Dual formulation of constrained solutions of the multi-state Choquard equation

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 induced 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

\[ -\Delta \phi + \phi -\left(\int\frac{ |\phi(y)|^2}{|x-y|} \,{\rm d}y\right)\phi(x)=0 \]

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

(1.1)\begin{equation} -\Delta \phi +\phi=\left(I_\alpha * |\phi|^p\right) |\phi|^{p-2} \phi \end{equation}

where

(1.2)\begin{equation} I_\alpha=A(\alpha) |x|^{\alpha-n} ; \quad A(\alpha):= \frac{ \Gamma\left( \frac{n-\alpha}{2}\right)}{2^\alpha\pi^{n/2} \Gamma\left(\alpha/2)\right) } \end{equation}

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

(1.3)\begin{equation} 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) \end{equation}

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 [6, 7, 10]. For existence, regularity and asymptotic behaviour 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

(1.4)\begin{equation} -i\partial_t\psi -\Delta\psi -a (I_\alpha*|\psi|^p)|\psi|^{p-2}\psi=0 \ . \end{equation}

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

(1.5)\begin{equation} -\Delta \phi -a\left(I_\alpha * |\phi|^p\right) |\phi|^{p-2} \phi-\lambda\phi =0 \end{equation}

which can be reduced to (1.1) by a proper scaling¹ . However, the solutions of the nonlinear equation (1.4) preserve the $\mathbb {L}^2$ norm, so it is natural to look for stationary solutions (1.5) 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 (1.5) 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 (1.5) 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 (1.5) takes the form

(1.6)\begin{equation} -\Delta \phi +W \phi -a \left(\int_{\mathbb{R}^n} \frac{|\phi(y)|^2}{|x-y|^{n-\alpha}} \,{\rm d}y\right) \phi-\lambda \phi=0, \ \ \ \|\phi\|_2=1 \ . \end{equation}

A solution of (1.6) is given by a minimizer of the functional

(1.7)\begin{equation} E^W_a(\phi):=\frac{1}{2} \int_{\mathbb{R}^n} \left(|\nabla \phi|^2 + W|\phi|^2\right)\,{\rm d}x - \frac{a}{4} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n} \frac{|\phi(x)|^2|\phi(y)|^2}{|x-y|^{n-\alpha}}{\rm d}x{\rm d}y \end{equation}

restricted to the $\mathbb {L}^2$ unit ball $\|\phi \|_2=1$.

In [8] the authors studied the equation (1.6) 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 (1.6) (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

(1.8)\begin{equation} -\Delta \bar\phi - \left(\int_{\mathbb{R}^n} \frac{|\bar\phi(y)|^2}{|x-y|^2} \,{\rm d}y\right) \bar\phi+\bar\phi=0. \end{equation}

The object of the present paper is two-fold.

The first object is to extend the $\mathbb {L}^2$-constraint Choquard equation (1.6) into a $k-$ state system

(1.9)\begin{align} & -\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}} \,{\rm d}y\right) \phi_j -\lambda_j\phi_j\nonumber\\ & \quad =0\ \ \|\phi_j\|_2=1, \ \ ; \ \ j=1\ldots k \end{align}

where $(\phi _1,\, \ldots \phi _k)$ constitutes an orthonormal $k-$sequence in $\mathbb {L}^2(\mathbb {R}^n)$ and

(1.10)\begin{equation} \beta_j>0, \ \ \ \ \sum_1^k \beta_j=1 \end{equation}

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

In § 1.2 we introduce the time dependent Heisenberg system which leads naturally to (1.9), while the steady state (1.9) and its constraint variational formulation are introduced in § 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 $E^W_a$ (1.7) for $\alpha =2$ on the constraint $\mathbb {L}^2$ sphere takes the form of the functional $V\mapsto \mathcal {H}^{W,\alpha }_a(V)$

\[ {\mathcal{H}}^{W,2}_a(V)= \frac{a}{2}\int_{\mathbb{R}^n} |\nabla V|^2 + \lambda_1(V) \]

over the unconstrained Beppo-Levi space $V\in \dot{\mathbb {H}}_1(\mathbb {R}^n)$ (c.f. §§ 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 (1.9) for $\alpha \in (0,\,2]$ is introduced in (1.28). In case $\alpha =2$ it takes the form

\[ {\mathcal{H}}^{W,2}_{\beta,a}(V)= \frac{a}{2}\int_{\mathbb{R}^n} |\nabla V|^2 + \sum_{j=1}^k\beta_j\lambda_j(V) \]

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 ( § 1.6):

Using the dual variational formulation we show the existence of a minimizer of ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ corresponding to a solution of (1.9) 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 ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ if $a< a^{(n)}_c(\beta )$ corresponding to a solution of (1.9), while ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ 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

(1.11)\begin{equation} i\frac{\partial R}{\partial t} =\left[ L^W -aV,R\right], \quad t\in \mathbb{R} \end{equation}

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 ${\rm e}^{itL^W}$ on $\mathbb {H}$ and $V$ is a non-linear operator.

In the context of mean-field system we consider $(\mathbb {H},\, \left \langle \cdot,\, \cdot \right \rangle )$ to be the Hilbert space $\mathbb {L}^2(\mathbb {R}^n)$ where $\left \langle \phi,\,\psi \right \rangle :=\int _{\mathbb {R}^n} \phi \bar \psi$ the canonical inner product. A density operator can be represented by a kernel $K_R$ acting on $\phi \in \mathbb {H}$ via $R(\phi )=\int _{\mathbb {R}^n}K_R(x,\,y)\phi (y)\,{\rm d}y$ 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

(1.12)\begin{equation} V(R):= I_\alpha *Tr(R ) \end{equation}

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

(1.13)\begin{equation} R({\cdot},t)= \exp\left({-}i\int_0^t (L^W-aV({\cdot}, s)){\rm d}s\right) R({\cdot}, 0) \exp\left(i\int_0^t (L^W-aV({\cdot}, s)){\rm d}s\right). \end{equation}

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

(1.14)\begin{equation} R(x,y,t)= \sum_1^k \beta_j\psi_j(x,t)\bar\psi_j(y,t) \end{equation}

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 (1.11) takes the form

(1.15)\begin{equation} i\frac{\partial \psi_j}{\partial t} = (L^W -aV)\psi_j\quad , j=1,2, \ldots k \end{equation}

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

(1.16)\begin{equation} Tr(R)(x,t)= \sum_{j=1}^k \beta_j|\psi_j(x,t)|^2, \end{equation}

and the potential $V$ is determined in terms of the solution $R$ by (1.12)

\[ V= \sum_{j=1}^k \beta_j I_\alpha* |\psi_j|^2 \ . \]

Consider now the Hamiltonian

\[ {\mathcal{E}}^{(\alpha)}_{\beta,a}(\vec{\psi}) := \frac{1}{2}\sum_{j=1}^k\beta_j \left[\left\langle L^W\psi_j, \psi_j\right\rangle - \frac{a}{2}\sum_{i=1}^k \beta_i \left\langle |\psi_j|^2, I_\alpha*|\psi_i|^2\right\rangle\right] \]

acting on $k-$ orthonormal frames $\vec {\psi }=(\psi _1,\, \ldots \psi _k)$. The system (1.11) (equivalently (1.15) ) is, in fact, an Hamiltonian system in the canonical variables $\{\psi _i,\, \bar {\psi }_j\}$:

(1.17)\begin{equation} i\frac{\partial \psi_j}{\partial t} ={-}\frac{1}{\beta_j} \delta_{\bar{\psi}_j}{\mathcal{E}}^{(\alpha)}_{\beta,a} ; \ \ \ i\frac{\partial \bar\psi_j}{\partial t} = \frac{1}{\beta_j} \delta_{\psi_j}{\mathcal{E}}^{(\alpha)}_{\beta,a}. \end{equation}

In particular, ${\mathcal {E}}^{(\alpha )}_{\beta,a}$ is constant along the solution of (1.11).

1.3. Steady states

The steady states of this system are given by $\psi _j={\rm e}^{-i\lambda _j}\phi _j$ where $\{\phi _j\}$ is an orthonormal sequence corresponding to eigenvalues $\lambda _j$ of the operator $L^W-aV$, satisfying

(1.18)\begin{equation} L^W\phi_j -a I_\alpha*\left(\sum_{i=1}^k \beta_i|\phi_i|^2\right)\phi_j -\lambda_j\phi_j =0. \end{equation}

Definition 1.1

\begin{align*} \mathbb{H}^1& :=\{ \phi\in \mathbb{L}^2(\mathbb{R}^n), \ ; \ \nabla\phi\in \mathbb{L}^2(\mathbb{R}^n); \ \|\phi\|_2=1, \quad \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\langle \phi_j, \phi_i\right\rangle=\delta_i^j , \quad i,j\in\{1, \ldots k\}\right\}. \end{align*}

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

\[ \left\langle \left\langle \phi,\phi\right\rangle\right\rangle_W:=\int_{\mathbb{R}^n} |\nabla\phi|^2 +W|\phi|^2. \]

Let

\[ {\mathcal{E}}^{(\alpha)}_{\beta,a}(\vec\phi):= \frac{1}{2} \sum_1^k\beta_j\left[ \left\langle \left\langle \phi_j,\phi_j\right\rangle\right\rangle_W -\frac{a }{2} \sum_{i=1}^k\beta_i\left\langle |\phi_j|^2, I_\alpha* |\phi_i|^2\right\rangle\right] \]

is defined over $\oplus ^k\mathbb {H}^1$ (c.f. Corollary 2.4 below).

We formally obtain from (1.17) that the steady states (1.18) are critical points of ${\mathcal {E}}^{(\alpha )}_{\beta,a}$ subject to the orthogonality constraints.

Proposition 1.2 Suppose $\beta _j\not =\beta _i$ for any $1\leq i\not = j\leq k$. Then any critical point of ${\mathcal {E}}^{(\alpha )}_{\beta,a}$ 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 $\bar V=I_\alpha *( \sum _1^k \beta _j|\bar \phi _j|^2)$.

For the proof of proposition 1.2 see the beginning of §2.

From now on we assume

(1.19)\begin{equation} \beta_1>\beta_2>\ldots > \beta_k> 0. \end{equation}

Formulation of the problem

: Consider the multi-state Choquard system satisfying the equivalent of (1.18):

(1.20)\begin{equation} (L^W -aV)\phi_j -\lambda_j\phi_j=0\ \ ; j=1, \ldots\ k \end{equation}

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

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

   (1.21)\begin{equation} W\in \mathbb{L}^\infty_{loc}(\mathbb{R}^n), \quad \lim_{|x|\rightarrow\infty} W(x)=\infty, \quad \inf_{x\in\mathbb{R}^n}W(x)=W(0)=0 \end{equation}

2. 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.

3. iii)

   (1.22)\begin{equation} V=\sum_{i=1}^k\beta_iI_\alpha *|\phi_i|^2 \end{equation}
   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

\[ \left\langle V_1, V_2\right\rangle_{\alpha/2}:= A(-\alpha)\int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{(V_1(x)-V_1(y)) (V_2(x)-V_2(y))}{|x-y|^{n+\alpha}}{\rm d}x{\rm d}y \]

where the constant $A(-\alpha )$ is defined as in (1.2). If $\alpha =2$

\[ \left\langle V_1, V_2\right\rangle_{(1)}:=\int_{\mathbb{R}^n}\nabla V_1\cdot\nabla V_2 \,{\rm d}x \ . \]

The closure of $C_0^\infty (\mathbb {R}^n)$ with respect to the norm induced by the inner product $\left \langle \cdot,\, \cdot \right \rangle _{\alpha /2}$ is denoted by $\dot{\mathbb {H}}^{\alpha /2}$. We denote the associated norm by $\||\cdot \||_{\alpha /2}$.² Recall that $\dot{\mathbb {H}}^{\alpha /2}$ is a Hilbert space so, in particular, is weakly locally compact.

Lemma 1.3 [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

\[ \left\langle V, \phi\right\rangle_{\alpha/2}=\left\langle (-\Delta)^{\alpha/2} V, \phi\right\rangle\ \forall\ \ \phi\in C_0^\infty(\mathbb{R}^n). \]

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

\[ (- \Delta)^{\alpha/2} V(x)= A(-\alpha) \int_{\mathbb{R}^n} \frac{V(x+y)-V(x)}{|y|^{n+\alpha} }\,{\rm d}y\ . \]

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 $\left \langle \cdot,\, \cdot \right \rangle _{\alpha /2}$ inner product:

(1.23)\begin{equation} \frac{1}{2}\left\langle I_\alpha*\rho,\rho\right\rangle:= \sup_{V\in C_0^\infty(\mathbb{R}^n)} \left\langle \rho, V\right\rangle-\frac{1}{2}\left\langle V, V\right\rangle_{\alpha/2}. \end{equation}

The Euler-Lagrange equation corresponding to the right hand side of (1.23) takes the form

(1.24)\begin{equation} (-\Delta)^{\alpha/2} V=\rho . \end{equation}

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

(1.25)\begin{equation} I_\alpha*(-\Delta)^{\alpha/2}V = V\ . \end{equation}

The pointwise representation of the kernel $I_\alpha$ is given by (1.2). Moreover

Lemma 1.4 [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

(1.26)\begin{equation} \frac{1}{2}\left\langle I^\Omega_\alpha(\rho),\rho\right\rangle:= \sup_{\phi\in C_0^\infty(\Omega)} \left\langle \rho, \phi\right\rangle-\frac{1}{2}\left\langle \phi, \phi\right\rangle_{\alpha/2}. \end{equation}

In the case $\alpha =2$ this definition induces the Green function of the Dirichlet problem $I_2^\Omega \equiv (-\Delta _\Omega )^{-1}$, that is, $V(x)= \int _\Omega I_2^\Omega (x,\,y)\rho (y)\,{\rm d}y$ is the solution of the Poisson problem

(1.27)\begin{equation} \Delta V+\rho=0\quad x\in \Omega; \ \ V=0 \ \ \text{on} \ \ \partial\Omega\ . \end{equation}

Not much is known³ 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 (1.26) :

Lemma 1.5 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 (1.21) implies, in particular

Proposition 1.6 Let $V\in C_0^\infty (\mathbb {R}^n)$ and $W$ satisfies (1.21). 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 1.7 see Lemma 2.7 in § 2.

Proposition 1.7 If $3\leq n\leq 5$, $0<\alpha \leq 2$, then proposition 1.6 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 1.7 allows us to define the dual functional on $\dot{\mathbb {H}}^{\alpha /2}(\mathbb {R}^n)$:

(1.28)\begin{equation} {\mathcal{H}}^{W,\alpha}_{\beta,a}(V):= \frac{a}{2}\left\langle V,V\right\rangle_{\alpha/2} + \sum_{j=1}^k \beta_j\lambda_j(V) \end{equation}

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 $\lambda _j(V)$⁴

1. [i] $V$ is a minimizer of ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ on $\dot{\mathbb {H}}_{\alpha /2}(\mathbb {R}^n)$ if and only if $(\phi _1(V),\, \ldots \phi _k(V))$ is a minimizer of ${\mathcal {E}}^{(\alpha )}_{\beta,a}$ 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 (1.20) .

2. [ii] If $\alpha \in (0,\,2]$ $3\leq n<2+\alpha$ then ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ is bounded from below on $\dot{\mathbb {H}}_{\alpha /2}(\mathbb {R}^n)$ for any $\beta$ satisfying (1.19) and any $a>0$, and there is a minimizer of ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ in $\dot{\mathbb {H}}_{\alpha /2}(\mathbb {R}^n)$.

3. [iii] If $\alpha =1,\, n=3$ or $\alpha =2,\, n=4$ then there exists $a_c(\beta )>0$ such that ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ 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 ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ in $\dot{\mathbb {H}}_{\alpha /2}(\mathbb {R}^n)$.

2. Proofs

We start by proving proposition 1.2

Proof. Let $\gamma _{i,j}$ be the Lagrange multiplier for the constraints $\left \langle \phi _i,\,\phi _j\right \rangle =\delta _{i,j}$. Then $\overrightarrow {\phi }$ is an unconstraint critical point of

\[ {\mathcal{E}}^{(\alpha)}_{\beta,a}(\vec{\phi}) + \sum_1^k\gamma_{j,j}\|\phi_j\|_{\mathbb{H}^1}^2 + \sum_{i\not= j} \gamma_{i,j}\left\langle \phi_i, \phi_j\right\rangle\ . \]

This implies

\[ \frac{\delta {\mathcal{E}}^{(\alpha)}_{\beta,a}}{\delta\bar\phi_j }+ 2\gamma_{j,j}\bar\phi_j + \sum_{i\not= j }\gamma_{i,j} \bar\phi_i= \beta_j (L^W-a\bar V)\bar\phi_j + 2\gamma_{j,j}\bar\phi_j + \sum_{i\not= j }\gamma_{i,j} \bar\phi_i= 0 \ . \]

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:

\[ \left\langle \beta_j\left[L^W-a\bar V\right]\bar\phi_j, \bar\phi_i\right\rangle +\gamma_{i,j}=0 \]

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

\[ \left\langle \beta_i\left[L^W-a\bar V\right]\bar\phi_j, \bar\phi_i\right\rangle +\gamma_{i,j}=0 \]

Subtracting the two inequalities we obtain $\left \langle (\beta _j-\beta _i)(L^W-a\bar V)\bar \phi _j,\, \bar \phi _i\right \rangle =0$ thus $\left \langle (L^W-a\bar V)\bar \phi _j,\, \bar \phi _i\right \rangle =0$ 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 $2< r< {2n}/{n-2}$. If ${n-2}/{n+\alpha } \leq {1}/{2} \leq {n}/{n+\alpha }$ 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 2.2 If $\max (0,\, n-4)\leq \alpha \leq n$ then the functional ${\mathcal {E}}^{(\alpha )}_{\beta,a}$ 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

(2.1)\begin{equation} {\bf H}^{(\alpha)}_\beta(\vec\phi, V)= \sum_1^k\beta_j \left\langle L^W\phi_j, \phi_j\right\rangle +a\left[ \left\langle V, V\right\rangle_{\alpha/2} -\left\langle V, \sum\beta_j|\phi_j|^2\right\rangle \right]\ . \end{equation}

By (1.23) we get (c.f definition 1.1)

\[ \inf_{V\in C_0^\infty(\mathbb{R}^n)} {\bf H}^{(\alpha)}_\beta(\vec\phi,V)=2 {\mathcal{E}}^{(\alpha)}_{\beta,a}(\vec\phi) \ . \]

Thus

\[ \inf_{V\in C_0^\infty} \inf_{\vec\phi\in{\oplus}^k\mathbb{H}^1} {\bf H}^{(\alpha)}_\beta(\vec\phi,V)\equiv \inf_{\vec\phi\in{\oplus}^k\mathbb{H}^1}\inf_{V\in C_0^\infty} {\bf H}^{(\alpha)}_\beta(\vec\phi,V)= \inf_{\vec\phi\in{\oplus}^k\mathbb{H}^1}{\mathcal{E}}^{(\alpha)}_{\beta,a}(\vec\phi) \ . \]

Let

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

From (1.23 , 2.1) we observe that

\[ {\mathcal{H}}^{W,\alpha}_{\beta,a}(V)=\frac{a}{2}\left\langle V,V\right\rangle_{\alpha/2} + \inf_{\vec{\phi}\in{\oplus}^k\mathbb{H}^1}\sum_{j=1}^k \beta_j \left\langle (L^W-aV)\phi_j, \phi_j\right\rangle\ . \]

Let

(2.2)\begin{equation} \inf_{\vec{\phi}\in{\oplus}^k\mathbb{H}^1}\sum_{j=1}^k \beta_j \left\langle (L^W-aV)\phi_j, \phi_j\right\rangle := G_{\beta, a}(V)\ . \end{equation}

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

(2.3)\begin{equation} {\mathcal{H}}^{W,\alpha}_{\beta,a}(V)=\frac{a}{2}\left\langle V,V\right\rangle_{\alpha/2}+G_{\beta, a}(V) \end{equation}

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

\[ G_{1,a}(V)= \inf_{\|\phi\|=1}\left\langle (L^W-aV)\phi,\phi\right\rangle= \lambda_1(V) \]

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 (1.19) then we claim that $V\mapsto \sum _{j=1}^k \beta _j \lambda _j(V)$ is concave. Indeed:

Lemma 2.3

(2.4)\begin{equation} G_{\beta,a}(V)=\sum_{j=1}^k \beta_j \lambda_j(V) \end{equation}

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 (2.2) 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 $V$:

\[ \partial_V G:= \left\{ \zeta\in (C_0^\infty)^{'}; \ \ G(Z)\leq G(V)+ \left\langle Z-V, \zeta\right\rangle\ \forall Z\in C_0^\infty\right\} \]

while $G$ is differentiable at $V$ if $\partial _V G$ is a singleton.

Corollary 2.4 The sup-gradient of the functional $G_{\beta, a}$ on $C_0^\infty (\mathbb {R}^n)$ is contained in $\mathbb {L}^1$. In fact $\partial _VG_{\beta, a}=a \sum _{j=1}^k \beta _j|\bar \phi _j|^2$ where $\bar \phi _j\in \mathbb {L}^2$ is a normalized eigenstate of $L^W-aV$ corresponding to the $j-$ eigenvalue. So, in particular, $\|\partial _VG_{\beta, a}\|_1=a$. If all eigenvalues of $L^W-aV$ are simple then $G_{\beta,a}$ is differentiable at $V$.

Proof. Since $V\mapsto \sum _{j=1}^k \beta _j \left \langle (L^W-aV)\phi _j,\, \phi _j\right \rangle$ is a linear functional, (2.2) 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 (2.2) to $\mathbb {H}^k_m:=\{ \vec \phi := (\phi _1,\, \ldots \phi _k),\, \phi _j\in \mathbb {H}_m\}\subset \mathbb {H}^k$.

Then

\[ \phi_j= \sum_{i=1}^m\left\langle \phi_j, \bar\phi_i\right\rangle\bar\phi_i , \quad (L^W-aV)\phi_j= \sum_{i=1}^m\lambda_i\left\langle \phi_j, \bar\phi_i\right\rangle\bar\phi_i. \]

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

(2.5)\begin{equation} \sum_{j=1}^k \beta_j \left\langle (L^W-aV)\phi_j, \phi_j\right\rangle =\sum_{i=1}^m\sum_{j=1}^m \beta_j\lambda_i|\left\langle \phi_j, \bar\phi_i\right\rangle|^2. \end{equation}

Denote now $\gamma _{i,j}:= |\left \langle \phi _j,\, \bar \phi _i\right \rangle |^2$. 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(2.5)

\[ \forall \vec\phi\in \mathbb{H}^k_m, \quad \sum_{j=1}^k \beta_j \left\langle (L^W-aV)\phi_j, \phi_j\right\rangle\geq \sum_{j=1}^m \beta_{\pi(j)} \lambda_j \]

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 obtained at the identity permutation $\pi (i)=i$, that is, at the identity matrix $\tilde \gamma _{i,j}:=\left \langle \phi _j,\,\bar \phi _i\right \rangle =\delta _{i,j}$. This implies that the eigenbasis $\bar \phi _1,\, \ldots \bar \phi _k$ of the $k$ leading eigenvalues is the minimizer of (2.2) on $\mathbb {H}_m^k$ for any $m\geq k$.

In particular, the minimizer of (2.2) 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 (2.2) is strictly smaller than its value on the first $k-$ leading eigenspace. Since the eigenstates of the Schrödinger operator under assumption (1.21) 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 (2.2) 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 2.4 and (1.24, 1.25) It follows that the Euler-Lagrange equation corresponding to ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ is

\[ (-\Delta)^{\alpha/2} V - \sum_{j=1}^k\beta_j|\phi_j|^2=0 \quad \Longleftrightarrow V=I_\alpha*\left( \sum_{j=1}^k\beta_j|\phi_j|^2 \right) \]

where $\phi _j$ are the normalized eigenfunction corresponding to $\lambda _j( V)$. In particular we obtain the proof of theorem 1.6-(i):

Corollary 2.5 If $\bar V$ is a minimizer of ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ 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 (1.20, 1.22).

Lemma 2.6 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 $\|\nabla \phi \|_2$ , $\int W|\phi |^2\,{\rm d}x$ 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

(2.6)\begin{equation} \|\nabla \phi\|_2^2 -a\int V|\phi|^2 \,{\rm d}x+\int W|\phi|^2\,{\rm d}x - \lambda=0. \end{equation}

By the critical Sobolev inequality (lemma 1.3) and and Holder inequality

(2.7)\begin{equation} \int V|\phi|^2\,{\rm d}x \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}} \end{equation}

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

(2.8)\begin{equation} \||\phi|^2\|_{p/2}= \|\phi\|^2_p\leq C^2(\theta)\| \nabla \phi\|_2^{2\theta}. \end{equation}

Let now $p/2={2n}/{n+\alpha }$, corresponding to $\theta =(n-\alpha )/4$, we obtain from (2.7)

\[ \int V|\phi|^2\,{\rm d}x \leq S_{n,\alpha} C^2(\theta) \||V\||_{\alpha/2}^2 \|\nabla \phi\|_2^{2\theta} \]

where $\theta <1$ if $3\leq n< 4+\alpha$. Substitute it in (2.6) we obtain the upper estimate on $\|\nabla \phi \|_2$ and $\int W|\phi |^2\,{\rm d}x$. Finally setting $\theta =1$ corresponding to $p=2n/(n-2)$ we obtain from (2.8) the estimate on $\|\phi \|_{2n/(n-2)}$.

Lemma 2.7 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 2.3, there exists $\vec {\phi }^{(V)}\in \mathbb {H}^k_1$ such that

\[ G_{\beta, a}(V) = \inf_{\vec{\phi}\in{\oplus}^k\mathbb{H}^1}\sum_{j=1}^k \beta_j \left\langle (L^W-aV)\phi_j, \phi_j\right\rangle\ . \]

Thus, for $\tilde V_1,\, \tilde V_2$ bounded in $\dot{\mathbb {H}}^{\alpha /2}$,

\begin{align*} G_{\beta,a}(\tilde V_1)-G_{\beta,a}(\tilde V_2)& \leq \sum_{j=1}^k \beta_j \left\langle (L^W-a\tilde V_1)\phi^{(\tilde V_2)}_j, \phi^{(\tilde V_2)}_j\right\rangle\\ & \quad - \sum_{j=1}^k \beta_j \left\langle (L^W-a\tilde V_2)\phi^{(\tilde V_2)}_j, \phi^{(\tilde V_2)}_j\right\rangle\\ & = a \sum_{j=1}^k \beta_j \left\langle (\tilde V_2-\tilde V_1)\phi^{(\tilde V_2)}_j, \phi^{(\tilde V^2)}_j\right\rangle\\ & \quad\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}} \ . \end{align*}

By lemma 2.6, $\||\phi _j^{(V)}|^2\|_p$ 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 $n/2< {2n}/{n-\alpha }$ 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 2.3.

2.1. Lower limit of the dual functional

Recall the Lieb-Thirring inequality for Schrodinger operator :

Theorem 2.8 [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 $\gamma \geq 0$

(2.9)\begin{equation} \sum_{j; \mu_j(V)<0} |\mu_j(V)|^\gamma\leq L_{\gamma,n}\int V_-^{n/2+\gamma} \,{\rm d}x \end{equation}

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

Proposition 2.9 The functional $V \mapsto {a}/{2}\left \langle V,\,V\right \rangle _{\alpha /2} + G_{\beta, a}(V)$ 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

(2.10)\begin{equation} \lim_{\||V\||_{\alpha/2}\rightarrow\infty} \frac{1}{2}\left\langle V,V\right\rangle_{\alpha/2} + G_{\beta, a}(V)=\infty. \end{equation}

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 $\gamma \geq 1$, $\gamma ^{'}=\gamma /(\gamma -1)$ and (2.9)

\[ G_{\beta,a}(V) \geq{-}\left( \sum_{j=1}^k |\beta_j|^{\gamma^{'}}\right)^{1/\gamma^{'}} \left( \sum_{j; \mu_j(aV)<0}| \mu_j(aV)|^\gamma\right)^{1/\gamma} \geq \]

\[{-}a^{1+n/2\gamma}L_{\gamma,n}^{1/\gamma} \|\vec{\beta}\|_{\gamma^{'}}\left(\int V_-^{n/2+\gamma} \,{\rm d}x\right)^{1/\gamma} \ . \]

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 $\gamma ^{'}_{n,\alpha }\geq 1$ and

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

Using the critical Sobolev inequality

\[ G_{\beta,a}(V)\geq{-} a^{\frac{4}{4+\alpha-n}} L_{\gamma,n}^{1/\gamma} \|\beta\|_{\gamma^{'}_{n,\alpha} }S_{n,\alpha}^{\frac{4}{(4+\alpha)-n}}\left\langle V,V\right\rangle_{\alpha/2} ^{\frac{2}{(4+\alpha)-n}} \]

hence

\[ \frac{a}{2} \left\langle V,V\right\rangle_{\alpha/2} +G_{\beta,a}(V) \geq \]

(2.11)\begin{equation} a\left\langle V,V\right\rangle_{\alpha/2} ^{\frac{2}{4+\alpha-n}}\left( \frac{1}{2} \left\langle V,V\right\rangle_{\alpha/2} ^{1-\frac{2}{4+\alpha-n}} -a^{\frac{n-\alpha}{4+\alpha-n}} L_{\gamma,n}^{1/\gamma}\|\beta\|_{\gamma^{'}_{n,\alpha} }S_{n,\alpha}^{\frac{4}{(4+\alpha)-n}}\right) \end{equation}

It follows that ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ is coersive for any $a>0$ if $3\leq n<2+\alpha$. If $n=2+\alpha$ then the functional is coersive if $a<\frac {S_{n,\alpha }^{\frac {4}{n-(4+\alpha )}}}{2L_{\gamma,n}^{1/\gamma }}|\beta |_{\gamma ^{'}_{n,\alpha }}^{-1}$. Note that $\gamma ^{'}_{n,\alpha }=\infty$ for $n=3,\, \alpha =1$ and $\gamma ^{'}_{n,\alpha }=2$ 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 strength $a_c(\beta )$ 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 2.3 we may obtain

(2.12)\begin{equation} 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) \end{equation}

where $O(1)$ stands for some constant independent of $V$.⁵

Substitute now now $V/\sqrt {a}$ for $V$. Then $\frac {a}{2}\left \langle V/\sqrt {a},\,V/\sqrt {a}\right \rangle _{\alpha /2} +G^0_{\beta, a}(V/\sqrt {a}) = \frac {1}{2}\left \langle V,\,V\right \rangle _{\alpha /2} +G^0_{\beta, a}(V/\sqrt {a})$. 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

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

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 $G^0_{\beta,a}(V_\delta )=\delta ^2G^0_{\beta,a}(V)$ so, by (2.12), $G_{\beta,a}(V_\delta )\leq \delta ^2 G^0_{\beta,a}(V)+\sum _{j=1}^{k\wedge k(V)}\beta _j\int W|\phi _{j, \delta }^0|^2 +O(1)$.

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

(2.14)\begin{align} \frac{a}{2}\left\langle V_\delta,V_\delta\right\rangle_{\alpha/2}+G_{\beta, a}(V_\delta) & \leq \delta^2\left[\frac{a}{2}\left\langle V,V\right\rangle_{\alpha/2} +G^0_{\beta, a}(V) \right]\nonumber\\ & \quad + \sum_{j=1}^{k\wedge k(V)}\beta_j\int W|\phi_{j, \delta}^0|^2 +O(1). \end{align}

By (1.21) we also get $\lim _{\delta \rightarrow \infty } \int W|\phi ^0_{j,\delta }|^2 =W(0)=0$, so, using (2.13) we obtain the existence of $V$ for which $\frac {a}{2}\left \langle V_\delta,\,V_\delta \right \rangle _{\alpha /2} +G_{\beta, a}(V_\delta )\rightarrow -\infty$ as $\delta \rightarrow \infty$, if $a>0$ is large enough.

Now let

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

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 ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ (2.3) 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 (§§ 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 2.10 Given $\vec \beta$ satisfying (1.19), $R>0$, $3\leq n<2+\alpha$ or either $n=3,\, \alpha =1$ or $n=4,\, \alpha =2$. Then there exists a minimizer $\bar {V}_R$ of ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ restricted to $\dot{\mathbb {H}}_{\alpha /2}(B_R^n)$. Moreover,

(2.15)\begin{equation} I_\alpha^{B_R^n}(\bar V_R)= \left(\sum_{j=1}^k \beta_j|\bar\phi^R_j|^2\right) \end{equation}

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 ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ . Since ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ is bounded from below by proposition 2.9 we get

\[ \lim_{n\rightarrow\infty} {\mathcal{H}}^{W,\alpha}_{\beta,a}(V_n)= \inf_{V\in \dot{\mathbb {H}}_{\alpha/2}(B^n_R)} {\mathcal{H}}^{W,\alpha}_{\beta,a}(V). \]

Since $\dot{\mathbb {H}}_{\alpha /2}(B^n_R)$ is weakly compact and the functional is coersive (2.10) 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 2.7

(2.16)\begin{equation} \lim_{n\rightarrow\infty} G_{\beta,a}(V_n)= G_{\beta,a}(\bar V_R). \end{equation}

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

\[ \lim_{n\rightarrow \infty} \left\langle V_n, V_n\right\rangle_{\alpha/2}\,{\rm d}x \geq \left\langle \bar V_R, \bar V_R\right\rangle_{\alpha/2}\ . \]

This and (2.16) imply that $\bar V_R$ is, indeed, a minimizer of ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ on $\dot{\mathbb {H}}_{\alpha /2}(B^n_R)$.

Finally, (2.15) follows from (1.26) while taking $\Omega =B_R^n$.

2.3. Proof of theorem 1.6(ii, iii)

Let $V_m$ be a minimizing sequence for ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ 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 ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ on $\dot{\mathbb {H}}_{\alpha /2}(B^n_R)$.

Since ${\mathcal {H}}^{W,\alpha }_{\beta,a}(\bar V_m) \leq {\mathcal {H}}^{W,\alpha }_{\beta,a}(V_m)$ then $\bar V_m$ is a minimizing sequence of ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ on $\dot{\mathbb {H}}_{\alpha /2}(\mathbb {R}^n)$ as well. Now, under the conditions of the Theorem we know by proposition 2.9 that ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ is bounded from below on $\dot{\mathbb {H}}_{\alpha /2}(\mathbb {R}^n)$ and coersive (2.10), 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 2.6 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 1.4 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 $\left \langle \bar V,\, \bar V\right \rangle _{\alpha /2}\leq \lim _{m\rightarrow \infty }\left \langle \bar V_m,\,\bar V_m\right \rangle _{\alpha /2}$.

In addition, lemma 2.7 implies that $G_{\beta,a}(\bar V_m)$ converges to $G_{\beta,a}(\bar V)$. This implies

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

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 1.5 and Corollary 2.5.

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 (1.8)).

1. (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 (1.8). 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 $V\leq (\frac {n-2}{2}) ^2|x|^{-2}$.

   As discussed in §§ 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

   (3.1)\begin{equation} 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 \ . \end{equation}
   Moreover, by Riesz's rearrangement theorem we can assume that this $\tilde \phi$ is radially symmetric.

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

   (3.2)\begin{equation} -\Delta-( A/2) V\not\geq 0. \end{equation}

   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

   (3.3)\begin{equation} r^{{-}3}\left( r^3 U^{'}\right)^{'} + \rho(r)=0. \end{equation}
   Let $m(r)=2\pi ^2\int _0^r s^3\rho (s){\rm d}s$. In particular, $m(\cdot )$ is non-decreasing on $\mathbb {R}_+$, $m(0)=0$, and $m(r)\leq 1$ by assumption. Integrating (3.3) we get
   \[ r^3U^{'}(r)={-}(2\pi^2)^{{-}1} m(r) \Longrightarrow U(r)= (2\pi)^2\int_r^\infty \frac{m(s)}{s^3}{\rm d}s\leq 2\pi^2 r^{{-}2} . \]
   Thus, taking $V=2\pi ^2r^{-2}$ in (3.2) we obtain a violation of the Hardy inequality if $\pi ^2 A$ is below the Hardy constant. Since the Hardy constant $(\frac {n-2}{2})^2=1$ 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 $c$.

2. (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 (1.10). Indeed, using Definition 1.1 and the polar inequality

   \[ \left\langle |\phi_j|^2, I_\alpha* |\phi_i|^2\right\rangle\leq \frac{1}{2}\left[ \left\langle |\phi_j|^2, I_\alpha* |\phi_j|^2\right\rangle + \left\langle |\phi_i|^2, I_\alpha* |\phi_i|^2\right\rangle\right] \]
   we obtain
   \[ {\mathcal{E}}^{(\alpha)}_{\beta,a}(\vec\phi)\geq \frac{1}{2} \sum_{j=1}^k\beta_j\left[ \left\langle \left\langle \phi_j,\phi_j\right\rangle\right\rangle_W -\frac{a }{2} \left\langle |\phi_j|^2, I_\alpha* |\phi_j|^2\right\rangle\right] = \sum_{j=1}^k \beta_j E^W_a(\phi_j) \]
   where $E^W_a$ as defined in (1.7). It follows that ${\mathcal {E}}^{(\alpha )}_{\beta,a}$ 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 ${\mathcal {H}}^{W,\alpha }_{\beta,a}$ is unbounded from below on $\dot{\mathbb {H}}_{\alpha /2}$. Using (2.11) 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 $k$.

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

   \[ I_\alpha =\frac{1}{\Gamma(\alpha)} \int_0^\infty t^{\alpha/2-1} {\rm e}^{t\Delta} \,{\rm d}t;(-\Delta)^{\alpha/2}= \frac{1}{\Gamma(-\alpha)} \int_0^\infty t^{-\alpha/2-1} \left({\rm e}^{t\Delta} -I\right)\,{\rm d}t \]
   where ${\rm e}^{t\Delta }$ is the heat kernel on $\mathbb {R}^n$:
   \[ {\rm e}^{t\Delta}= (4\pi t)^{{-}n/2} {\rm e}^{-{|x|^2}/{4t}}. \]

We may, at least formally, substitute the kernel ${\rm e}^{\Delta _\Omega t}$ 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$\ldots$) an explicit expressions for $I_\alpha ^\Omega$ and $(-\Delta _\Omega )^{\alpha /2}$, introduced implicitly in (1.26). Such a representation can provide some insight on the trace of $I^\Omega _\alpha$ for $\alpha <2$.