#  Dual formulation of constraint solutions for the multi-state Choquard
equation

Gershon Wolansky  Gershon Wolansky 111Department of Mathematics, Technion,
Israel Inst. of Technology

###### Abstract

The Choquard equation is a partial differential equation that has gained
significant interest and attention in recent decades. It is a nonlinear
equation that combines elements of both the Laplace and Schrödinger operators,
and it arises frequently in the study of numerous physical phenomena, from
condensed matter physics to nonlinear optics.

In particular, the steady states of the Choquard equation were thoroughly
investigated using a variational functional acting on the wave functions.

In this article, we introduce a dual formulation for the variational
functional in terms of the potential indiced by the wave function, and use it
to explore the existence of steady states of a multi-state version the
Choquard equation in critical and sub-critical cases.

##  1 Introduction

###  1.1 Background

The Choquard equation

$-\Delta\phi+\phi-\left(\int\frac{|\phi(y)|^{2}}{|x-y|}dy\right)\phi(x)=0$

in $\mathbb{R}^{3}$ was originally proposed by Ph. Choquard, as an
approximation to Hartree-Fock theory for a one component plasma. Equation of
similar types also appear to be a prototype of the so-called nonlocal
problems, which arise in many situations (see, e.g [17]) and as a model of
self-gravitating matter [11].

A generalized version in $\mathbb{R}^{n}$ takes the form

$-\Delta\phi+\phi=\left(I_{\alpha}*|\phi|^{p}\right)|\phi|^{p-2}\phi$ (1)

where

$I_{\alpha}=A(\alpha)|x|^{\alpha-n};\ \ \
A(\alpha):=\frac{\Gamma\left(\frac{n-\alpha}{2}\right)}{2^{\alpha}\pi^{n/2}\Gamma\left(\alpha/2)\right)}$
(2)

is the Rietz potential, $\alpha\in(0,n)$, $p\in(1,\infty)$ was considered by
many authors in the last decades, using its variational structure as a
critical point of the functional

$E_{p,\alpha}(\phi)=\frac{1}{2}\int_{\mathbb{R}^{n}}\left(|\nabla
u|^{2}+|\phi|^{2}-\frac{1}{2p}\left(I_{\alpha}*|\phi|^{p}\right)|\phi|^{p}\right)$
(3)

on an appropriate space. In particular, existence of solutions the case $p=2$
(and for more general singular interaction kernels) was studied by E.H. Lieb,
P.L Lions and G. Menzala [10, 7, 6]. For existence, regularity and asymptotic
behavior of solutions in the general case see [12, 13] and references therein.

The non-linear Schrödinger equation associated with $E_{p,\alpha}$ takes the
form

$-i\partial_{t}\psi-\Delta\psi-a(I_{\alpha}*|\psi|^{p})|\psi|^{p-2}\psi=0\ .$
(4)

The number $a\in\mathbb{R}$ is the strength of interaction. The case $a>0$
corresponds to the attractive, gravitation-like dynamics, and is related to
Choquard’s equation. The case $a<0$ is the repulsive, electrostatic case and
is related to the Hartree system (see, e.g. [18]). In this paper we deal with
the attractive case.

Considering an eigenmode $\psi=e^{-i\lambda t}\phi$ we get that $\phi$ satisfy
the non-linear eigenvalue problems

$-\Delta\phi-a\left(I_{\alpha}*|\phi|^{p}\right)|\phi|^{p-2}\phi-\lambda\phi=0$
(5)

which can be reduced to (1) by a proper scaling222Note that $\lambda<0$ is an
eigenvalue below the essential spectrum of $-\Delta$ . However, the solutions
of the nonlinear equation (4) preserve the $\mathbb{L}^{2}$ norm, so it is
natural to look for stationary solutions (5) under a prescribed
$\mathbb{L}^{2}$ norm (say, $\|\phi\|_{2}=1$). It is not difficult to see
that, in general, one can find a scaling
$\phi\mapsto\phi_{\epsilon}(x)=\epsilon^{-n/2}\phi(\epsilon/x)$ which
preserves the $\mathbb{L}^{2}$ norm and transform the strength of interaction
in (5) into $a=1$, making this parameter mathematically insignificant. There
is, however, an exceptional case $\alpha=n(p-1)-2$. In that case the first two
terms in (5) are transformed with equal coefficients under $\mathbb{L}^{2}$
preserving scaling, so the size of the interaction coefficient $a$ is
mathematically significant in that case.

In the case $p=2$ and in the presence of a prescribed, confining potential
$W$, the $\mathbb{L}^{2}-$ constraint version of (5) takes the form

$-\Delta\phi+W\phi-a\left(\int_{\mathbb{R}^{n}}\frac{|\phi(y)|^{2}}{|x-y|^{n-\alpha}}dy\right)\phi-\lambda\phi=0,\
\ \ \|\phi\|_{2}=1\ .$ (6)

A solution of (6) is given by a minimizer of the functional

$E^{W}_{a}(\phi):=\frac{1}{2}\int_{\mathbb{R}^{n}}\left(|\nabla\phi|^{2}+W|\phi|^{2}\right)dx-\frac{a}{4}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{|\phi(x)|^{2}|\phi(y)|^{2}}{|x-y|^{n-\alpha}}dxdy$
(7)

restricted to the $\mathbb{L}^{2}$ unit ball $\|\phi\|_{2}=1$.

In [8] the authors studied the equation (6) in the exceptional case
$\alpha=n-2$, for $n\geq 3$, $a>0$ and $W$ a prescribed function satisfying
$\lim_{x\rightarrow\infty}W(x)=\infty$. In particular, they showed the
existence of a critical strength $\bar{a}_{c}>0$, depending on $n$ but
independent of $W$, such that $E^{W}_{a}$ is bounded from below on the sphere
$\|\phi\|_{2}=1$ iff $a\leq\bar{a}_{c}$. Moreover, a minimizer of $E_{a}^{W}$
exists if $a<\bar{a}_{c}$, and is a solution of (6) (c.f. [8]). It was also
shown that $a_{c}=\|\bar{\phi}\|_{2}$, where $\bar{\phi}$ is the unique,
positive solution (c.f. [9]) of the equation of

$-\Delta\bar{\phi}-\left(\int_{\mathbb{R}^{n}}\frac{|\bar{\phi}(y)|^{2}}{|x-y|^{2}}dy\right)\bar{\phi}+\bar{\phi}=0\
.\ $ (8)

The object of the present paper is two-fold.

The first object is to extend the $\mathbb{L}^{2}$-constraint Choquard
equation (6) into a $k-$ state system

$-\Delta\phi_{j}+W\phi_{j}-a\left(\sum_{i=1}^{k}\beta_{i}\int_{\mathbb{R}^{n}}\frac{|\phi_{i}|^{2}(y)}{|x-y|^{n-\alpha}}dy\right)\phi_{j}-\lambda_{j}\phi_{j}=0\
\ \|\phi_{j}\|_{2}=1,\ \ ;\ \ j=1\ldots k$ (9)

where $(\phi_{1},\ldots\phi_{k})$ constitutes an orthonormal $k-$sequence in
$\mathbb{L}^{2}(\mathbb{R}^{n})$ and

$\beta_{j}>0,\ \ \ \ \sum_{1}^{k}\beta_{j}=1\ $ (10)

are the probabilities of occupation of the states $j=1\ldots k$,

In Section 1.2 we introduce the time dependent Heisenberg system which leads
naturally to (9), while the steady state (9) and its constraint variational
formulation are introduced in Section 1.3.

The second object is to introduce a dual approach to the $\mathbb{L}^{2}$
constraint Choquard problem in the case $p=2$. For the case of single state
$k=1$, the dual formulation of $E^{W}_{a}$ (7) for $\alpha=2$ on the
constraint $\mathbb{L}^{2}$ sphere takes the form of the functional
$V\mapsto{\cal H}^{W,\alpha}_{a}(V)$

${\cal H}^{W,2}_{a}(V)=\frac{a}{2}\int_{\mathbb{R}^{n}}|\nabla
V|^{2}+\lambda_{1}(V)$

over the unconstrained Beppo-Levi space
$V\in\dot{\mathbb{H}}_{1}(\mathbb{R}^{n})$ (c.f. section 1.4). Here the
functional $\lambda_{1}=\lambda_{1}(V)$ is the leading (minimal) eigenvalue of
the Schrödinger operator $-\Delta+W-aV$ on $\mathbb{R}^{n}$.

The extension of this dual formulation to the $k-$system (9) for
$\alpha\in(0,2]$ is introduced in (28). In case $\alpha=2$ it takes the form

${\cal H}^{W,2}_{\beta,a}(V)=\frac{a}{2}\int_{\mathbb{R}^{n}}|\nabla
V|^{2}+\sum_{j=1}^{k}\beta_{j}\lambda_{j}(V)$

where $\lambda_{1}(V)<\lambda_{2}(V)\leq\ldots\lambda_{k}(V)$ are the leading
$k$ eigenvalues of the Schrödinger operator, while
$\beta_{1}>\beta_{2}>\ldots\beta_{k}>0$.

The main result of this paper is summarized below ( Section 1.6):

Using the dual variational formulation we show the existence of a minimizer of
${\cal H}^{W,\alpha}_{\beta,a}$ corresponding to a solution of (9) in
$\mathbb{R}^{n}$ for any $a>0$ where $\alpha\in(0,2]$, $3\leq n<2+\alpha$ . In
the critical cases $\alpha=2,n=4$ and $\alpha=1,n=3$ we show the existence of
a critical interaction level $a^{(n)}_{c}(\beta)$ for which there is a
minimizer of ${\cal H}^{W,\alpha}_{\beta,a}$ if $a<a^{(n)}_{c}(\beta)$
corresponding to a solution of (9), while ${\cal H}^{W,\alpha}_{\beta,a}$ is
unbounded from below for any $a>a^{(n)}_{c}(\beta)$ for $n=3,4$.

###  1.2 Mean-field Heisenberg system

Consider the Von Neumann-Heisenberg equation

$i\frac{\partial R}{\partial t}=\left[L^{W}-aV,R\right]\ ,\ t\in\mathbb{R}$
(11)

on a Hilbert space $\mathbb{H}$. Here $R$ is a density operator, namely a
bounded linear operator on $\mathbb{H}$ which is self-adjoint, non-negative
and of trace equal one. $L^{W}$ is an Hermitian operator generating a norm
preserving group $e^{itL^{W}}$ on $\mathbb{H}$ and $V$ is a non-linear
operator.

In the context of mean-field system we consider
$(\mathbb{H},\left<\cdot,\cdot\right>)$ to be the Hilbert space
$\mathbb{L}^{2}(\mathbb{R}^{n})$ where
$\left<\phi,\psi\right>:=\int_{\mathbb{R}^{n}}\phi\bar{\psi}$ the canonical
inner product. A density operator can be represented by a kernel $K_{R}$
acting on $\phi\in\mathbb{H}$ via
$R(\phi)=\int_{\mathbb{R}^{n}}K_{R}(x,y)\phi(y)dy$ and $Tr(R)(x):=K_{R}(x,x)$.
In these terms we define $V(R)$ as the operator acting on $\phi\in\mathbb{H}$
by multiplication with

$V(R):=I_{\alpha}*Tr(R)\ \ $ (12)

Since $L^{W}-aV$ is hermitian for any prescribed potential $V$, all
observables along the orbit $t\mapsto R(\cdot,t)$ are unitary equivalent:

$R(\cdot,t)=\exp\left(-i\int_{0}^{t}(L^{W}-aV(\cdot,s))ds\right)R(\cdot,0)\exp\left(i\int_{0}^{t}(L^{W}-aV(\cdot,s))ds\right)\
.$ (13)

We restrict ourselves to a class of observables of a finite rank
$k\in\mathbb{N}$. Hence the kernel of $R$ can be represented as

$R(x,y,t)=\sum_{1}^{k}\beta_{j}\psi_{j}(x,t)\bar{\psi}_{j}(y,t)$ (14)

where $\beta_{j}>0$ are the eigenvalues of $R$, which are constant in time,
and $\psi_{j}(\cdot,t)\in\mathbb{H}$ constitute an orthonormal sequence for
any $t\in\mathbb{R}$. Under this representation (11) takes the form

$i\frac{\partial\psi_{j}}{\partial t}=(L^{W}-aV)\psi_{j}\ \ ,j=1,2,\ldots k$
(15)

The eigenvalues $\beta_{j}\in[0,1]$ are interpreted as the probability of
occupation of the $j-$ level satisfying $\sum_{j=1}^{k}\beta_{j}=1$. For any
$t\in\mathbb{R}$, the trace of $R$ conditioned on $x\in\mathbb{R}^{n}$ is

$Tr(R)(x,t)=\sum_{j=1}^{k}\beta_{j}|\psi_{j}(x,t)|^{2}\ ,$ (16)

and the potential $V$ is determined in terms of the solution $R$ by (12)

$V=\sum_{j=1}^{k}\beta_{j}I_{\alpha}*|\psi_{j}|^{2}\ .$

Consider now the Hamiltonian

${\cal
E}^{(\alpha)}_{\beta,a}(\vec{\psi}):=\frac{1}{2}\sum_{j=1}^{k}\beta_{j}\left[\left<L^{W}\psi_{j},\psi_{j}\right>-\frac{a}{2}\sum_{i=1}^{k}\beta_{i}\left<|\psi_{j}|^{2},I_{\alpha}*|\psi_{i}|^{2}\right>\right]$

acting on $k-$ orthonormal frames $\vec{\psi}=(\psi_{1},\ldots\psi_{k})$. The
system (11) (equivalently (15) ) is, in fact, an Hamiltonian system in the
canonical variables $\\{\psi_{i},\bar{\psi}_{j}\\}$:

$i\frac{\partial\psi_{j}}{\partial
t}=-\frac{1}{\beta_{j}}\delta_{\bar{\psi}_{j}}{\cal E}^{(\alpha)}_{\beta,a};\
\ \ i\frac{\partial\bar{\psi}_{j}}{\partial
t}=\frac{1}{\beta_{j}}\delta_{\psi_{j}}{\cal E}^{(\alpha)}_{\beta,a}\ .$ (17)

In particular, ${\cal E}^{(\alpha)}_{\beta,a}$ is constant along the solution
of (11).

###  1.3 Steady states

The steady states of this system are given by
$\psi_{j}=e^{-i\lambda_{j}}\phi_{j}$ where $\\{\phi_{j}\\}$ is an orthonormal
sequence corresponding to eigenvalues $\lambda_{j}$ of the operator
$L^{W}-aV$, satisfying

$L^{W}\phi_{j}-aI_{\alpha}*\left(\sum_{i=1}^{k}\beta_{i}|\phi_{i}|^{2}\right)\phi_{j}-\lambda_{j}\phi_{j}=0\
.$ (18)

######  Definition 1.1.

$\mathbb{H}^{1}:=\\{\phi\in\mathbb{L}^{2}(\mathbb{R}^{n}),\ ;\
\nabla\phi\in\mathbb{L}^{2}(\mathbb{R}^{n});\ \|\phi\|_{2}=1,\ \
\int_{\mathbb{R}^{n}}W|\phi|^{2}<\infty\ \\}$
$\oplus^{k}\mathbb{H}^{1}:=\left\\{\vec{\phi}=(\phi_{1},\ldots\phi_{k}),;\phi_{j}\in\mathbb{H}^{1};\
\left<\phi_{j},\phi_{i}\right>=\delta_{i}^{j},\ \ i,j\in\\{1,\ldots
k\\}\right\\}\ .$

$\left<\left<\phi,\phi\right>\right>_{W}$ is the quadratic form on
$\mathbb{H}^{1}$ defined by the completion of $\left<L^{W}\phi,\phi\right>$:

$\left<\left<\phi,\phi\right>\right>_{W}:=\int_{\mathbb{R}^{n}}|\nabla\phi|^{2}+W|\phi|^{2}\
.$

Let

${\cal
E}^{(\alpha)}_{\beta,a}(\vec{\phi}):=\frac{1}{2}\sum_{1}^{k}\beta_{j}\left[\left<\left<\phi_{j},\phi_{j}\right>\right>_{W}-\frac{a}{2}\sum_{i=1}^{k}\beta_{i}\left<|\phi_{j}|^{2},I_{\alpha}*|\phi_{i}|^{2}\right>\right]\
$

is defined over $\oplus^{k}\mathbb{H}^{1}$ (c.f. Corollary 2.2 below).

We formally obtain from (17) that the steady states (18) are critical points
of ${\cal E}^{(\alpha)}_{\beta,a}$ subject to the orthogonality constraints.

######  Proposition 1.

Suppose $\beta_{j}\not=\beta_{i}$ for any $1\leq i\not=j\leq k$. Then any
critical point of ${\cal E}^{(\alpha)}_{\beta,a}$ restricted to orthonormal
frames $\overrightarrow{\phi}=(\bar{\phi}_{1}\ldots\bar{\phi}_{k})$ is
composed of $k$ normalized eigenstates of the operator $L^{W}-a\bar{V}$ where
$\bar{V}=I_{\alpha}*\left(\sum_{1}^{k}\beta_{j}|\bar{\phi}_{j}|^{2}\right)$.

For the proof of Proposition 1 see the beginning of section 2.

From now on we assume

$\beta_{1}>\beta_{2}>\ldots\ >\beta_{k}>0\ .$ (19)

#### Formulation of the problem

: Consider the multi-state Choquard system satisfying the equivalent of (18):

$(L^{W}-aV)\phi_{j}-\lambda_{j}\phi_{j}=0\ \ ;j=1,\ldots\ k$ (20)

on $\mathbb{R}^{n}$. Here:

i)

$L^{W}=-\Delta+W$, $\Delta:=\sum_{i=1}^{n}\partial^{2}_{x_{i}}$ is the
Laplacian on $\mathbb{R}^{n}$ and

$W\in\mathbb{L}^{\infty}_{loc}(\mathbb{R}^{n}),\ \ \
\lim_{|x|\rightarrow\infty}W(x)=\infty,\ \
\inf_{x\in\mathbb{R}^{n}}W(x)=W(0)=0\ $ (21)

ii)

$\vec{\phi}=(\phi_{1},\ldots\phi_{k})\in\oplus^{k}\mathbb{H}^{1}$ are
normalized eigenfunctions of $L^{W}-aV$ and $\lambda_{j}\in\mathbb{R}$ are the
corresponding eigenvalues.

iii)

$V=\sum_{i=1}^{k}\beta_{i}I_{\alpha}*|\phi_{i}|^{2}$ (22)

where $\beta_{j}$ are the probabilities of occupation of the states $j$, thus
$\beta_{j}>0$ and $\sum_{1}^{k}\beta_{j}=1$.

iv)

$a>0$.

###  1.4 A crush review on Rietz kernels and its dual

Let us recall some definitions and theorems we use later (for more details see
[4]):

For $V_{1},V_{2}\in C^{\infty}_{0}(\mathbb{R}^{n})$ and $\alpha\in(0,n)$,
consider the quadratic form

$\left<V_{1},V_{2}\right>_{\alpha/2}:=A(-\alpha)\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\frac{(V_{1}(x)-V_{1}(y))(V_{2}(x)-V_{2}(y))}{|x-y|^{n+\alpha}}dxdy$

where the constant $A(-\alpha)$ is defined as in (2). If $\alpha=2$

$\left<V_{1},V_{2}\right>_{(1)}:=\int_{\mathbb{R}^{n}}\nabla V_{1}\cdot\nabla
V_{2}dx\ .$

The closure of $C_{0}^{\infty}(\mathbb{R}^{n})$ with respect to the norm
induced by the inner product $\left<\cdot,\cdot\right>_{\alpha/2}$ is denoted
by $\dot{\mathbb{H}}^{\alpha/2}$. We denote the associated norm by
$\||\cdot\||_{\alpha/2}$. 333Note that
$\dot{\mathbb{H}}^{\alpha/2}(\mathbb{R}^{n})$ does not contain
$\mathbb{L}^{2}(\mathbb{R}^{n})$. In case $\alpha=2$ it is sometimes called
Beppo-Levi space. Recall that $\dot{\mathbb{H}}^{\alpha/2}$ is a Hilbert space
so, in particular, is weakly locally compact.

######  Lemma 1.1.

[1] For $\alpha\in(0,2]$, $n>2$, the space $\dot{\mathbb{H}}^{\alpha/2}$ is
continuously embedded in $\mathbb{L}^{2n/(n-\alpha)}(\mathbb{R}^{n})$, so
there exists $S=S_{n,\alpha}>0$ such that

$\|V\|_{2n/(n-\alpha)}\leq S_{n,\alpha}\|V\||_{\alpha/2}\ .$

The fractional Laplacian $(-\Delta)^{\alpha/2}$, $0<\alpha<2$ is defined as a
distribution by

$\left<V,\phi\right>_{\alpha/2}=\left<(-\Delta)^{\alpha/2}V,\phi\right>\
\forall\ \ \phi\in C_{0}^{\infty}(\mathbb{R}^{n})\ .$

and the pointwise definition of the fractional Laplacian for $0<\alpha<2$ is
given in terms of the singular integral

$(-\Delta)^{\alpha/2}V(x)=A(-\alpha)\int_{\mathbb{R}^{n}}\frac{V(x+y)-V(x)}{|y|^{n+\alpha}}dy\
.$

For $\alpha=2$, the above definition is reduced to the classical, local
Laplacian $-\Delta=\sum_{j=1}^{n}\partial^{2}_{x_{j}}$.

The Rietz potential $I_{\alpha}$ is defined as a distribution via the
quadratic form induced by the dual of the
$\left<\cdot,\cdot\right>_{\alpha/2}$ inner product:

$\frac{1}{2}\left<I_{\alpha}*\rho,\rho\right>:=\sup_{V\in
C_{0}^{\infty}(\mathbb{R}^{n})}\left<\rho,V\right>-\frac{1}{2}\left<V,V\right>_{\alpha/2}\
.$ (23)

The Euler-Lagrange equation corresponding to the right hand side of (23) takes
the form

$(-\Delta)^{\alpha/2}V=\rho\ .$ (24)

In particular, $I_{\alpha}\equiv(-\Delta)^{-\alpha/2}$ corresponds to the
right inverse of the fractional Laplacian

$I_{\alpha}*(-\Delta)^{\alpha/2}V=V\ .$ (25)

The pointwise representation of the kernel $I_{\alpha}$ is given by (2).
Moreover

######  Lemma 1.2.

[16] For any $0<\alpha<n$, the Rietz potential is a bounded operator from
$\mathbb{L}^{p}(\mathbb{R}^{n})$ to $\mathbb{L}^{q}(\mathbb{R}^{n})$ iff
$1<p<n/\alpha$ and $1/q=1/p-\alpha/n$.

Our main results, described below, concern the Choquard problem on
$\mathbb{R}^{n}$. However, in order to overcome problems of lack of
compactness, we shell need to introduce a version of this problem in bounded
domain $\Omega\subset\mathbb{R}^{n}$. In order to handle this we need to
define the Green function corresponding to the fractional Laplacian
$(-\Delta)^{\alpha}$ in a bounded domain under homogeneous Dirichlet
condition. This is the motivation to define the local Rietz potential
$I_{\alpha}^{\Omega}$ on $\mathbb{L}^{p}(\Omega)$ by

$\frac{1}{2}\left<I^{\Omega}_{\alpha}(\rho),\rho\right>:=\sup_{\phi\in
C_{0}^{\infty}(\Omega)}\left<\rho,\phi\right>-\frac{1}{2}\left<\phi,\phi\right>_{\alpha/2}\
.$ (26)

In the case $\alpha=2$ this definition induces the Green function of the
Dirichlet problem $I_{2}^{\Omega}\equiv(-\Delta_{\Omega})^{-1}$, that is,
$V(x)=\int_{\Omega}I_{2}^{\Omega}(x,y)\rho(y)dy$ is the solution of the
Poisson problem

$\Delta V+\rho=0\ \ \ x\in\Omega;\ \ V=0\ \ \text{on}\ \ \partial\Omega\ .$
(27)

Not much is known444But see Section 3-c. on the Green function
$I_{\alpha}^{\Omega}$ for $\alpha<2$. In case $\alpha=2$ the maximum principle
implies immediately that for any $x,y\in\Omega$ the inequality $I_{2}(x-y)\geq
I_{2}^{\Omega}(x,y)$ holds, and that $I_{2}^{\Omega}(x,y)=0$ if
$x\in\Omega,y\in\partial\Omega$. In the general case we obtain from (26) :

######  Lemma 1.3.

For any $0<\alpha<2$, $\Omega_{2}\supset\Omega_{1}$ and
$supp(\rho)\subset\Omega_{1}$ then $I_{\alpha}^{\Omega_{1}}(\rho)\leq
I_{\alpha}^{\Omega_{2}}(\rho)\leq I_{\alpha}*\rho$.

In addition: Let $\Omega_{j}\subset\mathbb{R}^{n}$,
$\Omega_{j}\rightarrow\mathbb{R}^{n}$ is a monotone sequence of domains in
$\mathbb{R}^{n}$. If $\rho_{j}$ converges to $\rho$ in
$\mathbb{L}^{p}(\mathbb{R}^{n})$, $p\in(1,n/\alpha)$, and $\rho_{j}$ are
supported in $\Omega_{j}$ then

$\lim_{j\rightarrow\infty}I^{\Omega_{j}}_{\alpha}(\rho_{j})=I_{\alpha}*\rho\
\text{in}\ \ \mathbb{L}^{\frac{pn}{n-p\alpha}}(\mathbb{R}^{n})\ .$

###  1.5 Spectrum of the Schrödinger operator

One of the most celebrated results on the discreteness of spectrum for the
Schrödinger operator $-\Delta+W$ in $\mathbb{L}^{2}(\mathbb{R}^{n})$ with a
locally integrable potential is a result of K. Friedrichs [2] which ensures
the discreteness of spectrum if the potential $W$ grows at infinity at
arbitrary rate. This and (21) implies, in particular

######  Proposition 2.

Let $V\in C_{0}^{\infty}(\mathbb{R}^{n})$ and $W$ satisfies (21). Then the
spectrum of the operator $L^{W}-aV$ is composed of an infinite set of
eigenvalues $\lambda_{j}\rightarrow\infty$ and the corresponding normalized
eigenfunctions $\phi_{j}$ constitute a complete orthonormal base of
$\mathbb{L}^{2}(\mathbb{R}^{n})$.

For the proof of Proposition 3 see Lemma 2.4 in Section 2.

######  Proposition 3.

If $3\leq n\leq 5$, $0<\alpha\leq 2$, then Proposition 2 can be extended for
$V\in\dot{\mathbb{H}}^{\alpha/2}(\mathbb{R}^{n})$, and
$V\mapsto\lambda_{j}(V)$ is a continuous functional in the
$\||\cdot\||_{\alpha/2}$ norm.

Proposition 3 allows us to define the dual functional on
$\dot{\mathbb{H}}^{\alpha/2}(\mathbb{R}^{n})$:

${\cal
H}^{W,\alpha}_{\beta,a}(V):=\frac{a}{2}\left<V,V\right>_{\alpha/2}+\sum_{j=1}^{k}\beta_{j}\lambda_{j}(V)$
(28)

where
$\lambda_{1}(V)<\lambda_{2}(V)\leq\lambda_{3}(V)\ldots\leq\lambda_{k}(V)$ are
the lowest $k$ eigenvalues of the operator $L^{W}-aV$.

###  1.6 Main Theorem

For any $V\in\dot{\mathbb{H}}_{\alpha/2}(\mathbb{R}^{n})$ let $\phi_{j}(V)$ be
a normalized eigenstate corresponding to $\lambda_{j}(V)$ 555If
$\lambda_{j}(V)$ is degenerate, so
$\lambda_{j-1}(V)>\lambda_{j}(V)=\ldots=\lambda_{j+l}(V)>\lambda_{j+l+1}(V)$,
then $\\{\phi_{j}(V),\ldots\phi_{j+l}(V)\\}$ is any orthonormal base of the
eigenspace of $\lambda_{j}$.

[i] $V$ is a minimizer of ${\cal H}^{W,\alpha}_{\beta,a}$ on
$\dot{\mathbb{H}}_{\alpha/2}(\mathbb{R}^{n})$ if and only if
$(\phi_{1}(V),\ldots\phi_{k}(V))$ is a minimizer of ${\cal
E}^{(\alpha)}_{\beta,a}$ on $\oplus^{k}\mathbb{H}^{1}$. If this is the case
then $\\{\lambda_{j}(V),\phi_{j}(V)\\}_{1\leq j\leq k}$ is a solution of (20)
.

[ii] If $\alpha\in(0,2]$ $3\leq n<2+\alpha$ then ${\cal
H}^{W,\alpha}_{\beta,a}$ is bounded from below on
$\dot{\mathbb{H}}_{\alpha/2}(\mathbb{R}^{n})$ for any $\beta$ satisfying (19)
and any $a>0$, and there is a minimizer of ${\cal H}^{W,\alpha}_{\beta,a}$ in
$\dot{\mathbb{H}}_{\alpha/2}(\mathbb{R}^{n})$.

[iii] If $\alpha=1,n=3$ or $\alpha=2,n=4$ then there exists $a_{c}(\beta)>0$
such that ${\cal H}^{W,\alpha}_{\beta,a}$ is bounded from below on
$\dot{\mathbb{H}}_{\alpha/2}(\mathbb{R}^{n})$ if $a<a_{c}(\beta)$ and
unbounded from below if $a>a_{c}(\beta)$. If $a<a_{c}(\beta)$ there exists a
minimizer of ${\cal H}^{W,\alpha}_{\beta,a}$ in
$\dot{\mathbb{H}}_{\alpha/2}(\mathbb{R}^{n})$.

##  2 Proofs

We start by proving Proposition 1

###### Proof.

Let $\gamma_{i,j}$ be the Lagrange multiplier for the constraints
$\left<\phi_{i},\phi_{j}\right>=\delta_{i,j}$. Then $\overrightarrow{\phi}$ is
an unconstraint critical point of

${\cal
E}^{(\alpha)}_{\beta,a}(\vec{\phi})+\sum_{1}^{k}\gamma_{j,j}\|\phi_{j}\|_{\mathbb{H}^{1}}^{2}+\sum_{i\not=j}\gamma_{i,j}\left<\phi_{i},\phi_{j}\right>\
.$

This implies

$\frac{\delta{\cal
E}^{(\alpha)}_{\beta,a}}{\delta\bar{\phi}_{j}}+2\gamma_{j,j}\bar{\phi}_{j}+\sum_{i\not=j}\gamma_{i,j}\bar{\phi}_{i}=\beta_{j}(L^{W}-a\bar{V})\bar{\phi}_{j}+2\gamma_{j,j}\bar{\phi}_{j}+\sum_{i\not=j}\gamma_{i,j}\bar{\phi}_{i}=0\
.$

In particular, $Sp(\bar{\phi}_{1}\ldots\bar{\phi}_{k})$ is an invariant
subspace of $L^{W}-a\bar{V}$. Since $(\bar{\phi}_{1}\ldots\bar{\phi}_{k})$ is
an orthonormal sequence we get, after multiplying the above line by
$\bar{\phi}_{i}$ and taking the inner product:

$\left<\beta_{j}\left[L^{W}-a\bar{V}\right]\bar{\phi}_{j},\bar{\phi}_{i}\right>+\gamma_{i,j}=0$

for any $i\not=j$. switching $i$ with $j$ and taking into consideration that
$L+\bar{V}$ is self-adjoint, we also get

$\left<\beta_{i}\left[L^{W}-a\bar{V}\right]\bar{\phi}_{j},\bar{\phi}_{i}\right>+\gamma_{i,j}=0$

Subtracting the two inequalities we obtain
$\left<(\beta_{j}-\beta_{i})\left(L^{W}-a\bar{V}\right)\bar{\phi}_{j},\bar{\phi}_{i}\right>=0$
thus $\left<\left(L^{W}-a\bar{V}\right)\bar{\phi}_{j},\bar{\phi}_{i}\right>=0$
for any $i\not=j$. Since $Sp(\bar{\phi}_{1}\ldots\bar{\phi}_{k})$ is an
invariant subspace of $L^{W}-a\bar{V}$, this implies that $\bar{\phi}_{j}$ is
are eigenstates of $L^{W}-a\bar{V}$. ∎

The first part of the following Lemma follows from a compactness embedding
Theorem (c.f. Theorem XIII.67 in [15]). The second part from Sobolev and HLS
inequalities (see, e.g. [13], sec. 3.1.1)

######  Lemma 2.1.

For any $n\geq 3$, $\mathbb{H}^{1}$ is compactly embedded in $\mathbb{L}^{r}$
for $2<r<\frac{2n}{n-2}$. If
$\frac{n-2}{n+\alpha}\leq\frac{1}{2}\leq\frac{n}{n+\alpha}$ and
$\phi\in\mathbb{H}^{1}$ then
$|\phi|^{2}\in\mathbb{L}^{2n/(n+\alpha)}(\mathbb{R}^{n})\cap\mathbb{L}^{1}(\mathbb{R}^{n})$
and

$\int_{\mathbb{R}^{n}}(I_{\alpha}*|\phi|^{2})|\phi|^{2}\leq
C_{n,\alpha}\left(\int|\phi|^{4n/(n+\alpha)}\right)^{1+\alpha/n}$

In particular we obtain:

######  Corollary 2.1.

If $\max(0,n-4)\leq\alpha\leq n$ then the functional ${\cal
E}^{(\alpha)}_{\beta,a}$ is defined on $\oplus^{k}\mathbb{H}^{1}$.

Let $\alpha\in(0,2]$, $\vec{\phi}\in\oplus^{k}\mathbb{H}^{1}$ and $V\in
C_{0}^{\infty}(\mathbb{R}^{n})$. Define

${\bf
H}^{(\alpha)}_{\beta}(\vec{\phi},V)=\sum_{1}^{k}\beta_{j}\left<L^{W}\phi_{j},\phi_{j}\right>+a\left[\left<V,V\right>_{\alpha/2}-\left<V,\sum\beta_{j}|\phi_{j}|^{2}\right>\right]\
.$ (29)

By (23) we get (c.f Definition 1.1)

$\inf_{V\in C_{0}^{\infty}(\mathbb{R}^{n})}{\bf
H}^{(\alpha)}_{\beta}(\vec{\phi},V)=2{\cal
E}^{(\alpha)}_{\beta,a}(\vec{\phi})\ .$

Thus

$\inf_{V\in C_{0}^{\infty}}\inf_{\vec{\phi}\in\oplus^{k}\mathbb{H}^{1}}{\bf
H}^{(\alpha)}_{\beta}(\vec{\phi},V)\equiv\inf_{\vec{\phi}\in\oplus^{k}\mathbb{H}^{1}}\inf_{V\in
C_{0}^{\infty}}{\bf
H}^{(\alpha)}_{\beta}(\vec{\phi},V)=\inf_{\vec{\phi}\in\oplus^{k}\mathbb{H}^{1}}{\cal
E}^{(\alpha)}_{\beta,a}(\vec{\phi})\ .$

Let

${\cal
H}^{W,\alpha}_{\beta,a}(V)=\inf_{\vec{\phi}\in\oplus^{k}\mathbb{H}^{1}}{\bf
H}^{(\alpha)}_{\beta}(\vec{\phi},V)\ .$

From (23 , 29) we observe that

${\cal
H}^{W,\alpha}_{\beta,a}(V)=\frac{a}{2}\left<V,V\right>_{\alpha/2}+\inf_{\vec{\phi}\in\oplus^{k}\mathbb{H}^{1}}\sum_{j=1}^{k}\beta_{j}\left<(L^{W}-aV)\phi_{j},\phi_{j}\right>\
.$

Let

$\inf_{\vec{\phi}\in\oplus^{k}\mathbb{H}^{1}}\sum_{j=1}^{k}\beta_{j}\left<(L^{W}-aV)\phi_{j},\phi_{j}\right>:=G_{\beta,a}(V)\
.$ (30)

As the infimum over linear functionals, $V\mapsto G_{\beta,a}(V)$ is a concave
functional, so

${\cal
H}^{W,\alpha}_{\beta,a}(V)=\frac{a}{2}\left<V,V\right>_{\alpha/2}+G_{\beta,a}(V)\
$ (31)

is the sum of convex and concave functionals. In the case $k=1$
($\vec{\beta}=\beta_{1}=1$) we observe, by the Rayleigh-Ritz principle

$G_{1,a}(V)=\inf_{\|\phi\|=1}\left<(L^{W}-aV)\phi,\phi\right>=\lambda_{1}(V)$

and the supremum is obtained at the normalized ground state $\bar{\phi}_{1}$
satisfying $(L^{W}-aV-\lambda_{1})\bar{\phi}_{1}=0$. In particular we reassure
that $G_{1,a}(V)=\lambda_{1}(V)$ is a concave functional. In general, higher
eigenvalues $\lambda_{j}=\lambda_{j}(V)$ are not concave functions if $j>1$.
However, if $\vec{\beta}:=(\beta_{1},\ldots\beta_{k})$ satisfies (19) then we
claim that $V\mapsto\sum_{j=1}^{k}\beta_{j}\lambda_{j}(V)$ is concave. Indeed:

######  Lemma 2.2.

$G_{\beta,a}(V)=\sum_{j=1}^{k}\beta_{j}\lambda_{j}(V)\ $ (32)

where $\lambda_{j}(V)$ are the $k$ lowest eigenvalues of the operator
$L^{W}-aV$ arranged by order

$\lambda_{1}(V)<\lambda_{2}(V)\leq\ldots\leq\lambda_{k}(V)\ .$

Moreover, the minimum in (30) is obtained at the eigenfunction
$\bar{\phi}_{j}$ of $L^{W}-aV$ corresponding to $\lambda_{j}$.

Recall the definition of sup-gradient of a concave functional $G$ on a vector
space $C^{\infty}_{0}(\mathbb{R}^{n})$ at $V$:

$\partial_{V}G:=\left\\{\zeta\in(C_{0}^{\infty})^{{}^{\prime}};\ \ G(Z)\leq
G(V)+\left<Z-V,\zeta\right>\ \forall Z\in C_{0}^{\infty}\right\\}$

while $G$ is differentiable at $V$ if $\partial_{V}G$ is a singleton.

######  Corollary 2.2.

The sup-gradient of the functional $G_{\beta,a}$ on
$C_{0}^{\infty}(\mathbb{R}^{n})$ is contained in $\mathbb{L}^{1}$. In fact
$\partial_{V}G_{\beta,a}=a\sum_{j=1}^{k}\beta_{j}|\bar{\phi}_{j}|^{2}$ where
$\bar{\phi}_{j}\in\mathbb{L}^{2}$ is a normalized eigenstate of $L^{W}-aV$
corresponding to the $j-$ eigenvalue. So, in particular,
$\|\partial_{V}G_{\beta,a}\|_{1}=a$. If all eigenvalues of $L^{W}-aV$ are
simple then $G_{\beta,a}$ is differentiable at $V$.

###### Proof.

Since
$V\mapsto\sum_{j=1}^{k}\beta_{j}\left<(L^{W}-aV)\phi_{j},\phi_{j}\right>$ is a
linear functional, (30) would imply, in particular, that the functional
$G_{\beta,a}$ is, indeed, a concave one.

Let $\bar{\phi}_{j}$ be the normalized eigenvalues of $L^{W}-aV$ corresponding
to $\lambda_{j}(V)$. Fix some $m\geq j$ and let
$\mathbb{H}_{m}=Sp(\bar{\phi}_{1},\ldots\bar{\phi}_{m})$. Let us restrict the
supremum (30) to
$\mathbb{H}^{k}_{m}:=\\{\vec{\phi}:=(\phi_{1},\ldots\phi_{k}),\phi_{j}\in\mathbb{H}_{m}\\}\subset\mathbb{H}^{k}$.

Then

$\phi_{j}=\sum_{i=1}^{m}\left<\phi_{j},\bar{\phi}_{i}\right>\bar{\phi}_{i},\ \
\
(L^{W}-aV)\phi_{j}=\sum_{i=1}^{m}\lambda_{i}\left<\phi_{j},\bar{\phi}_{i}\right>\bar{\phi}_{i}\
\ .$

Define $\beta_{k+1}=\ldots=\beta_{m}=0$. Then we can write, for any
$\vec{\phi}\in\mathbb{H}^{k}_{m}$

$\sum_{j=1}^{k}\beta_{j}\left<(L^{W}-aV)\phi_{j},\phi_{j}\right>=\sum_{i=1}^{m}\sum_{j=1}^{m}\beta_{j}\lambda_{i}|\left<\phi_{j},\bar{\phi}_{i}\right>|^{2}\
.$ (33)

Denote now $\gamma_{i,j}:=|\left<\phi_{j},\bar{\phi}_{i}\right>|^{2}$. Then
$\\{\gamma_{i,j}\\}$ is $m\times m$, bi-stochastic matrix, i.e
$\sum_{i=1}^{m}\gamma_{i,j}=\sum_{j=1}^{m}\gamma_{i,j}=1$ for all $i,j=1\ldots
m$. Consider now the infimum of
$\sum_{i=1}^{m}\sum_{j=1}^{m}\tilde{\gamma}_{i,j}\lambda_{i}\beta_{j}$ over
all bi-stochastic martices $\\{\tilde{\gamma}_{i,j}\\}$. By Krain-Milman
theorem, the minimum is obtained on an extreme point in the convex set of bi-
stochastic matrices. By Birkhoff theorem, the extreme points are permutations
so, from(33)

$\forall\vec{\phi}\in\mathbb{H}^{k}_{m},\ \ \
\sum_{j=1}^{k}\beta_{j}\left<(L^{W}-aV)\phi_{j},\phi_{j}\right>\geq\sum_{j=1}^{m}\beta_{\pi(j)}\lambda_{j}$

for some permutation $\pi:\\{1,\ldots m\\}\mapsto\\{1,\ldots m\\}$. Now,
recall that $\beta_{j}$ are assumed to be in descending order while
$\lambda_{j}$ are in ascending order by definition. By the discrete
rearangment theorem of Hardy, Littelwood and Polya [3] we obtain that the
maximum on the right above is obrained at the identity permutation $\pi(i)=i$,
that is, at the identity matrix
$\tilde{\gamma}_{i,j}:=\left<\phi_{j},\bar{\phi}_{i}\right>=\delta_{i,j}$.
This implies thet the eigenbasis $\bar{\phi}_{1},\ldots\bar{\phi}_{k}$ of the
$k$ leading eigenvalues is the minimizer of (30) on $\mathbb{H}_{m}^{k}$ for
any $m\geq k$.

In particular, the minimizer of (30) in $\mathbb{H}_{m}^{k}$ is independent of
$m$, as long as $m\geq k$. Suppose there exists some
$\vec{\psi}\in\mathbb{H}^{k}$ which is not contained in and finite dimensional
subspace generated by eigenstates, for which (30) is strictly smaller than its
value on the first $k-$ leading eigenspace. Since the eigenstates of the
Schrödinger operator under assumption (21) generate the whole space we can
find, for a sufficiently large $m$, an orthonormal base in
$\mathbb{H}_{m}^{k}$ for on which the left side of (30) is strictly larger
than $\sum_{j=1}^{k}\beta_{j}\lambda_{j}(V)$, and we get a contradiction for
this value of $m$.

∎

From Corollary 2.2 and (24, 25) It follows that the Euler-Lagrange equation
corresponding to ${\cal H}^{W,\alpha}_{\beta,a}$ is

$(-\Delta)^{\alpha/2}V-\sum_{j=1}^{k}\beta_{j}|\phi_{j}|^{2}=0\ \ \
\Longleftrightarrow V=I_{\alpha}*(\sum_{j=1}^{k}\beta_{j}|\phi_{j}|^{2})$

where $\phi_{j}$ are the normalized eigenfunction corresponding to
$\lambda_{j}(V)$. In particular we obtain the proof of Theorem 1.6-(i):

######  Corollary 2.3.

If $\bar{V}$ is a minimizer of ${\cal H}^{W,\alpha}_{\beta,a}$ then
$\bar{V}=\sum_{j=1}^{k}\beta_{j}I_{\alpha}*|\bar{\phi}_{j}|^{2}$ where
$\bar{\phi}_{j}$ are the normalized eigenfunction corresponding to
$\lambda_{j}(\bar{V})$. In particular,
$\\{\lambda_{j}(\bar{V}),\bar{\phi}_{j}\\}$ is a solution of the Choquard
system (20, 22).

######  Lemma 2.3.

Suppose $\alpha\in(0,2]$, $3\leq n\leq 4+\alpha$ and $V\in
C_{0}^{\infty}(\mathbb{R}^{n})$ is bounded in $\dot{\mathbb{H}}^{\alpha/2}$.
If $\phi$ is a normalized eigenfunction of $a^{-1}L^{W}-V$ then
$\|\nabla\phi\|_{2}$ , $\int W|\phi|^{2}dx$ and $\|\phi\|_{2n/(n-2)}$ are
bounded in terms of $\||V\||_{\alpha/2}$ and the corresponding eigenvalue
$\lambda$.

###### Proof.

By assumption, $\|\phi\|_{2}=1$ and satisfy

$(-\Delta\phi+W)\phi-aV\phi-\lambda\phi=0\ .$

Multiply by $\phi$ and integrate to obtain

$\|\nabla\phi\|_{2}^{2}-a\int V|\phi|^{2}dx+\int W|\phi|^{2}dx-\lambda=0\ .$
(34)

By the critical Sobolev inequality (Lemma 1.1) and and Holder inequality

$\int
V|\phi|^{2}dx\leq\|V\|_{\frac{2n}{n-\alpha}}\|\phi|^{2}\|_{\frac{2n}{n+\alpha}}\leq
S_{n,\alpha}\||V\||_{\alpha/2}^{2}\|\phi|^{2}\|_{\frac{2n}{n+\alpha}}$ (35)

By the Gagliardo-Nirenberg interpolation inequality [14]

$\|\phi\|_{p}\leq C(\theta)\|\nabla\phi\|_{2}^{\theta}\|\phi\|_{2}^{1-\theta}$

where $C(\theta)$ is independent of $\phi$, $p=2n/(n-2\theta)$ whenever
$\theta\in[0,1]$. Since $\|\phi\|_{2}=1$ we get

$\||\phi|^{2}\|_{p/2}=\|\phi\|^{2}_{p}\leq
C^{2}(\theta)\|\nabla\phi\|_{2}^{2\theta}\ .$ (36)

Let now $p/2=\frac{2n}{n+\alpha}$, corresponding to $\theta=(n-\alpha)/4$, we
obtain from (35)

$\int V|\phi|^{2}dx\leq
S_{n,\alpha}C^{2}(\theta)\||V\||_{\alpha/2}^{2}\|\nabla\phi\|_{2}^{2\theta}\ $

where $\theta<1$ if $3\leq n<4+\alpha$. Substitute it in (34) we obtain the
upper estimate on $\|\nabla\phi\|_{2}$ and $\int W|\phi|^{2}dx$. Finally
setting $\theta=1$ corresponding to $p=2n/(n-2)$ we obtain from (36) the
estimate on $\|\phi\|_{2n/(n-2)}$. ∎

######  Lemma 2.4.

If $2<n<4+\alpha$, $0<\alpha\leq 2$ then $C_{0}^{\infty}(\mathbb{R}^{n})\ni
V\mapsto\lambda_{j}^{(V)}$ is continuous on bounded sets in
$\dot{\mathbb{H}}_{\alpha/2}(\mathbb{R}^{n})$ with respect to Lebesgue norms
$\mathbb{L}^{q}(\mathbb{R}^{n})$, where $n/2\leq q\leq\infty$. In particular
$V\mapsto\lambda_{j}^{(V)}$ can be extended as a continuous function on
$\dot{\mathbb{H}}_{\alpha/2}$.

###### Proof.

By Lemma 2.2, there exists $\vec{\phi}^{(V)}\in\mathbb{H}^{k}_{1}$ such that

$G_{\beta,a}(V)=\inf_{\vec{\phi}\in\oplus^{k}\mathbb{H}^{1}}\sum_{j=1}^{k}\beta_{j}\left<(L^{W}-aV)\phi_{j},\phi_{j}\right>\
.$

Thus, for $\tilde{V}_{1},\tilde{V}_{2}$ bounded in
$\dot{\mathbb{H}}^{\alpha/2}$,

$G_{\beta,a}(\tilde{V}_{1})-G_{\beta,a}(\tilde{V}_{2})\leq\sum_{j=1}^{k}\beta_{j}\left<(L^{W}-a\tilde{V}_{1})\phi^{(\tilde{V}_{2})}_{j},\phi^{(\tilde{V}_{2})}_{j}\right>-\sum_{j=1}^{k}\beta_{j}\left<(L^{W}-a\tilde{V}_{2})\phi^{(\tilde{V}_{2})}_{j},\phi^{(\tilde{V}_{2})}_{j}\right>$
$=a\sum_{j=1}^{k}\beta_{j}\left<(\tilde{V}_{2}-\tilde{V}_{1})\phi^{(\tilde{V}_{2})}_{j},\phi^{(\tilde{V}^{2})}_{j}\right>\leq
a\sum_{j=1}^{k}\beta_{j}\|\tilde{V}_{2}-\tilde{V}_{1}\|_{q}\||\phi^{(\tilde{V}_{2})}_{j}|^{2}\|_{\frac{q}{q-1}}\
.$

By Lemma 2.3, $\||\phi_{j}^{(V)}|^{2}\|_{p}$ is bounded in terms of the norm
of $\||V\||_{\alpha/2}$ for $1\leq p\leq n/(n-2)$. It follows that
$G_{\beta,a}(\tilde{V}_{1})-G_{\beta,a}(\tilde{V}_{2})$ is bounded in terms of
$\|\tilde{V}_{2}-\tilde{V}_{1}\|_{q}$ for $n/2\leq q\leq\infty$, so
$G_{\beta,a}$ is continuous in these norms. Since $n/2<\frac{2n}{n-\alpha}$ by
assumption and $\dot{\mathbb{H}}_{\alpha/2}(\mathbb{R}^{n})$ is embedded in
$\mathbb{L}^{2n/(n-\alpha)}(\mathbb{R}^{n})$, we obtain the continuous
extension of $G_{\beta,a}$ on $\dot{\mathbb{H}}_{\alpha/2}(\mathbb{R}^{n})$ .

Finally, to continuity of each eigenvalue $\lambda_{j}$ is obtained by
subtraction
$G_{(\beta_{1},\ldots\beta_{j}),a}(V)-G_{(\beta_{1},\ldots\beta_{j-1}),a}(V)\equiv\beta_{j}\lambda_{j}^{(V)}$
by Lemma 2.2.

∎

###  2.1 Lower limit of the dual functional

Recall the Lieb-Thirring inequality for Schrodinger operator :

######  Theorem 1.

[5] For the Schrödinger operator $-\Delta+V$ on $\mathbb{R}^{n}$ with a real
valued potential $V$ the numbers $\mu_{1}(V)\leq\mu_{2}(V)\leq\dots\leq 0$
denote the (not necessarily finite) sequence of its negative eigenvalues.
Then, for $n\geq 3$ and $\gamma\geq 0$

$\sum_{j;\mu_{j}(V)<0}|\mu_{j}(V)|^{\gamma}\leq L_{\gamma,n}\int
V_{-}^{n/2+\gamma}dx\ $ (37)

where $V_{-}=\max\\{0,-V\\}$ and $L_{\gamma,n}$ is independent of $V$.

######  Proposition 4.

The functional $V\mapsto\frac{a}{2}\left<V,V\right>_{\alpha/2}+G_{\beta,a}(V)$
is bounded from below on $\dot{\mathbb{H}}_{\alpha/2}$ for any $a>0$ if $3\leq
n<2+\alpha$. if $n=3,\alpha=1$ or $n=4,\alpha=2$ there exists
$a=a^{(n)}_{c}(\vec{\beta})>0$ independent of $W$ for which the functional is
bounded from below if $a<a^{(n)}_{c}(\vec{\beta})$ and unbounded if
$a>a^{(n)}_{c}(\vec{\beta})$.

Moreover, in the cases $n=3$ and $n=4,a<a^{(n)}_{c}(\vec{\beta})$ the
functional is coersive on $\dot{\mathbb{H}}_{\alpha/2}(\mathbb{R}^{n})$,
namely

$\lim_{\||V\||_{\alpha/2}\rightarrow\infty}\frac{1}{2}\left<V,V\right>_{\alpha/2}+G_{\beta,a}(V)=\infty\
.$ (38)

###### Proof.

Recall that $\lambda_{j}(V)$ are the eigenvalues of $L^{W}-aV=-\Delta+W-aV$.
Since $W\geq 0$ it follows that $\lambda_{j}(V)\geq\mu_{j}(aV)$. Hence
$G_{\beta,a}(V):=\sum_{j=1}^{k}\beta_{j}\lambda_{j}(V)\geq-\sum_{j;\mu_{j}(aV)<0}\beta_{j}|\mu_{j}(aV)|$.
By Holder inequality, for $\gamma\geq 1$,
$\gamma^{{}^{\prime}}=\gamma/(\gamma-1)$ and (37)

$G_{\beta,a}(V)\geq-\left(\sum_{j=1}^{k}|\beta_{j}|^{\gamma^{{}^{\prime}}}\right)^{1/\gamma^{{}^{\prime}}}\left(\sum_{j;\mu_{j}(aV)<0}|\mu_{j}(aV)|^{\gamma}\right)^{1/\gamma}\geq$
$-a^{1+n/2\gamma}L_{\gamma,n}^{1/\gamma}\|\vec{\beta}\|_{\gamma^{{}^{\prime}}}\left(\int
V_{-}^{n/2+\gamma}dx\right)^{1/\gamma}\ .$

Set now
$\gamma=\frac{2n}{n-\alpha}-n/2\equiv\frac{(4+\alpha)n-n^{2}}{2(n-\alpha)}$.
Then, if $2<n<2+\sqrt{1+3\alpha}$ we get $\gamma^{{}^{\prime}}_{n,\alpha}\geq
1$ and

$G_{\beta,a}(V)\geq-a^{\frac{4}{4+\alpha-n}}L_{\gamma,n}^{1/\gamma}\|\beta\|_{\gamma^{{}^{\prime}}_{n,\alpha}}\left(\int_{\mathbb{R}^{n}}V_{+}^{\frac{2n}{n-\alpha}}\right)^{\frac{2(n-\alpha)}{(4+\alpha)n-n^{2}}}\
.$

Using the critical Sobolev inequality

$G_{\beta,a}(V)\geq-a^{\frac{4}{4+\alpha-n}}L_{\gamma,n}^{1/\gamma}\|\beta\|_{\gamma^{{}^{\prime}}_{n,\alpha}}S_{n,\alpha}^{\frac{4}{(4+\alpha)-n}}\left<V,V\right>_{\alpha/2}^{\frac{2}{(4+\alpha)-n}}$

hence

$\frac{a}{2}\left<V,V\right>_{\alpha/2}+G_{\beta,a}(V)\geq$
$a\left<V,V\right>_{\alpha/2}^{\frac{2}{4+\alpha-n}}\left(\frac{1}{2}\left<V,V\right>_{\alpha/2}^{1-\frac{2}{4+\alpha-n}}-a^{\frac{n-\alpha}{4+\alpha-n}}L_{\gamma,n}^{1/\gamma}\|\beta\|_{\gamma^{{}^{\prime}}_{n,\alpha}}S_{n,\alpha}^{\frac{4}{(4+\alpha)-n}}\right)$
(39)

It follows that ${\cal H}^{W,\alpha}_{\beta,a}$ is coersive for any $a>0$ if
$3\leq n<2+\alpha$. If $n=2+\alpha$ then the functional is coersive if
$a<\frac{S_{n,\alpha}^{\frac{4}{n-(4+\alpha)}}}{2L_{\gamma,n}^{1/\gamma}}|\beta|_{\gamma^{{}^{\prime}}_{n,\alpha}}^{-1}$.
Note that $\gamma^{{}^{\prime}}_{n,\alpha}=\infty$ for $n=3,\alpha=1$ and
$\gamma^{{}^{\prime}}_{n,\alpha}=2$ for $n=4,\alpha=2$. Hence coersivity holds
if

  * •

$(\alpha,n)=(1,3)$:
$a<\frac{S_{3,1}^{-2}}{2L_{\gamma,3}^{1/\gamma}}|\beta|_{\infty}^{-1}$

  * •

$(\alpha,n)=(2,4)$:
$a<\frac{S_{4,12}^{-2}}{2L_{\gamma,4}^{1/\gamma}}|\beta|_{2}^{-1}$

We now prove the existence of a critical strengh $a_{c}(\beta)$ in both cases.
For a given, non-negative $V\in\dot{\mathbb{H}}^{\alpha/2}$, let $k(V)$ be the
number of negative eigenvalues of $-\Delta-aV$, enumerated by order
$\lambda_{1}^{0}(V)<\lambda_{2}^{0}(V)\leq\ldots\lambda_{k(V)}^{0}(V)<0$.
Denote $G^{0}_{\beta,a}(V)=\sum_{1}^{k\wedge
k(V)}\beta_{j}\lambda_{j}^{0}(V)$. Let $\bar{\phi}_{j}^{0}$ be the
corresponding eigenfunctions of $-\Delta-aV$. From the variational
characterization of $G_{\beta,a}$ introduced in Lemma 2.2 we may obtain

$G^{0}_{\beta,a}(V)\leq G_{\beta,a}(V)\leq
G^{0}_{\beta,a}(V)+\sum_{j=1}^{k\wedge k(V)}\beta_{j}\int
W|\phi_{j}^{0}|^{2}+O(1)$ (40)

where $O(1)$ stands for some constant independent of $V$. 666 We may estimate
this constant by $\sum_{k\wedge k(V)+1}^{k}\beta_{j}\lambda^{w}_{j}$ where
$\lambda_{j}^{w}$ is the $j-$ eigenvalue of $H_{0}=-\Delta+W$.

Substitute now now $V/\sqrt{a}$ for $V$. Then
$\frac{a}{2}\left<V/\sqrt{a},V/\sqrt{a}\right>_{\alpha/2}+G^{0}_{\beta,a}(V/\sqrt{a})=\frac{1}{2}\left<V,V\right>_{\alpha/2}+G^{0}_{\beta,a}(V/\sqrt{a})$.
By definition $G^{0}_{\beta,a}(V/\sqrt{a})=\sum_{1}^{k\wedge
k(V/\sqrt{a})}\beta_{j}\lambda_{j}^{0}(V/\sqrt{a})$, while
$\lambda_{j}^{0}(V/\sqrt{a})$ is a negative eigenvalue of
$-\Delta+W+\sqrt{a}V$. Thus, if $V<0$ somewhere then
$\lim_{a\rightarrow\infty}G^{0}_{\beta,a}(V/\sqrt{a})=-\infty$. In particular
it follows that

$\text{if}\ \ a>0\ \ \text{large enough}\ \text{then}\ \exists
V\in\dot{\mathbb{H}}_{\alpha/2}\ \text{for which}\
\frac{a}{2}\left<V,V\right>_{\alpha/2}+G^{0}_{\beta,a}(V)<0$ (41)

Now apply the transformation $V\mapsto V_{\delta}(x):=\delta^{2}V(\delta x)$,
where $\delta>0$. We obtain that
$\lambda^{0}_{j}(V_{\delta})=\delta^{2}\lambda^{0}_{j}(V)$ (in particular,
$k(V_{\delta})=k(V)$), while
$\phi_{j}^{0,\delta}:=\delta^{n/2}\phi_{j}^{0}(\delta x)$ is the corresponding
normalized eigenfunction. Hence
$G^{0}_{\beta,a}(V_{\delta})=\delta^{2}G^{0}_{\beta,a}(V)$ so, by (40),
$G_{\beta,a}(V_{\delta})\leq\delta^{2}G^{0}_{\beta,a}(V)+\sum_{j=1}^{k\wedge
k(V)}\beta_{j}\int W|\phi_{j,\delta}^{0}|^{2}+O(1)$.

Next, we obtain for both $n=3,\alpha=1$ and $n=4,\alpha=2$ cases that the
quadratic form scale the same:
$\left<V_{\delta},V_{\delta}\right>_{\alpha/2}=\delta^{2}\left<V,V\right>_{\alpha/2}$
so

$\frac{a}{2}\left<V_{\delta},V_{\delta}\right>_{\alpha/2}+G_{\beta,a}(V_{\delta})\leq\delta^{2}\left[\frac{a}{2}\left<V,V\right>_{\alpha/2}+G^{0}_{\beta,a}(V)\right]+\sum_{j=1}^{k\wedge
k(V)}\beta_{j}\int W|\phi_{j,\delta}^{0}|^{2}+O(1)\ .$ (42)

By (21) we also get $\lim_{\delta\rightarrow\infty}\int
W|\phi^{0}_{j,\delta}|^{2}=W(0)=0$, so, using (41) we obtain the existence of
$V$ for which
$\frac{a}{2}\left<V_{\delta},V_{\delta}\right>_{\alpha/2}+G_{\beta,a}(V_{\delta})\rightarrow-\infty$
as $\delta\rightarrow\infty$, if $a>0$ is large enough.

Now let

$a_{c}(\vec{\beta})=\inf\left\\{a>0;\ \
\inf_{V\in\dot{\mathbb{H}}^{\alpha/2}}\frac{a}{2}\left<V,V\right>_{\alpha/2}+G^{0}_{\beta,a}(V)<0\right\\}\
.$

It follows that $\infty>a_{c}(\beta)>0$ and is independent of $W$ for any
$\vec{\beta}$ in the cases $n=3,\alpha=1$ and $n=4,\alpha=2$. ∎

###  2.2 Existence of minimizers of the local problem

When attempting to prove the existence of minimizers to the functional ${\cal
H}^{W,\alpha}_{\beta,a}$ (31) we face the problem of compactness of the space
$\dot{\mathbb{H}}_{\alpha/2}$. So, we start by considering the subspace of
$\dot{\mathbb{H}}_{\alpha/2}(B^{n}_{R})\subset\dot{\mathbb{H}}_{\alpha/2}\mathbb{R}^{n})$,
obtained by the closure of $C_{0}^{\infty}$ of functions supported on the ball
$B_{R}^{n}:=\\{x\in\mathbb{R}^{n};\ |x|<R\\}$ under the induced
$\||\cdot\||_{\alpha/2}$ norm (section 1.4).

Note that, by this definition, $V\in\dot{\mathbb{H}}_{\alpha/2}(B^{n}_{R})$ is
defined over the whole of $\mathbb{R}^{n}$, and is identically zero on
$\mathbb{R}^{n}-B^{n}_{R}$.

By

######  Lemma 2.5.

Given $\vec{\beta}$ satisfying (19), $R>0$, $3\leq n<2+\alpha$ or either
$n=3,\alpha=1$ or $n=4,\alpha=2$. Then there exists a minimizer $\bar{V}_{R}$
of ${\cal H}^{W,\alpha}_{\beta,a}$ restricted to
$\dot{\mathbb{H}}_{\alpha/2}(B_{R}^{n})$. Moreover,

$I_{\alpha}^{B_{R}^{n}}(\bar{V}_{R})=\left(\sum_{j=1}^{k}\beta_{j}|\bar{\phi}^{R}_{j}|^{2}\right)$
(43)

where $\bar{\phi}_{j}^{R}$ are the normalized eigenstates of
$L^{W}-a\bar{V}_{R}$ in $\mathbb{R}^{n}$.

###### Proof.

Let $V_{n}\subset\dot{\mathbb{H}}_{\alpha/2}(B^{n}_{R})$ be a minimizing
sequence of ${\cal H}^{W,\alpha}_{\beta,a}$ . Since ${\cal
H}^{W,\alpha}_{\beta,a}$ is bounded from below by Proposition 4 we get

$\lim_{n\rightarrow\infty}{\cal
H}^{W,\alpha}_{\beta,a}(V_{n})=\inf_{V\in\dot{\mathbb{H}}_{\alpha/2}(B^{n}_{R})}{\cal
H}^{W,\alpha}_{\beta,a}(V)\ .$

Since $\dot{\mathbb{H}}_{\alpha/2}(B^{n}_{R})$ is weakly compact and the
functional is coersive (38) there exists a weak limit
$\bar{V}_{R}\in\dot{\mathbb{H}}_{\alpha/2}(B^{n}_{R})$ of this sequence.
Moreover, by Sobolev compact embedding, $V_{n}$ converges strongly to
$\bar{V}_{R}$ in $\mathbb{L}^{q}(B_{B}^{n})$ for any $1\leq q<2n/(n-\alpha)$.
Since $n<2+\alpha$ then $n/2<2n/(n-\alpha)$ and by Lemma 2.4

$\lim_{n\rightarrow\infty}G_{\beta,a}(V_{n})=G_{\beta,a}(\bar{V}_{R})\ .$ (44)

Since $V\mapsto\||V\||_{\alpha/2}^{2}$ is l.s.c , it follows that

$\lim_{n\rightarrow\infty}\left<V_{n},V_{n}\right>_{\alpha/2}dx\geq\left<\bar{V}_{R},\bar{V}_{R}\right>_{\alpha/2}\
.$

This and (44) imply that $\bar{V}_{R}$ is, indeed, a minimizer of ${\cal
H}^{W,\alpha}_{\beta,a}$ on $\dot{\mathbb{H}}_{\alpha/2}(B^{n}_{R})$.

Finally, (43) follows from (26) while taking $\Omega=B_{R}^{n}$. ∎

###  2.3 Proof of Theorem 1.6(ii, iii)

Let $V_{m}$ be a minimizing sequence for ${\cal H}^{W,\alpha}_{\beta,a}$ in
$\dot{\mathbb{H}}_{\alpha/2}(\mathbb{R}^{n})$. Since
$C^{\infty}_{0}(\mathbb{R}^{n})$ is dense in
$\dot{\mathbb{H}}_{\alpha/2}(\mathbb{R}^{n})$ by definition, we can assume
that there exists a sequence $R_{m}\rightarrow\infty$ such that $V_{m}$ is
supported in $B^{n}_{R_{m}}$.

Let $\bar{V}_{m}$ be the minimizers of ${\cal H}^{W,\alpha}_{\beta,a}$ on
$\dot{\mathbb{H}}_{\alpha/2}(B^{n}_{R})$.

Since ${\cal H}^{W,\alpha}_{\beta,a}(\bar{V}_{m})\leq{\cal
H}^{W,\alpha}_{\beta,a}(V_{m})$ then $\bar{V}_{m}$ is a minimizing sequence of
${\cal H}^{W,\alpha}_{\beta,a}$ on
$\dot{\mathbb{H}}_{\alpha/2}(\mathbb{R}^{n})$ as well. Now, under the
conditions of the Theorem we know by Proposition 4 that ${\cal
H}^{W,\alpha}_{\beta,a}$ is bounded from below on
$\dot{\mathbb{H}}_{\alpha/2}(\mathbb{R}^{n})$ and coersive (38), so
$\||\bar{V}_{m}\||_{\alpha/2}$ are uniformly bounded. Let $\bar{\phi}_{j}^{m}$
be the normalized eigenfunctions of $L^{W}-a\bar{V}_{m}$. By Lemma 2.3 we
obtain that $\|\nabla\bar{\phi}^{m}_{j}\|_{2}$ and
$\int_{\mathbb{R}^{n}}W|\bar{\phi}_{j}^{m}|^{2}$ and
$\||\bar{\phi}_{j}^{R}|^{2}\|_{n/(n-2)}$ are uniformly bounded on
$\mathbb{R}^{n}$. In addition, $\|\bar{\phi}_{j}^{m}\|_{2}=1$ by definition.
In particular, $\bar{\phi}_{j}^{m}$ are in the space $\mathbb{H}^{1}$ (c.f
Definition 1.1) . Using the first part of Lemma 2.1 we obtain a subsequence
(denoted by the index $m$ ) along which
$\bar{\rho}_{m}:=\sum_{j=1}^{k}\beta_{j}|\bar{\phi}_{j}^{m}|^{2}$ converges in
$\mathbb{L}^{p}(\mathbb{R}^{n})$ for any $p<n/(n-2)\equiv 2^{*}/2$, while
$\bar{V}_{m}=I^{B^{n}_{R}}_{\alpha}(\bar{\rho}_{m})$. Since $n/2\leq n/(n-2)$
for $n=3,4$, Lemma 1.2 implies the convergence of $\bar{V}_{m}$ to $\bar{V}$
in $\mathbb{L}^{q}(\mathbb{R}^{n})$ for any $1\leq q<\infty$. By lower semi
continuity we obtain that
$\bar{V}\in\dot{\mathbb{H}}_{\alpha/2}(\mathbb{R}^{n})$ and
$\left<\bar{V},\bar{V}\right>_{\alpha/2}\leq\lim_{m\rightarrow\infty}\left<\bar{V}_{m},\bar{V}_{m}\right>_{\alpha/2}$.

In addition, Lemma 2.4 implies that $G_{\beta,a}(\bar{V}_{m})$ converges to
$G_{\beta,a}(\bar{V})$. This implies

${\cal
H}^{W,\alpha}_{\beta,a}(\bar{V})\leq\inf_{V\in\dot{\mathbb{H}}_{\alpha/2}(\mathbb{R}^{n})}{\cal
H}^{W,\alpha}_{\beta,a}(V)$

so $\bar{V}\in\dot{\mathbb{H}}_{\alpha/2}(\mathbb{R}^{n})$ is, indeed, a
minimizer. The proof of Theorem 1.6 follows now from Lemma 1.3 and Corollary
2.3.

##  3 Further remarks

It is interesting to consider the dependence of the solution to the Choquard
system on the probability vector $\vec{\beta}$. In particular, the relation
between the critical interaction strength $a(\beta)$ at dimension $n-4$ and
the universal critical value $\bar{a}_{c}$ corresponding to the scalar case
$k=1$ (see (8)).

a)

Estimate on $\bar{a}_{c}$: In [8] the critical value in case $\alpha=n-2$ is
implicitly given as the $\mathbb{L}^{2}$ norm of the solution of equation (8).
However, these solutions are not known explicitly. Here we introduce an
estimate based on Hardy inequality

$\int_{\mathbb{R}^{n}}|\nabla
f|^{2}\geq\left(\frac{n-2}{2}\right)^{2}\int_{\mathbb{R}^{n}}\frac{|f|^{2}}{|x|^{2}}\
$

for any $f\in C_{0}^{\infty}(\mathbb{R}^{n})$. In particular it implies that
the operator $-\Delta-V$ is non-negative in $\mathbb{R}^{n}$ for any
$V\leq\left(\frac{n-2}{2}\right)^{2}|x|^{-2}$.

As discussed in section 1.1, the functional $E^{W}_{a}$ is bounded from below
on the unit ball of $\mathbb{L}^{2}$ iff $a\leq\bar{a}_{c}$. This implies, in
particular, that if $A>\bar{a}_{c}$ there exists
$\tilde{\phi}\in\mathbb{H}^{1}$ for which

$E^{0}_{a}(\tilde{\phi}):=\frac{1}{2}\int_{\mathbb{R}^{n}}|\nabla\tilde{\phi}|^{2}-\frac{A}{4}\int_{\mathbb{R}^{n}}\left(I_{n-2}*|\tilde{\phi}|^{2}\right)|\tilde{\phi}|^{2}<0\
.$ (45)

Moreover, by Riesz’s rearrangement theorem we can assume that this
$\tilde{\phi}$ is radially symmetric.

In particular, for any $V\geq I_{n-2}*|\tilde{\phi}|^{2}$

$-\Delta-(A/2)V\not\geq 0\ .$ (46)

In the special case $n=4$, $I_{2}=(-\Delta)^{-1}$ is the fundamental solution
of the Laplacian. Let $\rho:=|\tilde{\phi}|^{2}$ be this radial function. Then
$U:=I_{2}*\rho$ is a solution of $\Delta U+\rho=0$. Thus

$r^{-3}\left(r^{3}U^{{}^{\prime}}\right)^{{}^{\prime}}+\rho(r)=0\ .$ (47)

Let $m(r)=2\pi^{2}\int_{0}^{r}s^{3}\rho(s)ds$. In particular, $m(\cdot)$ is
non-decreasing on $\mathbb{R}_{+}$, $m(0)=0$, and $m(r)\leq 1$ by assumption.
Integrating (47) we get

$r^{3}U^{{}^{\prime}}(r)=-(2\pi^{2})^{-1}m(r)\Longrightarrow
U(r)=(2\pi)^{2}\int_{r}^{\infty}\frac{m(s)}{s^{3}}ds\leq 2\pi^{2}r^{-2}.$

Thus, taking $V=2\pi^{2}r^{-2}$ in (46) we obtain a violation of the Hardy
inequality if $\pi^{2}A$ is below the Hardy constant. Since the Hardy constant
$\left(\frac{n-2}{2}\right)^{2}=1$ for $n=4$ we get $A>\pi^{-2}$ for any
$A>\bar{a}_{c}$, that is

$\bar{a}_{c}\geq\frac{1}{\pi^{2}}$

if $n=4$.

It is not clear, at this point, if the above estimate holds for general
dimension, since $I_{n-2}=(-\Delta)^{-1}$ only if $n=4$. There is, indeed, an
estimate of the form

$|x;I_{\alpha}*\rho(x)|>t|\leq
c\left(\frac{c}{t}\|\rho\|_{2}\right)^{n/(n-\alpha)}$

(c.f [16], eq. (2.12)) which, if $\rho$ is radial, is equivalent to

$I_{\alpha}*\rho(r)\leq
c^{\alpha}\omega_{n}^{(n-\alpha)/n}\|\rho\|_{1}r^{\alpha-n}\ $

where $\omega_{n}$ is the surface area of the unit sphere $\mathbb{S}^{n-1}$.
This suggests a similar estimate for $\bar{a}_{c}$ in for general $n$ and
$\alpha=n-2$ using Hardy inequality. However, there is now known estimate (as
far as we know) for the constant $c$.

b)

Relation between $\bar{a}_{c}$ and $a_{\beta}$: The inequality
$a(\beta)\geq\bar{a}_{c}$ can be easily obtained for the critical case for any
$\alpha=n-2$, $n\geq 3$, and any $\vec{\beta}$ satisfying (10). Indeed, using
Definition 1.1 and the polar inequality

$\left<|\phi_{j}|^{2},I_{\alpha}*|\phi_{i}|^{2}\right>\leq\frac{1}{2}\left[\left<|\phi_{j}|^{2},I_{\alpha}*|\phi_{j}|^{2}\right>+\left<|\phi_{i}|^{2},I_{\alpha}*|\phi_{i}|^{2}\right>\right]$

we obtain

${\cal
E}^{(\alpha)}_{\beta,a}(\vec{\phi})\geq\frac{1}{2}\sum_{j=1}^{k}\beta_{j}\left[\left<\left<\phi_{j},\phi_{j}\right>\right>_{W}-\frac{a}{2}\left<|\phi_{j}|^{2},I_{\alpha}*|\phi_{j}|^{2}\right>\right]=\sum_{j=1}^{k}\beta_{j}E^{W}_{a}(\phi_{j})$

where $E^{W}_{a}$ as defined in (7). It follows that ${\cal
E}^{(\alpha)}_{\beta,a}$ is bounded on $\oplus^{k}\mathbb{H}^{1}$ if
$E^{W}_{a}$ is bounded on $\mathbb{H}^{1}$. Since $E^{W}_{a}$ is bounded from
below iff $a\leq\bar{a}_{c}$ ([8]), the inequality $a(\beta)\geq\bar{a}_{c}$
follows.

In the case $n=3,\alpha=1$ and $n=4,\alpha=2$ we can say more about
$a_{c}(\beta)$. By definition, $a>a_{c}(\beta)$ iff ${\cal
H}^{W,\alpha}_{\beta,a}$ is unbounded from below on
$\dot{\mathbb{H}}_{\alpha/2}$. Using (39) we obtain that
$a_{c}(\beta)>O(|\beta|_{\infty}^{-1})$ for $n=3$ and
$a_{c}(\beta)>O(|\beta|^{-1}_{2})$ for $n=4$.

For an interesting conclusion from the above estimate, let $\vec{\beta}$ be
the uniform vector $\vec{\beta}={\bf 1}_{k}:=k^{-1}(1,\ldots
1)\in\mathbb{R}^{k}$. Then $|\vec{\beta}|_{2}=k^{-1/2}$ (resp.
$|\vec{\beta}|_{\infty}=k^{-1}$) so

$n=4\Rightarrow a_{c}({\bf 1}_{k})\geq O(k^{1/2})\ \ \ \text{resp.}\
n=3\Rightarrow\ (a_{c}({\bf 1}_{k})\geq O(k)$

for large $k$.

[c] The following alternative definitions of $I_{\alpha}$ and
$(-\Delta)^{\alpha/2}$ is known [16, 4]:

$I_{\alpha}=\frac{1}{\Gamma(\alpha)}\int_{0}^{\infty}t^{\alpha/2-1}e^{t\Delta}dt\
\ \ \ ;\ \ \
(-\Delta)^{\alpha/2}=\frac{1}{\Gamma(-\alpha)}\int_{0}^{\infty}t^{-\alpha/2-1}\left(e^{t\Delta}-I\right)dt$

where $e^{t\Delta}$ is the heat kernel on $\mathbb{R}^{n}$:

$e^{t\Delta}=(4\pi t)^{-n/2}e^{-\frac{|x|^{2}}{4t}}\ .$

We may, at least formally, substitute the kernel $e^{\Delta_{\Omega}t}$ of the
killing, Dirichlet problem for the heat flow in a domain
$\Omega\subset\mathbb{R}^{n}$ in the above expression, and obtain (again, at
least formally…) an explicit expressions for $I_{\alpha}^{\Omega}$ and
$(-\Delta_{\Omega})^{\alpha/2}$, introduced implicitly in (26). Such a
representation can provide some insight on the trace of $I^{\Omega}_{\alpha}$
for $\alpha<2$.

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