution on
$\mathbb {R}^3$
. The pseudorelativistic operator
$\sqrt {-c^2\Delta +m^2c^4}$
is defined via multiplication in the Fourier space with the symbol
$\sqrt {c^2|\xi |^2+m^2c^4}$
for
$\xi \in \mathbb {R}^3$
, which describes the kinetic energy of a relativistic particle with mass
$m>0$
. The convolution kernel
$|x|^{-1}$
represents the Newtonian potential in appropriate physical units.
A great deal of work has been devoted to the pseudorelativistic Hartree equation. Fröhlich et al. [Reference Fröhlich, Jonsson and Lenzmann4] proved the existence of travelling solitary waves for (1.1) with
$c=1$
by using concentration-compactness arguments [Reference Lions10, Reference Lions11] and Lenzman considered local and global well-posedness for Equation (1.1) with
$c=1$
[Reference Lenzmann7]. Lenzmann [Reference Lenzmann8] and Guo and Zeng [Reference Guo and Zeng5] studied the uniqueness of the ground state for the pseudorelativistic Hartree energy using the nonrelativistic limit of (1.1). For further work on travelling wave solutions of Equation (1.1), we refer the reader to [Reference Elgart and Schlein3, Reference Herr and Lenzmann6, Reference Wang13].
We focus on travelling solitary waves of the form
$$ \begin{align} \psi(t,x)=e^{it\mu}\varphi_{c}(x-vt), \end{align} $$
with some
$\mu \in \mathbb {R}$
and travelling velocity
$v\in \mathbb {R}^3$
such that
$|v|<1$
. Substituting (1.2) into (1.1) yields
$$ \begin{align} (\sqrt{-c^2\Delta +m^2c^4}-mc^2)\varphi_{c}+i(v\cdot\nabla)\varphi_{c}-({|x|^{-1}}*|\varphi_{c}|^2)\varphi_{c}=-\mu\varphi_{c}, \end{align} $$
which can be viewed as an Euler–Lagrange equation for the minimising problem
$$ \begin{align} E_{c}(N):=\inf\bigg\lbrace \mathcal{E}_{c}(\psi):\psi\in H^{{1}/{2}}(\mathbb{R}^3), \mathcal{N}(\psi)=\int_{\mathbb{R}^3}|\psi(x)|^2\,dx=N\bigg\rbrace, \end{align} $$
where
$$ \begin{align*} \mathcal{E}_{c}(\psi):=\frac{1}{2}\langle\psi,(\sqrt{-c^2\Delta +m^2c^4}-mc^2)\psi\rangle+\frac{i}{2}\langle\psi,(v\cdot\nabla)\psi\rangle -\frac{1}{4}\int_{\mathbb{R}^3}\bigg( \frac{1}{|x|}*|\psi|^2\bigg)|\psi|^2\,dx, \end{align*} $$
and the space
$H^{1/2}(\mathbb {R}^3)$
is defined by
$H^{1/2}(\mathbb {R}^3):=\lbrace \psi \in L^2(\mathbb {R}^3):(1+|\xi |)^{{1}/{2}}\hat {\psi }\in L^2(\mathbb {R}^3)\rbrace $
, with the norm
$$ \begin{align*} \|\psi\|_{H^{1/2}(\mathbb{R}^3)}^2:=\int_{\mathbb{R}^3}(1+|\xi|)|\hat{\psi}(\xi)|^2\,d\xi<\infty. \end{align*} $$
We recall from [Reference Fröhlich, Jonsson and Lenzmann4] the following Gagliardo–Nirenberg type inequality: for any
$v\in \mathbb {R}^3$
with
$|v|<1$
and
$\psi \in H^{1/2}(\mathbb {R}^3)$
,
$$ \begin{align} \int_{\mathbb{R}^3}\bigg( \frac{1}{|x|}*|\psi|^2\bigg)|\psi|^2\,dx\le \frac{2}{N_*(v)}\langle\psi,(\sqrt{-\Delta}+iv\cdot\nabla)\psi\rangle\langle\psi,\psi\rangle, \end{align} $$
where
${2}/{N_*(v)}$
is the best constant and
$N_*(v)$
is given by
$$ \begin{align*} N_*(v):=\langle Q_{v},Q_{v}\rangle=\|Q_{v}\|_{L^2}^2. \end{align*} $$
As stated in [Reference Fröhlich, Jonsson and Lenzmann4], an optimiser
$Q_v$
of (1.5) with
$Q_{v}\in H^{1/2}(\mathbb {R}^3)$
and
$Q_{v}\not \equiv 0$
satisfies
$$ \begin{align*} \sqrt{-\Delta}Q_{v}+i(v\cdot\nabla)Q_{v}-\bigg(\frac{1}{|x|}*|Q_{v}|^2\bigg)Q_{v}=-Q_{v}. \end{align*} $$
The constant
$N_*(v)$
is subject to the bounds
$ (1-|v|)N_*(0)\le N_*(v)\le N_*(0)=N_*$
, say. From [Reference Fröhlich, Jonsson and Lenzmann4], for
$|v|<1$
there exists a critical constant
$N_*(v)$
such that travelling waves exist if
$0<N<N_*(v)$
with the light speed
$c=1$
. Lenzmann in [Reference Lenzmann8] has given the existence of the ground state for
$E_{c}(N)$
with
$v=0$
for
$0<N<cN_*$
. For
$v\ne 0$
, using similar arguments to [Reference Fröhlich, Jonsson and Lenzmann4], it is easy to show the existence of a minimiser of
$E_{c}(N)$
. This gives the following existence theorem.
Theorem 1.1. Assume that
$m>0$
,
$v\in \mathbb {R}^3$
and
$|v|<1$
. Then there exists a positive constant
$N_*(v)$
, depending only on v, such that, for
$0<N<cN_*(v)$
, the problem (1.4) has a minimiser
$\varphi _{c}\in H^{1/2}(\mathbb {R}^3)$
.
We are interested in the limiting behaviour of minimisers for (1.4) as we pass to the limit
$c\to \infty $
, which is called the nonrelativistic limit. We will show that the minimiser of (1.4) converges strongly in
$H^1(\mathbb {R}^3)$
to the minimiser of the problem
$$ \begin{align} E_{\infty}(N):=\inf\bigg\lbrace\mathcal{E}_{\infty}(\psi):\psi\in H^1({\mathbb{R}^3}), \mathcal{N}(\psi)=\int_{\mathbb{R}^3}|\psi(x)|^2\,dx=N\bigg\rbrace, \end{align} $$
where
$\mathcal {E}_{\infty }(\psi )$
is given by
$$ \begin{align*} \mathcal{E}_{\infty}(\psi):=\frac{1}{4m}\int_{\mathbb{R}^3}|\nabla\psi|^2\,dx+\frac{i}{2}\langle\psi,(v\cdot\nabla)\psi\rangle-\frac{1}{4}\int_{\mathbb{R}^3}\bigg( \frac{1}{|x|}*|\psi|^2\bigg)|\psi|^2\,dx. \end{align*} $$
Any minimiser
$\varphi _{\infty }$
for (1.6) must satisfy the corresponding Euler–Lagrange equation
$$ \begin{align} -\frac{1}{2m}\Delta\varphi_{\infty}+i(v\cdot\nabla)\varphi_{\infty}-\bigg(\frac{1}{|x|}*|\varphi_{\infty}|^2\bigg)\varphi_{\infty}=-\lambda\varphi_{\infty} \end{align} $$
for some Lagrange multiplier
$\lambda \in \mathbb {R}$
.
We first establish the existence of a minimiser for
$E_{\infty }(N)$
.
Theorem 1.2. Assume that
$v\in \mathbb {R}^3$
and
$|v|<1$
,
$m>0$
is sufficiently small and
$$ \begin{align*} \frac{1}{2m}\int_{\mathbb{R}^3}|\nabla \psi|^2\,dx+\langle\psi,iv\cdot\nabla \psi\rangle\ge0 \quad \mbox{for any } \psi\in H^1({\mathbb{R}^3}). \end{align*} $$
Then the problem (1.6) has at least one minimiser.
The next result shows the
$H^1$
convergence for the solution of (1.3) to a solution of the limit equation (1.7) as
$c\to \infty $
. This is the main theorem of this paper.
Theorem 1.3. Under the assumptions of Theorem 1.2, let
$\varphi _{c}$
be a minimiser of
$E_{c}(N)$
with fixed N satisfying
$0<N<cN_*(v)$
. Then, as
$c\to \infty $
,
$$ \begin{align} \varphi_{c}\to\varphi_{\infty}\quad \text{strongly in}\ H^1(\mathbb{R}^3), \end{align} $$
where
$\varphi _{\infty }$
is a minimiser of
$E_{\infty }(N)$
.
Remark 1.4. Theorem 1.2 ensures the existence of minimisers of
$E_\infty (N)$
in the nonrelativistic limit for small m. Thus, we need the assumption that
$m>0$
is sufficiently small in Theorem 1.3.
Lenzmann in [Reference Lenzmann8] considered the nonrelativistic limit of a solution to (1.3) with
$v\hspace{-0.5pt}=\hspace{-0.5pt}0$
. We have to handle an additional term
$v\cdot \nabla $
, which needs careful analysis. We note that radially symmetric solutions to (1.3) do not exist (see [Reference Melgaard and Zongo12]). Since
$\varphi _c$
is not a radial function, we cannot use the method in [Reference Lenzmann8] which invokes Newton’s theorem to derive the lower bound for the Lagrange multiplier
$-\mu $
. Inspired by the work of Choi et al. [Reference Choi, Seok and Hong2], we find a new way to deal with the problem. In a similar way to [Reference Choi, Seok and Hong2, Lemma 4.3], we obtain the lower bound
$H_c\ge B|\xi |$
for the operator
$H_c=\sqrt {c^2|\xi |^2+m^2c^4}-mc^2+\delta $
with
$\delta>0$
, where
$B=\min \lbrace {2\delta ^{1/2}}/{(2\sqrt {5}m)^{1/2}},{c}/{2}\rbrace $
. Based on this inequality and the Gagliardo–Nirenberg inequality, we deduce that
$\varphi _c$
is uniformly bounded in
$ H^{1/2}(\mathbb {R}^3)$
. Then we can derive the upper bound for
$\mu $
and the uniform boundedness of
$\|\varphi _c\|_{H^1(\mathbb {R}^3)}$
.
The organisation of the paper is as follows. In Section 2, we consider the nonrelativistic limit and complete the proof of Theorem 1.3. In Section 3, we give the existence result of the limit energy functional by using concentration-compactness arguments.
We use the following notation.
-
•
$\rightharpoonup $
denotes weak convergence. -
•
$\langle \,,\rangle $
denotes the
$L^2$
inner product. -
•
$f*h$
denotes the convolution on
$\mathbb {R}^3$
. -
•
$\hat {f}$
denotes the Fourier transform of the function f (see [Reference Lieb and Loss9]). -
• The value of the positive constant C is allowed to change from line to line and also in the same formula.
-
•
$X\lesssim Y\ (X\gtrsim Y)$
denotes
$X\leq CY$
(respectively,
$X\geq CY$
) for some appropriate positive constant C. -
•
$v\cdot \nabla ={\textstyle \sum _{k=1}^{3}}v_{k}\partial _{x_k}$
, where
$v\in \mathbb {R}^3$
is some fixed vector.
2 The nonrelativistic limit
Before considering the nonrelativistic limit, we prove some preliminary lemmas.
Lemma 2.1. Let
$H_c=\sqrt {c^2|\xi |^2+m^2c^4}-mc^2+\delta $
with
$\delta>0$
independent of c. Then
$ H_c\ge B|\xi |$
, where
$B=\min \lbrace {2\delta ^{1/2}}/{(2\sqrt {5}m)^{1/2}},{c}/{2}\rbrace $
is a constant.
Proof. Factorising out
$mc^2$
from the square root, we write
$$ \begin{align*} H_{c}=mc^2\bigg(\sqrt{1+\bigg|\frac{\xi}{mc}\bigg|^2}-1\bigg)+\delta=mc^2f\bigg(\bigg|\frac{\xi}{mc}\bigg|^2\bigg)+\delta, \end{align*} $$
where
$f(t)=\sqrt {1+t}-1$
. By a Taylor expansion, if
$0\le t\le 4$
, then there is some
$t_*\in [0,4]$
such that
$$ \begin{align*} f(t)=\sqrt{1+t}-1=f(0)+f'(t_*)t =\frac{t}{2\sqrt{1+t_*}}\ge\frac{t}{2\sqrt{5}}. \end{align*} $$
Hence, if
$|\xi |\le 2mc$
, then
$$ \begin{align*} H_{c}&=mc^2\bigg(\sqrt{1+\bigg|\frac{\xi}{mc}\bigg|^2}-1\bigg)+\delta=mc^2f\bigg(\bigg|\frac{\xi}{mc}\bigg|^2\bigg)+\delta \notag\\ &\ge mc^2\frac{|{\xi}/{mc}|^2}{2\sqrt{5}}+\delta=\frac{|\xi|^2}{2\sqrt{5}m}+\delta \ge\frac{2\delta^{1/2}}{(2\sqrt{5}m)^{1/2}}|\xi|, \end{align*} $$
using the fact that
$a^2+b^2\ge 2ab$
for the last inequality.
On the other hand, if
$|\xi |\ge 2mc$
, then
$$ \begin{align*} H_{c}=c|\xi|\sqrt{1+\bigg|\frac{mc}{\xi}\bigg|^2}-mc^2+\delta \ge c|\xi|-mc^2+\delta \ge c|\xi|-\frac{c|\xi|}{2}+\delta\geq\frac{c}{2}|\xi|. \end{align*} $$
This establishes Lemma 2.1.
Lemma 2.2. If
$\varphi _{c}$
is a minimiser of
$E_{c}(N)$
, then
$\{\varphi _{c}\}$
is uniformly bounded in
$H^{1/2}(\mathbb {R}^3)$
.
Proof. By (1.5),
$$ \begin{align*} &2\mathcal{E}_{c}(\varphi_{c}) \notag\\ &\quad =\langle\varphi_{c},(\sqrt{-c^2\Delta +m^2c^4}-mc^2)\varphi_{c}\rangle+i\langle\varphi_{c},(v\cdot\nabla)\varphi_{c}\rangle-\frac{1}{2}\int_{\mathbb{R}^3}\bigg( \frac{1}{|x|}*|\varphi_{c}|^2\bigg)|\varphi_{c}|^2\,dx \notag\\ &\quad\ge\langle\varphi_{c},(\sqrt{-c^2\Delta +m^2c^4}-mc^2)\varphi_{c}\rangle+i\langle\varphi_{c},(v\cdot\nabla)\varphi_{c}\rangle-\frac{N}{N_*(v)}\langle\varphi_{c},(\sqrt{-\Delta}+iv\cdot\nabla)\varphi_{c}\rangle \notag\\ &\quad=\int_{\mathbb{R}^3}(\sqrt{c^2|\xi|^2+m^2c^4}-mc^2)|\hat{\varphi}_c({\xi})|^2\,d\xi-\int_{\mathbb{R}^3}(v\cdot\xi)|\hat{\varphi}_c({\xi})|^2\,d\xi \notag\\ &\qquad -\frac{N}{N_*(v)}\int_{\mathbb{R}^3}(|\xi|-v\cdot\xi)|\hat{\varphi}_c({\xi})|^2\,d\xi. \end{align*} $$
It follows from Lemma 2.1 that
$$ \begin{align*} &2\mathcal{E}_{c}(\varphi_{c})+\delta N \notag\\ &\ge B\int_{\mathbb{R}^3}|\xi||\hat{\varphi}_c({\xi})|^2\,d\xi-\int_{\mathbb{R}^3}(v\cdot\xi)|\hat{\varphi}_c({\xi})|^2\,d\xi-\frac{N}{N_*(v)}\int_{\mathbb{R}^3}(|\xi|-v\cdot\xi)|\hat{\varphi}_c({\xi})|^2\,d\xi. \end{align*} $$
For
$c>1$
sufficiently large,
$B=\min \lbrace {2\delta ^{1/2}}/{(2\sqrt {5}m)^{1/2}},{c}/{2}\rbrace ={2\delta ^{1/2}}/{(2\sqrt {5}m)^{1/2}}$
.
Case I: Fix N with
$0<N<cN_*$
and suppose that
$0<N<N_*(v)<cN_*(v)$
. Let
$\delta ={\sqrt {5}m}/{2}$
. Then
$B=1$
and
$$ \begin{align} 2\mathcal{E}_{c}(\varphi_{c})+\frac{\sqrt{5}m}{2}N&\ge\bigg(1-\frac{N}{N_*(v)}\bigg)\int_{\mathbb{R}^3}(|\xi|-v\cdot\xi)|\hat{\varphi}_c({\xi})|^2\,d\xi \notag\\ &\ge\bigg(1-\frac{N}{N_*(v)}\bigg)(1-|v|)\int_{\mathbb{R}^3}|\xi|\,|\hat{\varphi}_c({\xi})|^2\,d\xi. \end{align} $$
In the last inequality, we use the fact that
$|\xi |-v\cdot \xi \ge (1-|v|)|\xi |$
. Since
$\varphi _{c}$
is a minimiser of
$\mathcal {E}_{c}(\psi )$
, the operator inequality
$\sqrt {-c^{2}\Delta +m^2c^{4}}-mc^2\le {-\Delta }/{2m}$
yields
$$ \begin{align} \mathcal{E}_{c}(\varphi_{c})\le \mathcal{E}_{c}(\varphi_{\infty})\le\mathcal{E}_{\infty}(\varphi_{\infty}). \end{align} $$
Combining (2.1) with (2.2) and noting that
$\mathcal {E}_{\infty }(\varphi _{\infty })<0$
gives
$$ \begin{align} \bigg(1-\frac{N}{N_*(v)}\bigg)(1-|v|)\int_{\mathbb{R}^3}|\xi|\,|\hat{\varphi}_c({\xi})|^2\,d\xi&\le2\mathcal{E}_{c}(\varphi_{c})+\frac{\sqrt{5}m}{2}N \notag\\ &\le2\mathcal{E}_{\infty}(\varphi_{\infty})+\frac{\sqrt{5}m}{2}N \le\frac{\sqrt{5}m}{2}N. \end{align} $$
Since
$|v|<1$
, we have
$(1-{N}/{N_*(v)})(1-|v|)>0$
.
Case II: Fix N with
$0<N<cN_*$
and suppose that
$0<N_*(v)\le N<cN_*(v)$
. We can take
$\delta =8\sqrt {5}m({N}/{N_*(v)})^2$
in Lemma 2.1. Then
$B={4N}/{N_*(v)}$
and, as in (2.1),
$$ \begin{align*} &2\mathcal{E}_{c}(\varphi_{c})+8\sqrt{5}m\bigg(\frac{N}{N_*(v)}\bigg)^2N \notag\\ &\quad\ge\frac{4N}{N_*(v)}\int_{\mathbb{R}^3}|\xi|\,|\hat{\varphi}_c({\xi})|^2\,d\xi-\int_{\mathbb{R}^3}|v||\xi|\,|\hat{\varphi}_c({\xi})|^2\,d\xi-\frac{N}{N_*(v)}\int_{\mathbb{R}^3}(|\xi|-v\cdot\xi)|\hat{\varphi}_c({\xi})|^2\,d\xi \notag\\ &\quad\ge\frac{N}{N_*(v)}(3-2|v|)\int_{\mathbb{R}^3}|\xi|\,|\hat{\varphi}_c({\xi})|^2\,d\xi. \end{align*} $$
Since
$|v|<1$
, we have
$({N}/{N_*(v)})(3-2|v|)>0$
and, as in (2.3), we obtain
$$ \begin{align} \frac{N}{N_*(v)}(3-2|v|)\int_{\mathbb{R}^3}|\xi|\,|\hat{\varphi}_c({\xi})|^2\,d\xi\le\frac{8\sqrt{5}mN^3}{N_*^2(v)}. \end{align} $$
By combining (2.3) and (2.4), we conclude that there exists a constant
$C_1>0$
, which is independent of c, such that
$$ \begin{align*} \int_{\mathbb{R}^3}|\xi|\,|\hat{\varphi}_c({\xi})|^2\,d\xi\le C_1. \end{align*} $$
This completes the proof of Lemma 2.2.
Lemma 2.3. If
$m>0$
,
$v\in \mathbb {R}^3$
and
$|v|<1$
, then
$E_{\infty }(N)<0$
.
Proof. Fix
$\psi (x)\in H^1(\mathbb {R}^3)$
with
$\int _{\mathbb {R}^3}|\psi |^2\,dx=N$
. Let
$\psi ^\lambda (x)=\lambda ^{{3}/{2}}\psi (\lambda x)$
with
$\lambda>0$
. Then
$\|\psi ^{\lambda }\|^2_{L^2}=\|\psi \|^2_{L^2}=N$
. By the definition of
$\mathcal {E}_{\infty }(\psi )$
,
$$ \begin{align*} \mathcal{E}_{\infty}(\psi^\lambda(x))=\frac{\lambda^2}{2}\bigg\langle\psi,\frac{-\Delta}{2m}\psi\bigg\rangle+\frac{\lambda i}{2}\langle\psi,(v\cdot\nabla)\psi\rangle-\frac{\lambda}{4}\int_{\mathbb{R}^3}\bigg( \frac{1}{|x|}*|\psi|^2\bigg)|\psi|^2\,dx. \end{align*} $$
Case I: If
$i \langle \psi ,(v\cdot \nabla )\psi \rangle <0$
and
$\lambda $
is small enough, then, clearly,
$\mathcal {E}_{\infty }(\psi ^\lambda )<0$
.
Case II: If
$i \langle \psi ,(v\cdot \nabla )\psi \rangle \geq 0$
, then
$$ \begin{align*} \mathcal{E}_{\infty}(\psi^\lambda(-x))=\frac{\lambda^2}{2}\bigg\langle\psi,\frac{-\Delta}{2m}\psi\bigg\rangle-\frac{\lambda i}{2}\langle\psi,(v\cdot\nabla)\psi\rangle-\frac{\lambda}{4}\int_{\mathbb{R}^3}\bigg( \frac{1}{|x|}*|\psi|^2\bigg)|\psi|^2\,dx. \end{align*} $$
If
$\lambda $
is small enough, then
$\mathcal {E}_{\infty }(\psi ^\lambda (-x))<0$
.
Combining Cases I and II gives
$E_{\infty }(N)<0$
. This completes the proof.
Lemma 2.4. Let
$\varphi _{c}$
be a minimiser of
$E_{c}(N)$
satisfying the assumptions of Theorem 1.1 and let
$\mu $
be the associated Lagrange multiplier to
$\varphi _{c}$
. Then there exists a constant
$K>0$
such that
$|\mu |\le K$
, where the constant
$K>0$
is independent of
$c>0$
.
Proof. First, we claim that
$\mu>0$
. The minimiser
$\varphi _{c}$
of
$E_{c}(N)$
satisfies the Euler–Lagrange equation (1.3). Multiplying by
$\varphi _{c}$
and integrating gives
$$ \begin{align*} -\mu N=2\mathcal{E}_{c}(\varphi_{c})-\frac{1}{2}\int_{\mathbb{R}^3}\bigg( \frac{1}{|x|}*|\varphi_{c}|^2\bigg)|\varphi_{c}|^2\,dx. \end{align*} $$
We recall the operator inequality
$$ \begin{align*} \sqrt{-c^2\Delta+m^2c^4} \le -\frac{1}{2m}\Delta +mc^2, \end{align*} $$
which follows directly in the Fourier domain and we note that
$\sqrt {1+t}\le t/2+1$
for all
$t\ge 0$
. Since
$\varphi _{c}$
is a minimiser of
$E_{c}(N)$
, we have
$\mathcal {E} _{c}(\varphi _{c})\le \mathcal {E}_{c}(\varphi _{\infty })\le \mathcal {E} _\infty (\varphi _{\infty })$
. Consequently, by Lemma 2.3,
$$ \begin{align*} -\mu N=2\mathcal{E}_{c}(\varphi_{c})-\frac{1}{2}\int_{\mathbb{R}^3}\bigg( \frac{1}{|x|}*|\varphi_{c}|^2\bigg)|\varphi_{c}|^2\,dx \le2\mathcal{E}_{c}(\varphi_{c})\le2\mathcal{E}_{\infty}(\varphi_{\infty})<0. \end{align*} $$
This implies that
$\mu>0$
.
Next, we prove the upper bound for
$\mu $
. By (1.3),
$$ \begin{align*} -\mu N=\langle\varphi_{c},(\sqrt{-c^2\Delta +m^2c^4}-mc^2)\varphi_{c}\rangle+i\langle\varphi_{c},(v\cdot\nabla)\varphi_{c}\rangle -\int_{\mathbb{R}^3}\bigg( \frac{1}{|x|}*|\varphi_{c}|^2\bigg)|\varphi_{c}|^2\,dx. \end{align*} $$
Since
$\sqrt {-c^2\Delta +m^2c^4}-mc^2>0$
, by (1.5),
$$ \begin{align*} -\mu N&\ge i\langle\varphi_{c},(v\cdot\nabla)\varphi_{c}\rangle-\int_{\mathbb{R}^3}\bigg( \frac{1}{|x|}*|\varphi_{c}|^2\bigg)|\varphi_{c}|^2\,dx\\ &\ge i\langle\varphi_{c},(v\cdot\nabla)\varphi_{c}\rangle-\frac{2N}{N_*(v)}\langle\varphi_{c},(\sqrt{-\Delta}+iv\cdot\nabla)\varphi_{c}\rangle. \end{align*} $$
Therefore,
$$ \begin{align*} \mu N\le\frac{2N}{N_*(v)}\langle\varphi_{c},(\sqrt{-\Delta}+iv\cdot\nabla)\varphi_{c}\rangle-i\langle\varphi_{c},(v\cdot\nabla)\varphi_{c}\rangle. \end{align*} $$
By a Fourier transform and Plancherel’s theorem [Reference Lieb and Loss9, Theorem 5.3], using similar arguments to those in the proof of [Reference Fröhlich, Jonsson and Lenzmann4, Lemma A.4],
$$ \begin{align*} i\langle\varphi_{c},(v\cdot\nabla)\varphi_{c}\rangle=-\int_{\mathbb{R}^3}(v\cdot \xi)|\hat{\varphi}_{c}(\xi)|^2\,d\xi. \end{align*} $$
Since
$\sqrt {-\Delta }+iv\cdot \nabla \le \sqrt {-\Delta }$
, this yields
$$ \begin{align*} \mu N\le\frac{2N}{N_*(v)}\langle\varphi_{c},\sqrt{-\Delta}\varphi_{c}\rangle+|v|\langle\varphi_{c},\sqrt{-\Delta}\varphi_{c}\rangle \lesssim\|\varphi_{c}\|_{H^{1/2}(\mathbb{R}^3)}, \end{align*} $$
where we use the fact that
$v\cdot \xi \le |v||\xi |$
. Since
$\varphi _{c}$
is uniformly bounded in
$H^{1/2}(\mathbb {R}^3)$
, we can find a constant
$K>0$
such that
$\mu <K$
.
This completes the proof of Lemma 2.4.
Lemma 2.5. If
$\varphi _{c}$
is a minimiser of
$E_{c}(N)$
, then there exists a constant
$M>0$
independent of c such that
$\|\varphi _{c}\|_{H^{1}(\mathbb {R}^3)}\le M$
.
Proof. Since
$||\varphi _{c}||_{L^2}^{2}=N$
, we only need to derive a uniform bound for
$||\nabla \varphi _{c}||_{L^2}$
. It follows from (1.3) that
$$ \begin{align*} &c^2\|\nabla\varphi_{c}\|_{L^2}^2+m^2c^4\|\varphi_{c}\|_{L^2}^2\\ &\quad=\langle\sqrt{-c^2\Delta +m^2c^4}\varphi_{c},\sqrt{-c^2\Delta +m^2c^4}\varphi_{c}\rangle\\ &\quad=\langle(-\mu+mc^2+|x|^{-1}*|\varphi_{c}|^2-iv\cdot\nabla)\varphi_{c},(-\mu+mc^2+|x|^{-1}*|\varphi_{c}|^2-iv\cdot\nabla)\varphi_{c}\rangle\\ &\quad=\mu^2N-2\mu mc^2N-2\mu\langle\varphi_{c},(|x|^{-1}*|\varphi_{c}|^2)\varphi_{c}\rangle+2\mu\langle\varphi_{c},iv\cdot\nabla\varphi_{c}\rangle+m^2c^4N\\ &\qquad +2mc^2\langle\varphi_{c},(|x|^{-1}*|\varphi_{c}|^2)\varphi_{c}\rangle-2mc^2\langle\varphi_{c},iv\cdot\nabla\varphi_{c}\rangle-\langle v.\nabla\varphi_{c},v\cdot\nabla\varphi_{c}\rangle\\ &\qquad +\langle(|x|^{-1}*|\varphi_{c}|^2)\varphi_{c},(|x|^{-1}*|\varphi_{c}|^2)\varphi_{c}\rangle-2\langle(|x|^{-1}*|\varphi_{c}|^2)\varphi_{c},iv\cdot\nabla\varphi_{c}\rangle. \end{align*} $$
To bound the terms on the right, we note that Kato’s inequality
$|x|^{-1}\le |\nabla |$
implies that
$$ \begin{align} \|\,|x|^{-1}*|\varphi_{c}|^2\|_{L^{\infty}}\lesssim\langle\varphi_{c},|\nabla|\varphi_{c}\rangle\lesssim\|\varphi_{c}\|_{L^2}\|\nabla\varphi_{c}\|_{L^2}. \end{align} $$
On the other hand, since
$v\cdot \nabla \le |v||\nabla |$
and
$|v|<1$
,
$$ \begin{align} |\langle\varphi_{c},iv\cdot\nabla\varphi_{c}\rangle|\le\langle\varphi_{c},|\nabla|\varphi_{c}\rangle\le\|\varphi_{c}\|_{L^2}\|\nabla\varphi_{c}\|_{L^2}. \end{align} $$
$$ \begin{align*} &c^2\|\nabla\varphi_{c}\|_{L^2}^2\le \mu^2N +2\mu N^{{1}/{2}}\|\nabla\varphi_{c}\|_{L^2}+2mc^2N^{{3}/{2}}\|\nabla\varphi_{c}\|_{L^2} \\ &\quad+ 2mc^2N^{{1}/{2}}\|\nabla\varphi_{c}\|_{L^2} +N^2\|\nabla\varphi_{c}\|_{L^2}^2+2N\|\nabla\varphi_{c}\|_{L^2}^2. \end{align*} $$
From Lemma 2.4,
$\mu $
is uniformly bounded. As
$c\to \infty $
, N is fixed and m is sufficiently small, we conclude that there exists a constant
$M>0$
such that
$||\nabla \varphi _{c}||_{L^2}\le M$
. By choosing
$M>0$
possibly larger, we arrive at the bound in the lemma.
Proof of Theorem 1.3.
First, we claim that
$\{\varphi _{c}\}$
is a minimising sequence of
$E_\infty (N)$
. Since
$\varphi _{c}$
is a ground state of
$E_{c}(N)$
,
$$ \begin{align} 0\le E_\infty(N)-E_{c}(N) & \le\mathcal{E}_\infty(\varphi_{c})-\mathcal{E}_{c}(\varphi_{c}) \notag \\ &=\frac{1}{2}\int_{\mathbb{R}^3}\bar{\varphi}_{c}\bigg(\frac{-\Delta}{2m}-\big(\sqrt{-c^{2}\Delta +m^2c^{4}}-mc^2\big)\bigg)\varphi_{c}\,dx. \end{align} $$
From the proof of [Reference Choi and Seok1, Lemma 6.1],
$$ \begin{align} \lim_{c\to \infty}\bigg \langle f,\bigg(\sqrt{-c^{2}\Delta +m^2c^{4}}-mc^2+\frac{1}{2m}\Delta\bigg)\varphi_{c} \bigg \rangle =0\quad \text{for all }f\in H^1(\mathbb{R}^3). \end{align} $$
This is easy to verify for a test function
$f\in C_{0}^{\infty }(\mathbb {R}^3)$
by taking the Fourier transform and observing that
$$ \begin{align*} \sqrt{c^{2}\xi^2 +m^2c^{4}}-mc^2-\frac{\xi^2}{2m}\to0 \quad\text{for every }\xi\in \mathbb{R}^3\text{ as }c\to\infty. \end{align*} $$
By a simple density argument, (2.8) extends to all
$f\in H^1(\mathbb {R}^3)$
. Therefore,
$$ \begin{align} \lim_{c\to \infty}\int_{\mathbb{R}^3}\bar{\varphi}_{c}\bigg[\frac{-\Delta}{2m}-(\sqrt{-c^{2}\Delta+m^2c^{4}}-mc^2)\bigg]\varphi_{c}\,dx=0. \end{align} $$
From (2.7) and (2.9), we conclude that, as
$c\to \infty $
,
$$ \begin{align*} E_{c}(N)\to E_\infty(N)\quad\text{and}\quad\mathcal{E}_\infty(\varphi_{c})\to E_\infty(N). \end{align*} $$
Hence,
$\{\varphi _{c}\}$
is a minimising sequence of
$E_\infty (N)$
. Combining this with the existence of a minimiser for
$E_{\infty }(N)$
gives (1.8) and completes the proof of Theorem 1.3.
3 The existence of a minimiser for
$E_\infty (N)$
In this section, we prove the existence of a minimiser for the limit energy
$E_{\infty }(N)$
.
Lemma 3.1. If
$\lbrace \varphi _{c}\rbrace $
is a minimising sequence for
$E_{\infty }(N)$
, then
$E_{\infty }(N)$
is a continuous function of N.
Proof. Let
$\{\varphi _{c}\}$
be a minimising sequence for
$E_\infty (N)$
such that
$\lim _{c\to \infty }\mathcal {E}_\infty (\varphi _{c})=E_\infty (N)$
with
$\|\varphi _{c}\|_{L^2}^2=N$
. For any
$N_1>0$
,
$$ \begin{align*} E_{\infty}(N_1) & \le\mathcal{E}_{\infty}\bigg(\sqrt{\frac{N_1}{N}}\varphi_{c}\bigg) \quad\mbox{since } \bigg\|\sqrt{\frac{N_1}{N}}\varphi_{c}\bigg\|_{L^2}^2=N_1 \\ & =\frac{1}{4m}\frac{N_1}{N}\int_{\mathbb{R}^3}|\nabla\varphi_{c}|^2\,dx+\frac{N_1}{2N}\langle\varphi_{c},iv\cdot\nabla\varphi_{c}\rangle -\frac{1}{4}\bigg(\frac{N_1}{N}\bigg)^2\int_{\mathbb{R}^3}\bigg( \frac{1}{|x|}*|\varphi_{c}|^2\bigg)|\varphi_{c}|^2\,dx\\ & =\mathcal{E}_{\infty}(\varphi_{c})+\frac{1}{4m}\bigg(\frac{N_1}{N}-1\bigg)\int_{\mathbb{R}^3}|\nabla\varphi_{c}|^2\,dx+\frac{1}{2}\bigg(\frac{N_1}{N}-1\bigg)\langle\varphi_{c},iv\cdot\nabla\varphi_{c}\rangle\\ &\quad -\frac{1}{4}\bigg[ \bigg(\frac{N_1}{N}\bigg)^2-1\bigg] \int_{\mathbb{R}^3}\bigg( \frac{1}{|x|}*|\varphi_{c}|^2\bigg)|\varphi_{c}|^2\,dx. \end{align*} $$
Since
$\{\varphi _{c}\}$
is uniformly bounded in
$H^{1}(\mathbb {R}^3)$
, the two integrals and
$|\langle \varphi _{c},iv\cdot \nabla \varphi _{c}\rangle |$
can be bounded by a constant
$C>0$
which is independent of the light speed c. Thus,
$$ \begin{align} E_\infty(N_1)-E_\infty(N)\leq C\bigg| \frac{N_1}{N}-1\bigg|. \end{align} $$
By similar arguments,
$$ \begin{align} E_\infty(N)-E_\infty(N_1)\leq C\bigg| \frac{N}{N_1}-1\bigg|. \end{align} $$
From (3.1) and (3.2), it follows that
$E_\infty (N_1)\to E_\infty (N)$
as
$N_1\to N$
. This completes the proof of Lemma 3.1.
Lemma 3.2. For
$m>0$
sufficiently small, we have the strict binding inequality
$$ \begin{align} E_{\infty}(N)<E_{\infty}(\alpha)+ E_{\infty}(N-\alpha) \end{align} $$
for
$0<\alpha <N$
.
Proof. For any
$\varepsilon>0$
, there exists
$Q\in H^1(\mathbb {R}^3)$
with
$\|Q\|_{L^2}^2=\lambda <N$
such that
$E_{\infty }(\lambda )\le \mathcal {E}_{\infty }(Q)\le E_{\infty }(\lambda )+\varepsilon $
. Choose
$\theta>1$
such that
$\theta \lambda \le N$
. Then
$$ \begin{align*} E_{\infty}(\theta\lambda) \le\mathcal{E}_{\infty}(\sqrt{\theta}Q) & =\frac{\theta}{4m}\int_{\mathbb{R}^3}|\nabla Q|^2\,dx+\frac{\theta}{2}\langle Q,iv\cdot\nabla Q\rangle-\frac{\theta^2}{4}\int_{\mathbb{R}^3}\bigg( \frac{1}{|x|}*|Q|^2\bigg)|Q|^2\,dx\\ & =\frac{1}{2}(\theta-\theta^2)\bigg[\frac{1}{2m}\int_{\mathbb{R}^3}|\nabla Q|^2\,dx+\langle Q,iv\cdot\nabla Q\rangle\bigg]+\theta^2\mathcal{E}_{\infty}(Q). \end{align*} $$
For
$m>0$
sufficiently small,
$$ \begin{align} \frac{1}{2m}\int_{\mathbb{R}^3}|\nabla Q|^2\,dx+\langle Q,iv\cdot\nabla Q\rangle\ge0. \end{align} $$
Since
$\theta>1$
, we have
$E_{\infty }(\theta \lambda )\le \theta ^2\mathcal {E}_{\infty }(Q)$
and, in addition,
$$ \begin{align} E_{\infty}(\theta\lambda)\le\theta^2(E_{\infty}(\lambda)+\varepsilon). \end{align} $$
Next, we claim that
$$ \begin{align} E_{\infty}(N)<\frac{N}{\alpha}E_{\infty}(\alpha) \quad\mbox{for } 0<\alpha<N. \end{align} $$
Indeed, if
$E_{\infty }(\alpha )\ge 0$
, (3.6) obviously holds since
$E_{\infty }(N)<0$
. If
$E_{\infty }(\alpha )<0$
, taking
$\theta =N/\alpha $
,
$\alpha =\lambda $
and
$\varepsilon <(\theta ^{-1}-1)E_{\infty }(\alpha )$
in (3.5) gives (3.6). In the same way, replacing
$\alpha $
with
$N-\alpha $
gives
$$ \begin{align} E_{\infty}(N)<\frac{N}{N-\alpha}E_{\infty}(N-\alpha). \end{align} $$
Combining (3.6) and (3.7) yields (3.3) and completes the proof of Lemma 3.2.
By Lemma 2.5, the minimising sequence
$\{\varphi _{c}\}$
is uniformly bounded in
$H^{1}(\mathbb {R}^3)$
. Consequently, there exists a subsequence
$\{\varphi _{c_k}\}$
such that
$\varphi _{c_k}\rightharpoonup \varphi _{\infty }$
. We now apply the concentration-compactness lemma.
Lemma 3.3. Let
$\{\varphi _c\}$
be a bounded sequence in
$H^1(\mathbb {R}^3)$
satisfying
$\|\varphi _c\|_{L^2}^2=N$
. Then, there exists a subsequence
$\{\varphi _{c_k}\}$
satisfying one of the following three possibilities.
-
(i) Compactness: there exists a sequence
$\{y_k\}$
in
$\mathbb {R}^3$
such that, for every
$\bar {\varepsilon }>0$
, there exists R,
$0<R<\infty $
, with
$$ \begin{align*} \int_{|x-y_k|<R}|\varphi_{c_k}|^2\,dx\geq N-\bar{\varepsilon}. \end{align*} $$
-
(ii) Vanishing: for all
$R>0$
,
$$ \begin{align*} \lim_{k\to\infty}\sup_{y\in \mathbb{R}^3}\int_{|x-y|<R}|\varphi_{c_k}(x)|^2\,dx= 0. \end{align*} $$
-
(iii) Dichotomy: there exists
$\alpha \in (0, N)$
such that, for every
$\bar {\varepsilon }>0$
, there exist two bounded sequences
$\{\varphi _{k}^1\}$
and
$\{\varphi _{k}^2\}$
in
$H^1({\mathbb {R}^3})$
and
$k_0\ge 0$
such that, for all
$k\ge k_0$
, with
$$ \begin{align*} \|\varphi_{c_k}-(\varphi_{k}^1+\varphi_{k}^2)\|_{p}\le\delta_{p}(\bar{\varepsilon}) \quad \text{for } 2\le p<6, \end{align*} $$
$\delta _{p}(\bar {\varepsilon })\to 0$
as
$\bar {\varepsilon }\to 0$
, and, as
$k\to \infty $
,
$\mathrm {dist}(\mathrm {supp}\varphi _{k}^1,\mathrm {supp}\varphi _{k}^2)\to \infty $
,
$$ \begin{align*} \bigg|\!\int_{\mathbb{R}^3}|\varphi_{k}^1|^2\,dx-\alpha\bigg|\le\bar{\varepsilon} \quad\text{and}\quad \bigg|\!\int_{\mathbb{R}^3}|\varphi_{k}^2|^2\,dx-(N-\alpha)\bigg|\le\bar{\varepsilon}. \end{align*} $$
Invoking Lemma 3.3, we obtain a suitable subsequence
$\varphi _{c_k}$
with
$\varphi _{c_k}\rightharpoonup \varphi _{\infty }$
, which satisfies either (i), (ii) or (iii). We rule out (ii) and (iii) as follows.
Vanishing does not occur. If vanishing occurs, it follows from [Reference Fröhlich, Jonsson and Lenzmann4, Lemma A.1] that
$$ \begin{align*} \lim_{k\to\infty}\int_{\mathbb{R}^3}\bigg( \frac{1}{|x|}*|\varphi_{c_k}|^2\bigg)|\varphi_{c_k}|^2\,dx=0. \end{align*} $$
A similar statement can be found in [Reference Lions10, Reference Lions11] in the context of other variational problems. By (3.4), we deduce that
$$ \begin{align*} E_{\infty}(N)=\lim_{k\to\infty}\mathcal{E}_{\infty}(\varphi_{c_k})=\lim_{k\to\infty}\bigg(\frac{1}{4m}\int_{\mathbb{R}^3}|\nabla\varphi_{c_k}|^2\,dx+\frac{1}{2}\langle\varphi_{c},iv\cdot\nabla\varphi_{c}\rangle\bigg)\ge0, \end{align*} $$
which contradicts
$E_{\infty }(N)<0$
. Thus, vanishing does not occur.
Dichotomy does not occur. If (iii) is true for
$\varphi _{c_k}$
, by the same arguments as in [Reference Fröhlich, Jonsson and Lenzmann4],
$$ \begin{align*} E_{\infty}(N)\ge E_{\infty}(\alpha)+ E_{\infty}(N-\alpha) \end{align*} $$
for
$0<\alpha <N$
. This contradicts the strict binding inequality. Thus, dichotomy does not occur. Therefore, we have compactness.
Proof of Theorem 1.2.
From the above arguments, we have shown that there exists a subsequence
$\varphi _{c_k}$
such that Lemma 3.3(i) holds for some sequence
$\{y_k\}$
in
$\mathbb {R}^3$
. We now define the sequence
$$ \begin{align*}\tilde{\varphi}_{k}:=\varphi_{c_k}(\cdot+y_k).\end{align*} $$
Since
$\{\tilde {\varphi }_{k}\}$
is uniformly bounded in
$H^1(\mathbb {R}^3)$
, we can pass to a subsequence, still denoted by
$\{\tilde {\varphi }_{k}\}$
, such that
$\{\tilde {\varphi }_{k}\}$
converges weakly in
$H^1(\mathbb {R}^3)$
to some
$\varphi _{\infty }\in H^1(\mathbb {R}^3)$
as
$k\to \infty $
. Moreover, thanks to the Rellich-type theorem for
$H^1(\mathbb {R}^3)$
(see [Reference Lieb and Loss9, Theorem 8.6]),
$\tilde {\varphi }_{k}\to \varphi _{\infty }$
strongly in
$L_{loc}^p(\mathbb {R}^3)$
as
$k\to \infty $
for
$2\le p<6$
. Since
$$ \begin{align*} \int_{|x|<R}|\tilde{\varphi}_{k}|^2\,dx\geq N-\bar{\varepsilon}, \end{align*} $$
for every
$\bar {\varepsilon }>0$
and suitable
$R=R(\bar {\varepsilon })<\infty $
, we conclude that
$ \tilde {\varphi }_{k}\to \varphi _{\infty }$
strongly in
$L^p(\mathbb {R}^3)$
as
$k\to \infty $
for
$2\le p<6$
. By the same arguments as in [Reference Fröhlich, Jonsson and Lenzmann4],
$$ \begin{align*} \lim_{k\to\infty}\int_{\mathbb{R}^3}\bigg( \frac{1}{|x|}*|\tilde{\varphi}_{k}|^2\bigg)|\tilde{\varphi}_{k}|^2\,dx=\int_{\mathbb{R}^3}\bigg( \frac{1}{|x|}*|\varphi_{\infty }|^2\bigg)|\varphi_{\infty}|^2\,dx. \end{align*} $$
By weak lower semicontinuity, we conclude that
$$ \begin{align*} E_{\infty}(N)\le\mathcal{E}_{\infty}(\varphi_{\infty })\le\liminf_{k\to\infty}\mathcal{E}_{\infty}(\tilde{\varphi}_{k})=E_{\infty}(N). \end{align*} $$
This implies that
$\varphi _{\infty }$
is a minimiser of
$E_{\infty }(N)$
.
