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 [Reference Wiegel17]) and as a model of self-gravitating matter [Reference Moroz, Penrose and Tod11].

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 [Reference Lieb6, Reference Lions7, Reference Menzala10]. For existence, regularity and asymptotic behaviour of solutions in the general case see [Reference Moroz and Van Schaftingen12, Reference Moroz and Van Schaftingen13] 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. [Reference Markowich, Rein. and Wolansky18]). 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 scalingFootnote ¹ . 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 [Reference Yinbin, Lu and Wei8] 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. [Reference Yinbin, Lu and Wei8]). It was also shown that $a_c=\|\bar \phi \|_2$, where $\bar \phi$ is the unique, positive solution (c.f. [Reference Ma and Zhao9]) 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.

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 [Reference Kwas̀nicki4]):

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 $\||\cdot \||_{\alpha /2}$.Footnote ² Recall that $\dot{\mathbb {H}}^{\alpha /2}$ is a Hilbert space so, in particular, is weakly locally compact.

Lemma 1.3 [Reference DiNezzaa, Palatuccia and Valdinoci1]

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

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 knownFootnote ³ 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 [Reference Friedrichs2] which ensures the discreteness of spectrum if the potential $W$ grows at infinity at arbitrary rate. This and (1.21) implies, in particular

For the proof of proposition 1.7 see Lemma 2.7 in § 2.

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 $\lambda _j(V)$Footnote 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 $\dot{\mathbb {H}}_{\alpha /2}(\mathbb {R}^n)$.