Optimal decay and regularity for a Thomas--Fermi type variational problem

TL;DR

This paper investigates the existence, uniqueness, and qualitative properties of minimizers for a nonlocal Thomas--Fermi type energy functional, revealing how decay, positivity, and sign-changing behavior depend on parameters and potential decay.

Contribution

It provides new results on existence, regularity, and decay of minimizers for a nonlocal energy functional, including conditions for positivity and sign-changing behavior.

Findings

Minimizers exist and are unique under broad conditions.

Decay at infinity depends on parameters α and q.

Sign-changing minimizers can occur with fast-decaying potentials.

Abstract

We study existence and qualitative properties of the minimizers for a Thomas--Fermi type energy functional defined by $$E_\alpha(\rho):=\frac{1}{q}\int_{\mathbb{R}^d}|\rho(x)|^q dx+\frac{1}{2}\iint_{\mathbb{R}^d\times\mathbb{R}^d}\frac{\rho(x)\rho(y)}{|x-y|^{d-\alpha}}dx dy-\int_{\mathbb{R}^d}V(x)\rho(x)dx,$$ where $d\ge 2$, $\alpha\in (0,d)$ and $V$ is a potential. Under broad assumptions on $V$ we establish existence, uniqueness and qualitative properties such as positivity, regularity and decay at infinity of the global minimizer. The decay at infinity depends in a non--trivial way on the choice of $\alpha$ and $q$. If $\alpha\in (0,2)$ and $q>2$ the global minimizer is proved to be positive under mild regularity assumptions on $V$, unlike in the local case $\alpha=2$ where the global minimizer has typically compact support. We also show that if $V$ decays sufficiently fast the global minimizer is sign--changing even if $V$ is non--negative. In such regimes we establish a relation between the positive part of the global minimizer and the support of the minimizer of the energy, constrained on the non--negative functions. Our study is motivated by recent models of charge screening in graphene, where sign--changing minimizers appear in a natural way.

Equations

$${\mathcal{E}}_{\alpha }(\rho ):=\frac{1}{q}{\int }_{{\mathbb{R}}^d}|\rho (x){|}^{q}dx+\frac{1}{2}{\iint }_{{\mathbb{R}}^d\times {\mathbb{R}}^d}\frac{\rho (x)\rho (y)}{|x-y{|}^{d-\alpha }}dxdy-{\int }_{{\mathbb{R}}^d}V(x)\rho (x)dx,$$

$${\mathcal{E}}_{\alpha }^{TF}(\rho )=\frac{1}{q}{\int }_{{\mathbb{R}}^d}|\rho (x){|}^{q}dx-{\int }_{{\mathbb{R}}^d}\rho (x)V(x)dx+\frac{1}{2}{\int }_{{\mathbb{R}}^d}(I_{\alpha }\ast \rho )(x)\rho (x)dx$$

$$I_{\alpha }(x)=\frac{A_{\alpha }}{|x{|}^{d-\alpha }},A_{\alpha }=\frac{\Gamma (\frac{d-\alpha }{2})}{\Gamma (\frac{\alpha }{2}){\pi }^{d/2}{2}^{\alpha }}.$$

$${\mathcal{E}}_1^{TF}(\rho )=\frac{2}{3}{\int }_{{\mathbb{R}}^2}|\rho (x){|}^{\frac{3}{2}}dx-{\int }_{{\mathbb{R}}^2}\rho (x)V(x)dx+\frac{1}{4\pi }{\int }_{{\mathbb{R}}^2}{\int }_{{\mathbb{R}}^2}\frac{\rho (x)\rho (y)}{|x-y|}dxdy.$$

$$\frac{2d}{d+\alpha }<q<\infty ,$$

$${\Vert \rho \Vert }_{{\mathring{H}}^{-\frac{\alpha }{2}}({\mathbb{R}}^d)}^2:={\int }_{{\mathbb{R}}^d}{(2\pi |\xi |)}^{-\alpha }|\hat{\rho }(\xi ){|}^{2}d\xi <\infty ,$$

$${\mathcal{D}}_{\alpha }(\rho ,\rho )=A_{\alpha }{\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}\frac{\rho (y)\rho (x)}{|x-y{|}^{d-\alpha }}dxdy={\Vert \rho \Vert }_{{\mathring{H}}^{-\frac{\alpha }{2}}({\mathbb{R}}^d)}^2.$$

$${\mathcal{E}}_{\alpha }^{TF}(\rho ):=\frac{1}{q}{\int }_{{\mathbb{R}}^d}|\rho (x){|}^{q}dx-⟨\rho ,V⟩+\frac{1}{2}{\Vert \rho \Vert }_{{\mathring{H}}^{-\frac{\alpha }{2}}({\mathbb{R}}^d)}^2$$

$${\mathcal{H}}_{\alpha }:=L^q({\mathbb{R}}^d)\cap {\mathring{H}}^{-\frac{\alpha }{2}}({\mathbb{R}}^d),$$

$$⟨\rho ,V⟩:=⟨\rho ,V_1⟩+{\int }_{{\mathbb{R}}^d}\rho (x)V_2(x)dx,$$

$${\Vert V\Vert }_{L^{q^{\prime }}({\mathbb{R}}^d)+{\mathring{H}}^{\frac{\alpha }{2}}({\mathbb{R}}^d)}=\inf \left\{{\Vert V_1\Vert }_{{\mathring{H}}^{\frac{\alpha }{2}}({\mathbb{R}}^d)}+{\Vert V_2\Vert }_{L^{q^{\prime }}({\mathbb{R}}^d)}:V=V_1+V_2,V_1\in {\mathring{H}}^{\frac{\alpha }{2}}({\mathbb{R}}^d),V_2\in L^{q^{\prime }}({\mathbb{R}}^d)\right\}.$$

$${\int }_{{\mathbb{R}}^d}\text{sign}({\rho }_V(x))|{\rho }_V(x){|}^{q-1}\phi (x)dx-⟨\phi ,V⟩+⟨{\rho }_V,\phi {⟩}_{{\mathring{H}}^{-\frac{\alpha }{2}}({\mathbb{R}}^d)}=0\forall \phi \in {\mathcal{H}}_{\alpha }.$$

$$\text{sign}({\rho }_V)|{\rho }_V{|}^{q-1}=V-(-\Delta {)}^{-\frac{\alpha }{2}}{\rho }_V\text{in}{\mathcal{D}}^{\prime }({\mathbb{R}}^d),$$

$$(-\Delta {)}^{\frac{\alpha }{2}}u={\rho }_V.$$

$$u_V(x)=V(x)-A_{\alpha }{\int }_{{\mathbb{R}}^d}\frac{{\rho }_V(y)}{|x-y{|}^{d-\alpha }}dy\text{in}{\mathbb{R}}^d,$$

$$u_V:=\text{sign}({\rho }_V)|{\rho }_V{|}^{q-1}.$$

$$(-\Delta {)}^{\frac{\alpha }{2}}u+\text{sign}(u)|u{|}^{\frac{1}{q-1}}=(-\Delta {)}^{\frac{\alpha }{2}}V\text{in}{\mathring{H}}^{-\frac{\alpha }{2}}({\mathbb{R}}^d)$$

$${\mathcal{H}}_{\alpha }^{+}:=\left\{f\in {\mathcal{H}}_{\alpha }:f\ge 0\right\}.$$

$${\int }_{{\mathbb{R}}^d}{({\rho }_V^{+})}^{q-1}\phi -⟨\phi ,V⟩+⟨{\rho }_V^{+},\phi {⟩}_{{\mathring{H}}^{-\frac{\alpha }{2}}({\mathbb{R}}^d)}\ge 0\forall \phi \in {\mathcal{H}}_{\alpha }^{+},$$

$${\rho }_V^{+}={\left[V-(-\Delta {)}^{-\frac{\alpha }{2}}{\rho }_V^{+}\right]}_{+}^{\frac{1}{q-1}}\text{in}{\mathcal{D}}^{\prime }({\mathbb{R}}^d).$$

$$[{\rho }_V{]}_{+}\ge {\rho }_V^{+}\text{in}{\mathbb{R}}^d,$$

$$\underset{|x|\to +\infty }{\mathrm{limsup}}|x{|}^{d+\alpha }(-\Delta {)}^{\frac{\alpha }{2}}V(x)<+\infty ,$$

$$0<\underset{|x|\to +\infty }{\mathrm{liminf}}|x{|}^{\frac{d-\alpha }{q-1}}{\rho }_V(x)\le \underset{|x|\to +\infty }{\mathrm{limsup}}|x{|}^{\frac{d-\alpha }{q-1}}{\rho }_V(x)<+\infty ;$$

$$0<\underset{|x|\to +\infty }{\mathrm{liminf}}|x{|}^d(\log |x|{)}^{\frac{d}{\alpha }}{\rho }_V(x)\le \underset{|x|\to +\infty }{\mathrm{limsup}}|x{|}^d(\log |x|{)}^{\frac{d}{\alpha }}{\rho }_V(x)<+\infty ;$$

$$0<\underset{|x|\to +\infty }{\mathrm{liminf}}|x{|}^{\frac{\alpha }{2-q}}{\rho }_V(x)\le \underset{|x|\to +\infty }{\mathrm{limsup}}|x{|}^{\frac{\alpha }{2-q}}{\rho }_V(x)<+\infty ;$$

$$0<\underset{|x|\to +\infty }{\mathrm{liminf}}|x{|}^{d+\alpha }(\log |x|{)}^{-\frac{d+\alpha }{\alpha }}{\rho }_V(x)\le \underset{|x|\to +\infty }{\mathrm{limsup}}|x{|}^{d+\alpha }(\log |x|{)}^{-\frac{d+\alpha }{\alpha }}{\rho }_V(x)<+\infty ;$$

$$0<\underset{|x|\to +\infty }{\mathrm{liminf}}|x{|}^{d+\alpha }{\rho }_V(x)\le \underset{|x|\to +\infty }{\mathrm{limsup}}|x{|}^{d+\alpha }{\rho }_V(x)<+\infty .$$

$$\lim_{|x|\to+\infty}|x|^{\frac{d-\alpha}{q-1}}\rho_{V}(x)=\left[A_{\alpha}\left(\big{|}\!\big{|}(-\Delta)^{\frac{\alpha}{2}}V\big{|}\!\big{|}_{L^{1}(\mathbb{R}^{d})}-\left\|\rho_{V}\right\|_{L^{1}(\mathbb{R}^{d})}\right)\right]^{\frac{1}{q-1}}>0.$$

$$\mathcal{I}:=\underset{u\in {\mathring{H}}^{\frac{\alpha }{2}}({\mathbb{R}}^d)}{\inf }\left\{[u{]}_{{\mathring{H}}^{\frac{\alpha }{2}}({\mathbb{R}}^d)}^2:u\ge {\chi }_{B_1}\right\}$$

$$\underset{|x|\to +\infty }{\mathrm{liminf}}|x{|}^{d+\alpha }{\rho }_V(x)>0.$$

Full text

[TABLE]

where $d\geq 2$, $\alpha\in(0,d)$ and $V$ is a potential. Under broad assumptions on $V$ we establish existence, uniqueness and qualitative properties such as positivity, regularity and decay at infinity of the global minimizer. The decay at infinity depends in a non–trivial way on the choice of $\alpha$ and $q$. If $\alpha\in(0,2)$ and $q>2$ the global minimizer is proved to be positive under mild regularity assumptions on $V$, unlike in the local case $\alpha=2$ where the global minimizer has typically compact support. We also show that if $V$ decays sufficiently fast the global minimizer is sign–changing even if $V$ is non–negative. In such regimes we establish a relation between the positive part of the global minimizer and the support of the minimizer of the energy, constrained on the non–negative functions. Our study is motivated by recent models of charge screening in graphene, where sign–changing minimizers appear in a natural way.

Keywords: Thomas-Fermi energy, fractional Laplacian, Riesz potential, Asymptotic decay.

2020 Mathematics Subject Classification: 35J61, 35B09, 35B33, 35B40

1. 1 Introduction
2. 2 Preliminaries

1 Introduction

In this paper we analyse minimizers of a Thomas–Fermi type energy of the form

[TABLE]

where $d\geq 2$, $0<\alpha<d$, $\rho:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is a measurable function, $V:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is a potential and $I_{\alpha}$ is defined as

[TABLE]

The quantity $I_{\alpha}:{\mathbb{R}}^{d}\setminus\left\{0\right\}\to{\mathbb{R}}$ is the Riesz potential of order $\alpha$ and the choice of the normalisation constant $A_{\alpha}$ ensures that the convolution kernel $I_{\alpha}$ satisfies the semigroup property $I_{\alpha+\beta}=I_{\alpha}*I_{\beta}$ for each $\alpha,\beta\in(0,d)$ such that $\alpha+\beta<d$ (see for example [Stein]*eq. $(6.6)$, p. $118$). Furthermore, under suitable assumptions, the Riez potential operator can be associated to the inverse fractional Laplace operator $(-\Delta)^{-\frac{\alpha}{2}}$ (see [Stein]*Lemma $2$).

Our study is motivated by the Thomas–Fermi type model of charge screening in graphene [graphene] (see also [Katsnelson, orbital]) that corresponds to the special two–dimensional case $q=\tfrac{3}{2}$ and $\alpha=1$,

[TABLE]

The function $\rho(x)$ here has a meaning of a charge density of fermionic quasiparticles (electrons and holes) in a two–dimensional graphene layer. In general (and unlike in the classical Thomas–Fermi models of atoms and molecules [Lieb-Loss]), the density $\rho$ is a sign–changing function, with $\rho>0$ representing electrons and $\rho<0$ representing holes. The first term in (1.3) is an approximation of the kinetic energy of a uniform gas of non–interacting particles. The middle term is the interaction with an external potential $V$, and the last quantity is a non–local Coulombic interaction between quasi–particles. In [graphene], the authors prove the existence, uniqueness, and decay of the minimizer of the energy (1.3). In particular, they establish a log–correction to the decay rates of the minimizer, previously identified by M. Katsnelson in [Katsnelson].

The main goal of this paper is to deduce existence, uniqueness and qualitative properties of the minimizer in the general admissible range

[TABLE]

where $\frac{2d}{d+\alpha}$ is a critical exponent with respect to the Hardy-Littlewood-Sobolev inequality (see [Lieb-Loss]*Theorem $4.3$, p. $106$). Our main result is Theorem 1.5 where we establish five different asymptotic regimes for the minimizer, depending on the values of $d$, $\alpha$ and $q$. Two of the regimes are entirely new and not visible in the “local” case $\alpha=2$, when the non–local term is the standard Newtonian potential.

Before presenting our results in details, let’s emphasise the main differences between our case and the classical three dimensional TF–theory of atoms and molecules [lieb_3, lieb_dim3, Lieb-Simon] (see also [goldestein, Rieder] for generalisations of the mentioned three dimensional models).

Unlike in [lieb_3, lieb_dim3, Lieb-Simon], we are looking at

• •

global minimizers without a mass constraint since there is no a priori reason for the density function $\rho(x)$ to be integrable;

• •

potentially sign–changing profiles $\rho(x)$, since electrons and holes could co–exist in a graphene layer;

• •

general range of $\alpha\in(0,d)$, as for istance in (1.3) $\alpha=1$.

We further mention that several difficulties arise if sign–changing profiles $\rho(x)$ are allowed. To explain why, we first introduce the Banach space $\mathring{H}^{-\frac{\alpha}{2}}(\mathbb{R}^{d})$ which can be defined as the space of tempered distributions such that $\hat{\rho}\in L^{1}_{loc}(\mathbb{R}^{d})$ and

[TABLE]

where $\hat{\rho}$ stands for the Fourier transform of $\rho$ (see [classicbook]*Definition 1.31). It’s easy to see (e.g., Sect. 2, eq. (LABEL:riesz_energy)) that for smooth functions we have the equality

[TABLE]

However, it can be proved that there exist Cauchy sequences in $\mathring{H}^{-\frac{\alpha}{2}}(\mathbb{R}^{d})$ of smooth sign–changing functions $\left\{\rho_{n}\right\}_{n}$ that do not converge to a signed measure (see [orbital]*Example $4.1$, [cartan] and references therein). This can not happen when considering a smooth sequence of non–negative functions (see [landkof]*Theorem $1.18$, p. $90$) or a sequence in $L^{\frac{2d}{d+\alpha}}({\mathbb{R}}^{d})$ (this follows from [Lieb-Loss]*Theorem $4.3$, p. $106$). Hence, by only requiring the density $\rho$ to be a regular distribution belonging to $\mathring{H}^{-\frac{\alpha}{2}}(\mathbb{R}^{d})$ (see Section 2 for the definition of regular distributions) without prescribing any sign or fast decay condition we don’t expect the integral in (1.6) to be convergent. We show that these issues do not arise for a certain class of non–negative functions (not necessarily smooth or fast decaying) because the Brezis–Browder type results for Sobolev spaces of integer order (see [potential]*Theorems $3.4.1$, $3.4.2$) ensure Lebesgue integrability. We infact extend this type of results to the case of fractional Sobolev spaces (see Corollary LABEL:Lq_loc and LABEL:prod_integr in Sect. 2) proving that $\mathcal{D}_{\alpha}(\rho,\rho)$ is well defined as a Lebesgue integral. See also [Fukushima]*Example $2.2.1$. However, we emphasise again that in general $\mathcal{D}_{\alpha}$ can not be extended as an integral to the whole of $\mathring{H}^{-\frac{\alpha}{2}}({\mathbb{R}}^{d})$ and in the sequel we replace it by $\left\|\cdot\right\|^{2}_{\mathring{H}^{-\frac{\alpha}{2}}(\mathbb{R}^{d})}$.

In what follows we consider the energy functional

[TABLE]

on the domain $\mathcal{H}_{\alpha}$ defined as follows

[TABLE]

and endowed with the norm $\left\|\cdot\right\|_{\mathcal{H}_{\alpha}}:=\left\|\cdot\right\|_{\mathring{H}^{-\frac{\alpha}{2}}(\mathbb{R}^{d})}+\left\|\cdot\right\|_{L^{q}(\mathbb{R}^{d})}$. In (1.7) the function $V$ belongs to the dual space $(\mathcal{H}_{\alpha})^{\prime}$ and $\langle\cdot,\cdot\rangle$ denotes the duality between $\mathcal{H}_{\alpha}$ and $(\mathcal{H}_{\alpha})^{\prime}$. We recall that $(\mathcal{H}_{\alpha})^{\prime}$ identifies with $L^{q^{\prime}}(\mathbb{R}^{d})+\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d})$. More precisely, for every function $V=V_{1}+V_{2}\in L^{q^{\prime}}(\mathbb{R}^{d})+\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d})$, $V_{1}\in\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d})$ and $V_{2}\in L^{q^{\prime}}(\mathbb{R}^{d})$ we define

[TABLE]

where in (1.9) by $\langle\cdot,\cdot\rangle$ we identify the duality between $\mathring{H}^{\frac{\alpha}{2}}({\mathbb{R}}^{d})$ and $\mathring{H}^{-\frac{\alpha}{2}}({\mathbb{R}}^{d})$. 222We further recall that the space $L^{q^{\prime}}(\mathbb{R}^{d})+\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d})$ is a Banach space endowed with the norm

$\left\|V\right\|_{L^{q^{\prime}}(\mathbb{R}^{d})+\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d})}\!=\!\inf\left\{\left\|V_{1}\right\|_{\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d})}\!+\!\left\|V_{2}\right\|_{L^{q^{\prime}}(\mathbb{R}^{d})}:V=V_{1}+V_{2},V_{1}\in\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d}),V_{2}\in L^{q^{\prime}}(\mathbb{R}^{d})\right\}.$

For the precise definitions and properties of the fractional homogeneous Sobolev space $\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d})$ we refer to Section 2.

The first result is to establish existence and uniqueness of a minimizer for $\mathcal{E}^{TF}_{\alpha}$.

Theorem 1.1**.**

Assume $q>\frac{2d}{d+\alpha}$. Then, for every $V\in L^{q^{\prime}}(\mathbb{R}^{d})+\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d})$, the functional $\mathcal{E}^{TF}_{\alpha}$ admits a unique minimizer $\rho_{V}\in\mathcal{H}_{\alpha}$ that satisfies

[TABLE]

An equivalent way to read (1.10) is by the following relation

[TABLE]

where by $(-\Delta)^{-\frac{\alpha}{2}}\rho_{V}$ we understand the unique element of $\mathring{H}^{\frac{\alpha}{2}}({\mathbb{R}}^{d})$ weakly solving

[TABLE]

See Section 2 for all the details.

If, e.g., $\rho_{V}\in L^{q}(\mathbb{R}^{d})$ with $q<\frac{d}{\alpha}$, the Riesz potential of $\rho_{V}$ is well defined (see [Stein]*Theorem $1$, Sect. $5$) and identifies with $(-\Delta)^{-\frac{\alpha}{2}}\rho_{V}$ (cf. [mazya]*Lemma 1.8). Moreover, without any restriction on $q$, if $\alpha\in(0,2]$ and $\rho_{V}\geq 0$, the Riesz potential $I_{\alpha}*\rho_{V}$ can be identified again with the operator $(-\Delta)^{-\frac{\alpha}{2}}\rho_{V}$ (see [Fukushima]*Example $2.2.1$). Thus, in the above cases, (1.11) reads also pointwisely as

[TABLE]

where $u_{V}$ is defined by

[TABLE]

Such function $u_{V}$ plays an important role for the analysis of the sign and decay at infinity of the minimizer. As a matter of fact, if $V\in\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d})$ we prove that the Euler–Lagrange equation (1.10) is equivalent to the fractional semilinear PDE

[TABLE]

in the sense that $u_{V}$ weakly solves it. We further prove that (1.14) satisfies a weak comparison principle provided $\alpha\in(0,2]$. See for example [Brezis_1]*Lemma $2$, $(27)$, p. $8$ for some general results in the case $\alpha=2$.

Section LABEL:sec.4 is devoted to the study of the regularity of the minimizer. The way in which we obtain the regularity is based on a classical bootstrap argument similar to the one in [Carrillo-CalcVar]*Theorem $8$. However, some differences occur. The main one is the presence of the function $V$ which clearly might be an obstruction for the regularity of the minimizer. Indeed, from (1.11), it’s clear that in general the best regularity that $u_{V}$ can achieve is the same regularity as $V$. Such regularity is achieved for example in Corollary LABEL:non_local_regularity. We further summarise below the basic regularity result for the minimizer $\rho_{V}$. We stress that such regularity can be improved and specialised according to the regularity assumptions we impose on $V$.

Theorem 1.2**.**

Assume $q>\frac{2d}{d+\alpha}$. If $V\in\mathring{H}^{\frac{\alpha}{2}}({\mathbb{R}}^{d})\cap C_{b}(\mathbb{R}^{d})$ then $\rho_{V}\in C({\mathbb{R}}^{d})$.

In Section LABEL:sec.5 we focus on the relation between unconstrained minimizers and minimizers of $\mathcal{E}_{\alpha}^{TF}$ subject to contraint to non–negative functions, defined by

[TABLE]

Then, we establish existence and uniqueness of a minimizer in $\mathcal{H}^{+}_{\alpha}$. Namely, denoted by $[x]_{+}:=\leavevmode\nobreak\ \max\left\{0,x\right\}$, we prove the following:

Theorem 1.3**.**

Assume $q>\frac{2d}{d+\alpha}$. Then, for every $V\in L^{q^{\prime}}(\mathbb{R}^{d})+\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d})$ the function $\mathcal{E}^{TF}_{\alpha}$ admits a unique minimizer $\rho^{+}_{V}$ belonging to $\mathcal{H}^{+}_{\alpha}$. Moreover, the minimizer $\rho^{+}_{V}$ satisfies

[TABLE]

and

[TABLE]

Next, we discuss the relation between the non–negative and free minimizer. We also prove that the equality $[\rho_{V}]_{+}=\rho^{+}_{V}$ holds only in the trivial case in which $\rho_{V}\geq 0$ (see Theorem LABEL:minimi_diversi) while we obtain the inequality $[\rho_{V}]_{+}\geq\rho^{+}_{V}$ provided the potential $V$ is regular and decays sufficiently fast (see e.g., Remark LABEL:not_sharp_V). Namely, the following holds:

Theorem 1.4**.**

Assume $d\geq 2$, $0<\alpha\leq 2$. Let $V\in\mathring{H}^{\frac{\alpha}{2}}({\mathbb{R}}^{d})\cap C(\mathbb{R}^{d})$ be compactly supported and not identically zero. Then,

[TABLE]

the inequality being strict in a set of positive measure.

In Section LABEL:sec.6 we study decay properties of $\rho_{V}$ in the fractional framework $\alpha\in(0,2)$. First of all we study positivity of $\rho_{V}$. If $q\leq 2$, the total power of $u$ in (1.14) is bigger than $1$ and equation (1.14) is superlinear. Then, the positivity follows from Proposition LABEL:u_pos. If $q>2$ then equation (1.14) is sublinear which implies that studying the positivity of $\rho_{V}$ is more delicate and strongly relates to the non–local nature of the fractional Laplacian (see Proposition LABEL:superlinear and LABEL:lower_bound_48). Indeed, in the local case $\alpha=2$ the support of a non–negative solution of (1.14) could be compact (see [maximum]*Theorem $1.1.2$ and [Diaz]*Corollary $1.10$, Remark $1.5$).

Next, in Theorem 1.5 below, we outline the decay of the minimizer $\rho_{V}$ for $q\leq 2$. However, for the convenience of the reader, we state the result by requiring that $V$ satisfies the stronger assumption (1.18) even though such assumption is not sharp and can be weakened in several ranges of $q$ (see Remark LABEL:restrizione_deca).

Theorem 1.5**.**

Assume $d\geq 2$, $0<\alpha<2$ and $\frac{2d}{d+\alpha}<q\leq 2$ . If $V$ is a non–indentically zero element of $\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d})\cap C_{b}(\mathbb{R}^{d})$ and $(-\Delta)^{\frac{\alpha}{2}}V$ is a locally integrable non–negative function such that

[TABLE]

then $\rho_{V}\in L^{1}({\mathbb{R}}^{d})$. Moreover, the following cases occur:

• $(i)$

If $\frac{2d}{d+\alpha}<q<\frac{2d-\alpha}{d}$ then

[TABLE]

• $(ii)$

If $q=\frac{2d-\alpha}{d}$ and, either $1<\alpha<2$ and $d>\alpha+1$ or $q=\tfrac{3}{2}$, then

[TABLE]

• $(iii)$

If $\frac{2d-\alpha}{d}<q<\frac{2d+\alpha}{d+\alpha}$ then

[TABLE]

• $(iv)$

If $q=\frac{2d+\alpha}{d+\alpha}$ then

[TABLE]

• $(v)$

If $\frac{2d+\alpha}{d+\alpha}<q\leq 2$ then

[TABLE]

In particular, if $\frac{2d}{d+\alpha}<q<\frac{2d-\alpha}{d}$ then

[TABLE]

Before showing some applications of Theorem 1.5, we briefly underline how the proof develops. We intend to construct positive super and subsolutions of (1.14) in exterior domains of the form $\Omega=\overline{B_{R}}^{c}$. The scenario is the following:

• •

If q\in\big{(}\frac{2d}{d+\alpha},\frac{2d+\alpha}{d+\alpha}\big{)}\setminus\left\{\frac{2d-\alpha}{d}\right\}, we employ polynomial–decaying functions as barriers. Namely, we use a linear combination of the functions $f$ and $g$ where $f(x)=(1+|x|^{2})^{-\beta}$, for some suitable $\beta>0$, and $g$ is defined as the optimizer for the following problem

[TABLE]

(see Lemma LABEL:range_1, eq. (LABEL:v_intro)). Decay estimates for such funtions were derived in [Ferrari]*equation $(4.4)$ and [optimal]*Proposition $3.6$;

• •

If $q\in\left\{\frac{2d-\alpha}{d},\frac{2d+\alpha}{d+\alpha}\right\}$, polynomial–behaving barriers are not enough to obtain sharp estimates. Indeed, we apply [Ferrari]*Theorem $1.1$ to deduce asymptotic decay for functions with logarithmic behaviour (see Lemma LABEL:alfa_grande and LABEL:2_limit);

• •

If q\in\big{(}\frac{2d+\alpha}{d+\alpha},2\big{)}, by applying again [Ferrari]*Theorem $1.1$, we employ polynomial–behaving functions with fast decay (Lemma LABEL:max_decay);

• •

If $q=2$, the conclusion is essentially a consequence of [positivesolutions]*Lemmas 4.2–4.3.

Note also that, differently from the local case $\alpha=2$, the exponent $q=\frac{2d+\alpha}{d+\alpha}$ represents a critical value. Furthermore, if $q=\frac{2d-\alpha}{d}$ we have the slowest decay and if $\frac{2d+\alpha}{d+\alpha}<q\leq 2$ the decay is the same as in [positivesolutions]. We also recall that log–corrections appear at the critical values $\frac{2d-\alpha}{d}$ and $\frac{2d+\alpha}{d+\alpha}$.

Subsequently, we focus our attention on the case $q>2$. As we have already stressed above, positivity of the minimizer in this regime is not trivial. We provide two independent ways to prove it. The first result is Proposition LABEL:superlinear where we prove positivity of the minimizer by requiring further regularity of the potential $V$. The second result is Proposition LABEL:lower_bound_48 where, by requiring positivity of $(-\Delta)^{\frac{\alpha}{2}}V$ on the unit ball, we prove positivity of the minimizer and we find a lower bound on its decay matching the one of the linear case $q=2$. In particular, under this assumption, we deduce that

[TABLE]

We stress again that (1.20) is in contrast to what happens in the local case $\alpha=2$ where the support is compact (cf. [maximum]*Theorem $1.1.2$). Let’s further recall that when $\alpha=2$ it’s well known that the decay of non–negative solutions of (1.14) is polynomial provided $q<2$ and exponential provided $q=2$ (see [Veron]).

In Figure 1, we graphically show what Theorem 1.5 and Proposition LABEL:lower_bound_48 tell us in terms of decay. In particular, we stress that for $q>2$ the line is represented as dashed because of the lack of an upper bound matching the lower bound given by (1.20).

The idea behind the proof of (1.20) (see Proposition LABEL:lower_bound_48) is to find a positive function, behaving like $|x|^{-(d+\alpha)(q-1)}$ at infinity, which is a subsolution of (1.14) in ${\mathbb{R}}^{d}$.

An important example of a potential satisfying the assumptions of Theorem 1.5, Propositions LABEL:superlinear and LABEL:lower_bound_48 is the one of the form

[TABLE]

where $A_{\alpha}$ is defined in (1.2).

Then, if we consider $V_{Z}(x):=ZV(x)$, for some $Z>0$, we prove that:

• $(i)$

If $\frac{2d}{d+\alpha}<q<\frac{2d-\alpha}{d}$ then

[TABLE]

• $(ii)$

If $\frac{2d-\alpha}{d}\leq q<\frac{2d+\alpha}{d}$ then

[TABLE]

For the precise statements we refer to Corollaries LABEL:L_1_bound and LABEL:V_z.

Finally, we study asymptotic decay of sign–changing minimizers where we don’t expect the existence of a universal lower bound on the absolute value of the minimizer $|\rho_{V}|$. For this reason, we obtain a similar version of Theorem 1.5 with upper bounds only. Note that, in view of Remark LABEL:remark_soglia1, case $(ii)$ of Theorem 1.6 requires less restrictions than case $(ii)$ of Theorem 1.5.

Theorem 1.6**.**

Assume $d\geq 2$, $0<\alpha<2$ and $\frac{2d}{d+\alpha}<q\leq 2$. Assume that $V$ belongs to $\mathring{H}^{\frac{\alpha}{2}}({\mathbb{R}}^{d})\cap C_{b}(\mathbb{R}^{d})$, $(-\Delta)^{\frac{\alpha}{2}}V\in L^{\frac{2d}{d+\alpha}}_{loc}({\mathbb{R}}^{d})$ and

[TABLE]

If $|\rho_{V}|\in\mathring{H}^{-\frac{\alpha}{2}}({\mathbb{R}}^{d})$ then the following possibilities hold:

• $(i)$

If $\frac{2d}{d+\alpha}<q<\frac{2d-\alpha}{d}$ then

[TABLE]

• $(ii)$

If $q=\frac{2d-\alpha}{d}$ and, either $1<\alpha<2$ or $q=\tfrac{3}{2}$, then

[TABLE]

• $(iii)$

If $\frac{2d-\alpha}{d}<q<\frac{2d+\alpha}{d+\alpha}$ then

[TABLE]

• $(iv)$

If $q=\frac{2d+\alpha}{d+\alpha}$ then

[TABLE]

• $(v)$

If $\frac{2d+\alpha}{d+\alpha}<q\leq 2$ then

[TABLE]

The proof is essentialy based on two facts:

• •

similarly to the proof of Theorem 1.4, we employ a Kato–type inequality of the form

[TABLE]

which allows us to conclude that the function $|u_{V}|$ is a non–negative weak subsolution of the semilinear PDE

[TABLE]

• •

we employ the same barriers used for the proof of Theorem 1.5 as supersolutions of (1.23) in exterior domains of the form $\Omega=\overline{B_{R}}^{c}$.

Organization of the paper. In Section 2 we introduce some definitions useful for the rest of the paper. In Sections LABEL:sec.3 and LABEL:sec.5 we prove respectively Theorem 1.1 and 1.3. Section LABEL:sec.4 is entirely dedicated to prove regularity of the minimizer. In Section LABEL:sec.6 we study the asymptotic behaviour of the minimizer and we prove Theorems 1.5, 1.6. As a result, in Corollary LABEL:L_1_bound we also estimate the $L^{1}$–norm of the minimizer in terms of the $L^{1}$–norm of $(-\Delta)^{\frac{\alpha}{2}}V$. Finally, in the Appendix LABEL:Appendix, we derive asymptotic estimates for the fractional Laplacian of power–type functions.

Notations. Sets:

• •

For every $x\in{\mathbb{R}}^{d}$ and $R>0$, by $B_{R}(x)$ we understand the set defined as

[TABLE]

We will often set ${B_{R}:=B_{R}(0)}$. As it is customary we denote by $\omega_{d}$ the measure of the unit ball in ${\mathbb{R}}^{d}$.

• •

Let $\Omega$ be a subset of ${\mathbb{R}}^{d}$. By $|\Omega|$, $\Omega^{c}$ and $\overline{\Omega}$ we understand respectively the $d$–dimensional Lebsegue measure, the complementary and the closure of $\Omega$. Moreover, we say that $\Omega$ is smooth if its boundary $\partial\Omega$ locally identifies with a smooth function.

Function spaces: Let $\Omega$ be an open set in ${\mathbb{R}}^{d}$.

• •

$C^{\infty}_{c}(\Omega)$ denotes the space of smooth functions having compact support contained in $\Omega$;

• •

If $k\in{\mathbb{N}}$, the space $C^{k}(\Omega)$ (respectively $C^{k}_{b}(\Omega)$) represents the space of functions whose derivatives are continuous up to order $k$ on $\Omega$ (respectively continuous and bounded);

• •

$\mathcal{D}^{\prime}(\Omega)$ denotes the dual space of $C^{\infty}_{c}(\Omega)$. Moreover, if $T\in\mathcal{D}^{\prime}(\Omega)$, by the the notation $T\geq 0$ in $\mathcal{D}^{\prime}(\Omega)$ we understand the following inequality

[TABLE]

where by the symbol $\langle\cdot,\cdot\rangle$ we denote the action of $T$ against the test function $\varphi$;

• •

If $\gamma\in(0,1]$ and $k\in{\mathbb{N}}$, we define $C^{k,\gamma}({\Omega})$ (respectively $C^{k,\gamma}_{loc}({\Omega}))$ by

[TABLE]

endowed with the norm

[TABLE]

• •

$L^{p}(\Omega)$ with $p\in[1,+\infty]$ is the standard Lebesgue space;

• •

If $\Omega={\mathbb{R}}^{d}$, $\mathscr{S}({\mathbb{R}}^{d})$ denotes the class of Schwartz functions while by $\mathscr{S}^{\prime}({\mathbb{R}}^{d})$ we denote the space of tempered distributions.

Asymptotic notations:

Let $f,g:\mathbb{R}^{d}\rightarrow\mathbb{R}$. We write

• •

$f(x)\lesssim g(x)$ in $\Omega$ if there exists $C>0$ such that $f(x)\leq Cg(x)$ for every $x\in\Omega.$

• •

$g(x)\simeq f(x)$ in $\Omega$ if both $f(x)\lesssim g(x)$ and $g(x)\lesssim f(x)$ in $\Omega$.

In particular, by writing $f(x)\lesssim g(x)$ as $|x|\to+\infty$ we mean that there exists $C>0$ such that $f(x)\leq Cg(x)$ for every $x$ sufficiently large and by writing $f(x)\simeq g(x)$ as $|x|\to+\infty$ we understand that $f(x)\lesssim g(x)$ and $g(x)\lesssim f(x)$ as $|x|\to+\infty$. Similarly we write $f(x)\sim g(x)$ as $|x|\to+\infty$ if

[TABLE]

2 Preliminaries

Fractional Sobolev spaces

Let $d\geq 2$ and $|\alpha|<d$. The fractional Laplacian $(-\Delta)^{\frac{\alpha}{2}}$ on $\mathbb{R}^{d}$ is defined by means of the Fourier transform (cf. [Samko]*eq $(25.2)$, p. $486$, [mazya]*eq $(1.1$) and [book_triebel]*eq $(6.11)$, p. $2$)

[TABLE]

As usual, if $u$ is smooth enough, the Fourier transform is defined by the formula

[TABLE]

where by $x\cdot\xi$ we denote the standard inner product in $\mathbb{R}^{d}$. Moreover, the Fourier transform operator can be naturally extended to the class of tempered distributions (see [classicbook]*Definition $1.20$, Proposition $1.23$).

Let $\alpha$ be any real number such that $0<\alpha<d$. The homogeneous fractional Sobolev space $\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d})$ can be defined as the completion of $C^{\infty}_{c}({\mathbb{R}}^{d})$ under the norm

[TABLE]

(see [classicbook]*Definition $1.31$ and [multiplier]*p.10). The space $\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d})$ defined above is a Hilbert space ([classicbook]*Proposition 1.34) with inner product given by

[TABLE]

and it is continuosly embedded into $L^{\frac{2d}{d-\alpha}}(\mathbb{R}^{d})$ provided $\alpha>0$ ([classicbook]*Theorems $1.38$–$1.43$). The dual space to $\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d})$ identifies with $\mathring{H}^{-\frac{\alpha}{2}}(\mathbb{R}^{d})$ ([classicbook]*Proposition $1.36$) and, by the Riesz representation theorem, for every $T\in\mathring{H}^{-\frac{\alpha}{2}}(\mathbb{R}^{d})$ there exists a unique element $\mathcal{U}_{T}\in\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d})$ such that

[TABLE]

where by $\langle\cdot,\cdot\rangle$ we understand the duality between $\mathring{H}^{-\frac{\alpha}{2}}(\mathbb{R}^{d})$ and $\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d})$. Moreover,

[TABLE]

so that the duality (2.4) is an isometry. In particular, for every $T,\varphi\in\mathring{H}^{-\frac{\alpha}{2}}({\mathbb{R}}^{d})$ we have

[TABLE]

The function $\mathcal{U}_{T}$ solving (2.4) is called a weak solution of the equation

[TABLE]

In what follows, we will often replace the notation $\mathcal{U}_{T}$ with the more explicit one $(-\Delta)^{-\frac{\alpha}{2}}T$. Assume now $0<\alpha<2$. If $u$ is a smooth function, we have a pointwise representation of the operator $(-\Delta)^{\frac{\alpha}{2}}$ given by

[TABLE]

or by the singular integral

[TABLE]

where

[TABLE]

(see [Valdinoci]*Sect. $3$, (3.1)–(3.2)). Next, we introduce two useful spaces.

If $\varepsilon>0$ is sufficiently small, we define the space $C^{\alpha+\varepsilon}_{loc}({\mathbb{R}}^{d})$ as

[TABLE]

and

[TABLE]

By the Sobolev embedding, we further have

[TABLE]

In particular, if $u\in\mathcal{L}^{1}_{\alpha}(\mathbb{R}^{d})$, $\alpha>0$, the distributional fractional Laplacian $(-\Delta)^{\frac{\alpha}{2}}u$ defined by

[TABLE]

is a tempered distribution. See e.g., [garofalo]*Corollary $2.16$ for further details. In view of (2.12), we emphasise that if $\mathcal{U}_{T}$ solves (2.6) then it also solves

[TABLE]

Indeed, by combining (2.3) with (2.4), for every $\varphi\in C^{\infty}_{c}({\mathbb{R}}^{d})$ we have

[TABLE]

where $\langle\cdot,\cdot\rangle$ denotes the duality between $\mathring{H}^{\frac{\alpha}{2}}({\mathbb{R}}^{d})$ and $\mathring{H}^{-\frac{\alpha}{2}}({\mathbb{R}}^{d})$ and the last equality follows from the inclusion (2.11).

Moreover, the following facts are classical:

• •

if $u\in\mathcal{L}^{1}_{\alpha}(\mathbb{R}^{d})\cap C^{\alpha+\varepsilon}_{loc}(\mathbb{R}^{d})$, $\alpha>0$, then the operator $(-\Delta)^{\frac{\alpha}{2}}$ can be understood pointwise since the integral (2.8) is convergent, and $(-\Delta)^{\frac{\alpha}{2}}u\in C(\mathbb{R}^{d})$ ([garofalo]*Proposition $2.15$);

• •

if $u\in\mathcal{L}^{1}_{-\alpha}(\mathbb{R}^{d})$, $\alpha>0$, then the Riesz potential of $u$ is a convergent integral. Conversely, if $u\notin\mathcal{L}^{1}_{-\alpha}(\mathbb{R}^{d})$ then $(I_{\alpha}*|u|)(x)=\infty$ for a.e. $x\in{\mathbb{R}}^{d}$ ([landkof]*equation $(1.3.10)$).

If $0<\alpha<2$, instead of using the characterisation of the $\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d})$–norm given by (2.2), it’s often more convenient to employ the fractional Gagliardo seminorm defined as

[TABLE]

see [classicbook]*Proposition $1.37$. Namely, the space $\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d})$ can be defined as the completion of $C^{\infty}_{c}(\mathbb{R}^{d})$ with respect to $[u]_{\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d})}$. In particular

[TABLE]

and the inner product is now given by

[TABLE]

The expression of the Gagliardo seminorm (2.14) allows us to deduce some useful properties. As a matter of fact, if we denote by $[\cdot]_{+},[\cdot]_{-}$ respectively positive and negative part, we have that

[TABLE]

(cf. [Musina]*page $3$).

Finally, let $\Omega\subset\mathbb{R}^{d}$ be an open and connected set and $0<\alpha<d$. We define the completion of $C^{\infty}_{c}(\Omega)$ with respect to the norm $\|\cdot\|_{\mathring{H}^{\frac{\alpha}{2}}(\mathbb{R}^{d})}$. Namely, we define $\mathring{H}^{\frac{\alpha}{2}}_{0}(\Omega)$ as

[TABLE]

Note that, for a general domain $\Omega$, the space $\mathring{H}^{\frac{\alpha}{2}}_{0}(\Omega)$ defined by (2.18) is different from

[TABLE]

Nevertheless, our domains of interest will be either $\mathbb{R}^{d}$, or $\overline{B_{R}}^{c}$ with $R>0$, and in these cases the two spaces coincide.

Remark 2.1**.**

To be precise, in [chandler]*Theorem 3.24 or [h_0]*Theorem 3.29, it has been proved that if $\Omega$ is a domain (even unbounded) with continuous boundary then

[TABLE]

where $H^{\frac{\alpha}{2}}$ denotes the inhomogeneous fractional Sobolev space. However, the result remains valid in the homogeneous case by a density argument. Indeed, every $u\in\widetilde{\mathring{H}^{\frac{\alpha}{2}}}(\Omega)$ can be approximated by functions in $\widetilde{{H}^{\frac{\alpha}{2}}}(\Omega)$ by simply multiplying it with a suitable smooth function $\varphi_{\lambda}$ (see Lemma LABEL:converging_H for the precise definition).