Fractional superharmonic functions and the perron method

Universite de Nımes

Equipe MIPAUniversite de Nımes, Site des Carmes

Place Gabriel Peri, 30021 Nımes, Francehttp://mipa.unimes.fr

Fractional superharmonic functions

and the Perron method

for nonlinear integro-di↵erential equations

by

Janne Korvenp

¨

a

¨

a, Tuomo Kuusi

and Giampiero Palatucci

April 2016

Fractional superharmonic functions and the Perron

method for nonlinear integro-di↵erential equations

Janne Korvenpaa · Tuomo Kuusi ·Giampiero Palatucci

Abstract We deal with a class of equations driven by nonlocal, possiblydegenerate, integro-di↵erential operators of di↵erentiability order s 2 (0, 1)and summability growth p > 1, whose model is the fractional p-Laplacianwith measurable coe�cients. We state and prove several results for the cor-responding weak supersolutions, as comparison principles, a priori bounds,lower semicontinuity, and many others. We then discuss the good definitionof (s, p)-superharmonic functions, by also proving some related properties. Wefinally introduce the nonlocal counterpart of the celebrated Perron method innonlinear Potential Theory.

Keywords Quasilinear nonlocal operators · fractional Sobolev spaces ·fractional Laplacian · nonlocal tail · Caccioppoli estimates · obstacle problem ·

The first author has been supported by the Magnus Ehrnrooth Foundation (grant no.ma2014n1, ma2015n3). The second author has been supported by the Academy of Finland.The third author is a member of Gruppo Nazionale per l’Analisi Matematica, la Probabilitae le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica “F. Severi”(INdAM), whose support is acknowledged.

J. Korvenpaa, T. KuusiDepartment of Mathematics and Systems Analysis, Aalto UniversityP.O. Box 110000076 Aalto, FinlandTelefax: +358 9 863 2048E-mail: [email protected] E-mail: [email protected]

G. PalatucciDipartimento di Matematica e Informatica, Universita degli Studi di ParmaCampus - Parco Area delle Scienze, 53/AI-43124 Parma, ItalyTel: +39 521 90 21 11

Laboratoire MIPA, Universite de NımesSite des Carmes - Place Gabriel PeriF-30021 Nımes, FranceTel: +33 466 27 95 57E-mail: [email protected]

Preliminary version – April 13, 2016 – 11:55

2 J. Korvenpaa, T. Kuusi, G. Palatucci

comparison estimates · fractional superharmonic functions · the PerronMethod

Mathematics Subject Classification (2000) Primary: 35D10 · 35B45;Secondary: 35B05 · 35R05 · 47G20 · 60J75

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 Class of (s, p)-superharmonic functions . . . . . . . . . . . . . . . . . . . . . . 51.2 Dirichlet boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1 Algebraic inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 Some recent results on nonlocal fractional operators . . . . . . . . . . . . . . 15

3 Properties of the fractional weak supersolutions . . . . . . . . . . . . . . . . . . . . 183.1 A priori bounds for weak supersolutions . . . . . . . . . . . . . . . . . . . . . 183.2 Comparison principle for weak solutions . . . . . . . . . . . . . . . . . . . . . 223.3 Lower semicontinuity of weak supersolutions . . . . . . . . . . . . . . . . . . . 233.4 Convergence results for weak supersolutions . . . . . . . . . . . . . . . . . . . 25

4 (s, p)-superharmonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.1 Bounded (s, p)-superharmonic functions . . . . . . . . . . . . . . . . . . . . . 314.2 Pointwise behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 Summability of (s, p)-superharmonic functions . . . . . . . . . . . . . . . . . . 364.4 Convergence properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.5 Unbounded comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

5 The Perron method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.1 Poisson modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2 Perron solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

1 Introduction

The Perron method (also known as the PWB method, after Perron, Wiener,and Brelot) is a consolidated method introduced at the beginning of the lastcentury in order to solve the Dirichlet problem for the Laplace equation in agiven open set ⌦ with arbitrary boundary data g; that is,

(

Lu = 0 in ⌦

u = g on the boundary of ⌦,(1)

when L = �. Roughly speaking, the Perron method works by finding the leastsuperharmonic function with boundary values above the given values g. Underan assumption g 2 H1(⌦), the so-called Perron solution coincides with the de-sired Dirichlet energy solution. However, for general g energy methods do notwork and this is precisely the motivation of the Perron method. The methodworks essentially for many other partial di↵erential equations whenever a com-parison principle is available and appropriate barriers can be constructed toassume the boundary conditions. Thus, perhaps surprisingly, it turns out that


Nonlinear integro-di↵erential equations 3

the method extends to the case when the Laplacian operator in (1) is replacedby the p-Laplacian operator (��

p

) (see e.g. [14]) or even by more generalnonlinear operators. Consequently, the Perron method has become a funda-mental tool in nonlinear Potential Theory, as well as in the study of severalbranches of Mathematics and Mathematical Physics when problems as in (1),and the corresponding variational formulations arising from di↵erent contexts.The nonlinear Potential Theory covers a classical field having grown a lot dur-ing the last three decades from the necessity to understand better propertiesof supersolutions, potentials and obstacles. Much has been written about thistopic and the connection with the theory of degenerate elliptic equations; werefer the reader to the exhaustive book [16] by Heinonen, Kilpelainen andMartio, and to the useful lecture notes [31] by Lindqvist.

However – though many important physical contexts can be surely mod-eled using potentials satisfying the Laplace equation or via partial di↵erentialequations as in (1) with the leading term given by a nonlinear operator asfor instance the p-Laplacian with coe�cients – other contexts, as e. g. fromBiology and Financial Mathematics, are naturally described by the fractionalcounterpart of (1), that is, the fractional Laplacian operator (��)s. Recently,a great attention has been focused on the study of problems involving frac-tional Sobolev spaces and corresponding nonlocal equations, both from a puremathematical point of view and for concrete applications, since they naturallyarise in many context when the interactions coming from far are determinant1.

More in general, one can consider a class of fractional Laplacian-type op-erators with nonlinear growth together with a natural inhomogeneity. Accord-ingly, we deal with an extended class of nonlinear nonlocal equations, whichinclude as a particular case some fractional Laplacian-type equations,

Lu(x) =Z

Rn

K(x, y)|u(x)� u(y)|p�2

�

u(x)� u(y)�

dy = 0, x 2 Rn, (2)

where, for any s 2 (0, 1) and any p > 1, K is a suitable symmetric kernel oforder (s, p) with merely measurable coe�cients. The integral may be singularat the origin and must be interpreted in the appropriate sense. We immedi-ately refer to Section 2 for the precise assumptions on the involved quantities.However, in order to simplify, one can just keep in mind the model case whenthe kernel K = K(x, y) coincides with the Gagliardo kernel |x� y|�n�sp; thatis, when the equation in (1) reduces to

(��)sp

u = 0 in Rn,

where the symbol (��)sp

denotes the usual fractional p-Laplacian operator,though in such a case the di�culties arising from having merely measurablecoe�cients disappear.

1 For an elementary introduction to this topic and for a quite wide, but still limited, listof related references we refer to [11].



Let us come back to the celebrated Perron method. To our knowledge,especially in the nonlinear case when p 6= 2, the nonlocal counterpart seemsbasically missing2, and even the theory concerning regularity and related re-sults for the operators in (2) appears to be rather incomplete. Nonetheless,some partial results are known. It is worth citing the higher regularity con-tributions in the case when s is close to 1 proven in the interesting paper [2],recently extended to some extent in [3, 10] for any s 2 (0, 1); together withthe viscosity approach in the recent paper [27], and [5] for related existenceand uniqueness results in the case when p goes to infinity. Also, we would liketo mention the related results involving measure data, as seen in [1, 23, 26],and the fine analysis in the papers [4, 13, 24, 25, 28] where various results forfractional p-eigenvalues have been proven.

First, the main di↵erence with respect to the local case is that for nonlocalequations the Dirichlet condition has to be taken in the whole complementRn\⌦ of the domain, instead of only on the boundary @⌦. This comes from thevery definition of the fractional operators in (2), and it is strictly related tothe natural nonlocality of those operators, and the fact that the behavior ofa function outside the set ⌦ does a↵ect the problem in the whole space (andparticularly on the boundary of ⌦), which is indeed one of the main featurewhy those operators naturally arise in many contexts. On the other hand, sucha nonlocal feature is also one of the main di�culties to be handled when dealingwith fractional operators. For this, some sophisticated tools and techniqueshave been recently developed to treat the nonlocality, and to achieve manyfundamental results for nonlocal equations. As firstly seen in the breakthroughpaper [17] by Kassmann, where he revisited classical Harnack inequalities in acompletely new nonlocal form by incorporating some precise nonlocal terms.This is also the case here, and indeed we have to consider a special quantity,the nonlocal tail of a function u in the ball of radius r > 0 centered in z 2 Rn,given by

Tail(u; z, r) :=

✓

rspZ

Rn\Br(z)

|u(x)|p�1|x� z|�n�sp dx

◆

1p�1

. (3)

The nonlocal tail will be a key-point in the proofs when a fine quantitativecontrol of the long-range interactions, naturally arising when dealing withnonlocal operators as in (2), is needed. This quantity has been introducedin [10] and has been subsequently used in several recent results on the topic(see Section 2 for further details).

In clear accordance with the definition in (3), for any p > 1 and anys 2 (0, 1), we consider the corresponding tail space Lp�1

sp

(Rn) given by

Lp�1

sp

(Rn) :=n

f 2 Lp�1

loc

(Rn) : Tail(f ; 0, 1) < 1o

. (4)

2 As we were finishing this manuscript, we became aware of very recent manuscript [29]having an independent and di↵erent approach to the problem.



In particular, if f 2 Lp�1

sp

(Rn), then Tail(u; z, r) < 1 for all z 2 Rn andr 2 (0,1). It is worth noticing that the two definitions above are very natu-ral, by involving essentially only the leading parameters defining the nonlocalnonlinear operators; i. e., their di↵erentiability order s and their summabilityexponent p. Said this, we can now approach the nonlocal counterpart of thePerron method, which, as is well-known, relies on the concept of superharmonicfunctions. A good definition of nonlocal nonlinear superharmonic functions isneeded3. We thus introduce the (s, p)-superharmonic functions, by stating andproving also their main properties (see Section 4). The (s, p)-superharmonicfunctions constitute the nonlocal counterpart of the p-superharmonic functionsconsidered in the important paper [30]. As expected, in view of the nonlocalityof the involved operators L, this new definition will require to take into accountthe nonlocal tail in (3), in the form of the suitable tail space Lp�1

sp

(Rn). Thisis in clear accordance with the theory encountered in all the aforementionedpapers, when nonlocal operators have to be dealt with in bounded domains.

1.1 Class of (s, p)-superharmonic functions

Definition 1 We say that a function u : Rn ! [�1,1] is an (s, p)-super-harmonic function in an open set ⌦ if it satisfies the following four assump-tions:

(i) u < +1 almost everywhere and u > �1 everywhere in ⌦,

(ii) u is lower semicontinuous (l. s. c.) in ⌦,

(iii) u satisfies the comparison in ⌦ against solutions bounded from above; thatis, if D b ⌦ is an open set and v 2 C(D) is a weak solution in D such thatmax{v, 0} 2 L1(Rn) and u � v on @D and almost everywhere on Rn \D,then u � v in D,

(iv) u� belongs to Lp�1

sp

(Rn).

We say that a function u is (s, p)-subharmonic in⌦ if�u is (s, p)-superharmonicin ⌦; and when both u and �u are (s, p)-superharmonic, we say that u is (s, p)-harmonic.

Remark 1 An (s, p)-superharmonic function is locally bounded from below in⌦ as the lower semicontinuous function attains its minimum on compact setsand it cannot be �1 by the definition.

Remark 2 From the definition it is immediately seen that the pointwise mini-mum of two (s, p)-superharmonic functions is (s, p)-superharmonic as well.

3 We take the liberty to call superharmonic functions appearing in this context as (s, p)-superharmonic emphasizing the (s, p)-order of the involved Gagliardo kernel.



Remark 3 In the forthcoming paper [18] it is shown that the class of (s, p)-superharmonic functions is precisely the class of viscosity supersolutions for (2)(for a more restricted class of kernels).

The next theorem describes the basic properties of (s, p)-superharmonicfunctions, which all seem to be necessary for the theory.

Theorem 1 Suppose that u is (s, p)-superharmonic in an open set ⌦. Then

it has the following properties:

(i) Pointwise behavior.

u(x) = lim infy!x

u(y) = ess lim infy!x

u(y) for every x 2 ⌦.

(ii) Summability. For

t :=

(

(p�1)n

n�sp

, 1 < p < n

s

,

+1, p � n

s

,q := min

⇢

n(p� 1)

n� s, p

�

,

and h 2 (0, s), t 2 (0, t) and q 2 (0, q), u 2 Wh,q

loc

(⌦)\Lt

loc

(⌦)\Lp�1

sp

(Rn).

(iii) Unbounded comparison. If D b ⌦ is an open set and v 2 C(D) is a

weak solution in D such that u � v on @D and almost everywhere on Rn \D, then u � v in D.

(iv) Connection to weak supersolutions. If u is locally bounded in ⌦ or

u 2 W s,p

loc

(⌦), then it is a weak supersolution in ⌦.

As the property (iv) of the above Theorem states, the (s, p)-superharmonicfunctions are very much connected to fractional weak supersolutions, whichby the definition belong locally to the Sobolev space W s,p (see Section 2).Consequently, we prove very general results for the supersolutions u to (2), ase. g. the natural comparison principle given in forthcoming Lemma 6 whichtakes into account what happens outside ⌦, the lower semicontinuity of u (seeTheorem 9), the fact that the truncation of a supersolution is a supersolution aswell (see Theorem 7), the pointwise convergence of sequences of supersolutions(Theorem 10). Clearly, the aforementioned results are expected, but furthere↵orts and a somewhat new approach to the corresponding proofs are neededdue to the nonlocal nonlinear framework considered here (see the observationsat the beginning of Section 1.3 below).

As said before, for the nonlocal Perron method the (s, p)-superharmonicand (s, p)-subharmonic functions are the building blocks. Thus are now in aposition to introduce this concept.



1.2 Dirichlet boundary value problems

As in the classical local framework, in order to solve the boundary value prob-lem, we have to construct two class of functions leading to the upper Perronsolution and the lower Perron solution.

Definition 2 (Perron solutions) Let ⌦ be an open set. Assume that g 2Lp�1

sp

(Rn). The upper class Ug

of g consists of all functions u such that

(i) u is (s, p)-superharmonic in ⌦,

(ii) u is bounded from below in ⌦,

(iii) lim inf⌦3y!x

u(y) � ess lim supRn\⌦3y!x

g(y) for all x 2 @⌦,

(iv) u = g almost everywhere in Rn \⌦.

The lower class is Lg

:= {u : �u 2 U�g

}. The function Hg

:= inf {u : u 2 Ug

}is the upper Perron solution with boundary datum g in ⌦, where the infimum istaken pointwise in ⌦, and H

g

:= sup {u : u 2 Lg

} is the lower Perron solution

with boundary datum g in ⌦.

A few important observations are in order.

Remark 4 Notice that when g is continuous in a vicinity of the boundary of⌦, we can replace ess lim sup

y!x

g(y) with g(x) in Definition 2(iii) above.

Remark 5 We could also consider more general Perron solutions by droppingthe conditions (ii)–(iii) in Definition 2 above. However, in such a case it doesnot seem easy to exclude the possibility that the corresponding upper Perronsolution is identically �1 in ⌦ even for simple boundary value functions suchas constants.

In the case of the fractional Laplacian, we have the Poisson formula for thesolution u in a unit ball with boundary values g as

u(x) = cn,s

�

1� |x|2�

s

Z

Rn\B1(0)

g(y)�

|y|2 � 1��s |x� y|�n dy,

for every x 2 B1

(0); see e. g. [17], and also [34, 39] for related applications,and [12] for explicit computations. Using the Poisson formula one can considersimple examples in the unit ball.

Example 1 Taking the function g(x) =�

�|x|2�1�

�

s�1

, g 2 L1

2s

(Rn), as boundaryvalues in the Poisson formula above, the integral does not converge. Thisexample suggests that in this case H

g

⌘ Hg

⌘ +1 in B1

(0). The exampletells that one can not expect bounded solutions for all g 2 L1

2s

(Rn).



Example 2 Let us consider the previous example with g reflected to the neg-ative side in the half space, i. e.

g(x) :=

8

>

>

<

>

>

:

�

�|x|2 � 1�

�

s�1

, xn

> 0,

0, xn

= 0,

��

�|x|2 � 1�

�

s�1

, xn

< 0.

Then the “solution” via Poisson formula, for x 2 B1

, is

u(x) =

8

>

<

>

:

+1 xn

> 0,

0, xn

= 0,

�1, xn

< 0,

which is suggesting that we should now have Hg

⌘ +1 and Hg

⌘ �1 inB

1

(0). In view of this example it is reasonable to conjecture that the resolu-tivity fails in the class L1

2s

(Rn).

In accordance with the classical Perron theory, one can prove that the upperand lower nonlocal Perron solutions act in the expected order (see Lemma 17),and that the boundedness of the boundary values assures that the nonlocalPerron classes are non-empty (see Lemma 18). Then, we prove one of themain results, which is the nonlocal counterpart of the fundamental alternativetheorem for the classical nonlinear Potential Theory.

Theorem 2 The Perron solutions Hg

and Hg

can be either identically +1in ⌦, identically �1 in ⌦, or (s, p)-harmonic in ⌦, respectively.

Finally, we approach the problem of resolutivity in the nonlocal framework.We state and prove a basic, hopefully useful, existence and regularity result forthe solution to the nonlocal Dirichlet boundary value problem, under suitableassumptions on the boundary values and the domain ⌦ (see Theorem 17). Wethen show that if there is a solution to the nonlocal Dirichlet problem then itis necessarily the nonlocal Perron solution (see Lemma 19).

1.3 Conclusion

As one can expect, the main issues when dealing with the wide class of op-erators L in (2) whose kernel K satisfies fractional di↵erentiability for any

s 2 (0, 1) and p-summability for any p > 1, lie in their very definition, whichcombines the typical issues given by its nonlocal feature together with theones given by its nonlinear growth behavior; also, further e↵orts are neededdue to the presence of merely measurable coe�cient in the kernel K. As aconsequence, we can make use neither of some very important results recently



introduced in the nonlocal theory, as the by-now classical s-harmonic exten-sion framework provided by Ca↵arelli and Silvestre in [6], nor of various toolsas, e. g., the strong three-commutators estimates introduced in [7,8] to deducethe regularity of weak fractional harmonic maps (see also [40]), the strongbarriers and density estimates in [37, 39], the pseudo-di↵erential commutatorand energy estimates in [35, 36], and many other successful techniques whichseem not to be trivially adaptable to the nonlinear framework considered here.Increased di�culties are due to the non-Hilbertian structure of the involvedfractional Sobolev spaces W s,p when p is di↵erent than 2.

Although some of our complementary results are well-known in the linearnonlocal case, i.e. when L reduces to the pure fractional Laplacian operator(��)s, all our proofs are new even in this case. Indeed, since we actually dealwith very general operators with measurable coe�cients, we have to change theapproach to the problem. As a concrete example, for instance, let us mentionthat the proof that the supersolutions can be chosen to be lower semicontinu-ous functions will follow by a careful interpolation of the local and the nonlocalcontributions via a new supremum estimate with tail (see Theorem 4). On thecontrary, in the purely fractional Laplacian case when p = 2, the proof of thesame result is simply based on a characterization of supersolutions somewhatsimilar to the super mean value formula for classical superharmonic functions(see, e. g., [38, Proposition A4]), which is not available in our general nonlin-ear nonlocal framework due to the presence of possible irregular coe�cients inthe kernel K. While in the purely (local) case when s = 1, for the p-Laplaceequation, the same result is a consequence of weak Harnack estimates (see,e. g., [16, Theorem 3.51-3.63]).

All in all, in our opinion, the contribution in the present paper is twofold.We introduce the nonlocal counterpart of the Perron method, by also introduc-ing the concept of (s, p)-superharmonic functions, and extending very generalresults for supersolutions to the nonlocal Dirichlet problem in (2), hence estab-lishing a powerful framework which could be useful for developing a completefractional nonlinear Potential Theory; in this respect, we could already referto the forthcoming papers [18–20], where all the machinery, and in particularthe good definition of fractional superharmonic functions developed here, havebeen required in order to deal with the nonlocal obstacle problem as well asto investigate di↵erent notions of solutions to nonlinear fractional equations ofp-Laplace type. Moreover, since we derive all those results for a general classof nonlinear integro-di↵erential operators with measurable coe�cients via ourapproach by also taking into account the nonlocal tail contributions, we obtainalternative proofs that are new even in the by-now classical case of the purefractional Laplacian operator (��)s.

The paper is organized as follows. Firstly, an e↵ort has been made to keepthe presentation self-contained, so that in Section 2 we collect some prelimi-



nary observations, and very recent results for fractional weak supersolutionsadapted to our framework. In Section 3, we present some independent generalresults to be applied here and elsewhere when dealing with nonlocal nonlinearoperators (Section 3.1), and we state and prove the most essential propertiesof fractional weak supersolutions (Sections 3.2–3.4). Section 4 is devoted tothe concept of (s, p)-superharmonic functions: we prove Theorem 1 and otherrelated results, by also investigating their connection to the fractional weak su-persolutions. Finally, in Section 5 we focus on the nonlocal Dirichlet boundaryvalue problems and collect some useful tools, introducing the natural nonlocalPoisson modification (Section 5.1), as well as the nonlocal Perron method, byproving the corresponding properties and the main related results as the onesin Theorem 2 and the resolutivity presented in forthcoming Lemma 19; seeSection 5.2.

Acknowledgments. This paper was partially carried out while Giampiero Pa-latucci was visiting the Department of Mathematics and Systems Analysis atAalto University School of Science in Helsinki, supported by the Academy ofFinland. The authors would like to thank Professor Juha Kinnunen for thehospitality and the stimulating discussions. A special thank also to AgneseDi Castro for her useful observations on a preliminary version of this paper.

Finally, we would like to thank Erik Lindgren, who has kindly informed usof his paper [29] in collaboration with Peter Lindqvist, where they deal witha general class of fractional Laplace equations with bounded boundary data,in the case when the operators L in (2) does reduce to the pure fractional p-Laplacian (��)s

p

without coe�cients. This very relevant paper contains severalimportant results, as a fractional Perron method and a Wiener resolutivitytheorem, together with the subsequent classification of the regular points, insuch a nonlinear fractional framework. It could be interesting to compare thoseresults together with the ones presented here.

2 Preliminaries

In this section, we state the general assumptions on the quantities we aredealing with. We keep these assumptions throughout the paper.

First of all, we recall that the class of integro-di↵erential equations in whichwe are interested is the following

Lu(x) = P.V.

Z

Rn

K(x, y)|u(x)� u(y)|p�2

�

u(x)� u(y)�

dy = 0, x 2 ⌦. (5)

The nonlocal operator L in the display above (being read a priori in the prin-cipal value sense) is driven by its kernel K : Rn ⇥ Rn ! [0,1), which is a



measurable function satisfying the following property:

⇤�1 K(x, y)|x� y|n+sp ⇤ for a. e. x, y 2 Rn, (6)

for some s 2 (0, 1), p > 1, ⇤ � 1. We immediately notice that in the specialcase when p = 2 and ⇤ = 1 we recover (up to a multiplicative constant) thewell-known fractional Laplacian operator (��)s.

Moreover, notice that the assumption on K can be weakened as follows

⇤�1 K(x, y)|x� y|n+sp ⇤ for a. e. x, y 2 Rn s. t. |x� y| 1, (7)

0 K(x, y)|x� y|n+⌘ M for a. e. x, y 2 Rn s. t. |x� y| > 1, (8)

for some s, p, ⇤ as above, ⌘ > 0 and M � 1, as seen, e. g., in the recent seriesof papers by Kassmann (see for instance the more general assumptions in thebreakthru paper [17]). In the same sake of generalizing, one can also considerthe operator L = L

�

defined by

L�

u(x) = P.V.

Z

Rn

K(x, y)�(u(x)� u(y)) dy, x 2 ⌦, (9)

where the real function � is assumed to be continuous, satisfying �(0) = 0together with the monotonicity property

��1|t|p �(t)t �|t|p for every t 2 R \ {0},

for some � > 1, and some p as above (see, for instance, [23]).However, for the sake of simplicity, we will take �(t) = |t|p�2t and we

will work under the assumption in (6), since the assumptions in (7)–(9) wouldbring no relevant di↵erences in all the forthcoming proofs. Moreover, let usremark that we will assume that the kernel K is symmetric, and once againthis is not restrictive, in view of the weak formulation presented in forthcomingDefinition 3, since one may always define the corresponding symmetric kernelKsym given by

Ksym(x, y) :=1

2

⇣

K(x, y) +K(y, x)⌘

.

We now call up the definition of the nonlocal tail Tail(f ; z, r) of a function

f in the ball of radius r > 0 centered in z 2 Rn. We have

Tail(f ; z, r) :=

✓

rspZ

Rn\Br(z)

|f(x)|p�1|x� z|�n�sp dx

◆

1p�1

, (10)

for any function f initially defined in Lp�1

loc

(Rn). As mentioned in the intro-duction, this quantity will play an important role in the rest of the paper.The nonlocal tail has been introduced in [10], and, as seen subsequently in



several recent papers (see e. g., [3, 4, 9, 15, 22–25] and many others4), it hasbeen crucial in order to control in a quantifiable way the long-range interac-tions which naturally appear when dealing with nonlocal operators of the typeconsidered here in (5). In the following, when the center point z will be clearfrom the context, we shall use the shorter notation Tail(f ; r) ⌘ Tail(f ; z, r).In accordance with (10), we recall the definition of the tail space Lp�1

sp

givenin (4), and we immediately notice that one can use the following equivalentdefinition

Lp�1

sp

(Rn) =n

f 2 Lp�1

loc

(Rn) :

Z

Rn

|f(x)|p�1(1 + |x|)�n�sp dx < 1o

.

As expected, one can check that L1(Rn) ⇢ Lp�1

sp

(Rn) and W s,p(Rn) ⇢Lp�1

sp

(Rn), where we denoted by W s,p(Rn) the usual fractional Sobolev spaceof di↵erentiability order s 2 (0, 1) and summability exponent p � 1, which isdefined as follows

W s,p(Rn) :=

⇢

v 2 Lp(Rn) :|v(x)� v(y)||x� y|

np +s

2 Lp(Rn ⇥Rn)

�

;

i. e., an intermediary Banach space between Lp(Rn) and W 1,p(Rn) endowedwith the natural norm

kvkW

s,p(Rn

)

:= kvkL

p(Rn

)

+ [v]W

s,p(Rn

)

=

✓

Z

Rn

|v|p dx◆

1p

+

✓

Z

Rn

Z

Rn

|v(x)� v(y)|p

|x� y|n+sp

dxdy

◆

1p

.

In a similar way, it is possible to define the fractional Sobolev spaces W s,p(⌦)in a domain ⌦ ⇢ Rn. By W s,p

0

(⌦) we denote the closure of C10

(⌦) inW s,p(Rn). Conversely, if v 2 W s,p(⌦0) with ⌦ b ⌦0 and v = 0 outside of⌦ almost everywhere, then v has a representative in W s,p

0

(⌦) as well. For thebasic properties of these spaces and some related topics we refer to [11] andthe references therein.

Let us denote the positive part and the negative one of a real valued func-tion u by u

+

:= max{u, 0} and u� := max{�u, 0}, respectively. We are nowready to provide the definitions of sub- and supersolutions u to the class ofintegro-di↵erential problems we are interested in.

Definition 3 A function u 2 W s,p

loc

(⌦) such that u� belongs Lp�1

sp

(Rn) is afractional weak p-supersolution of (5) ifZ

Rn

Z

Rn

|u(x)� u(y)|p�2

�

u(x)� u(y)��

⌘(x)� ⌘(y)�

K(x, y) dxdy � 0 (11)

4 When needed, our definition of Tail can also be given in a more general way by replacingthe ball Br and the corresponding rsp term by an open bounded set E ⇢ Rn and its rescaledmeasure |E|sp/n, respectively. This is not the case in the present paper.



for every nonnegative ⌘ 2 C10

(⌦). A function u is a fractional weak p-

subsolution if �u is a fractional weak p-supersolution, and u is a fractional

weak p-solution if it is both fractional weak p-sub- and p-supersolution.

We often suppress p from notation and say simply that u is a weak superso-lution in ⌦. Above ⌘ 2 C1

0

(⌦) can be replaced by ⌘ 2 W s,p

0

(D) with everyD b ⌦. Furthermore, it can be extended to a W s,p-function in the wholeRn (see, e. g., Section 5 in [11]). It is worth noticing that the summabilityassumption of u� belonging to the tail space Lp�1

sp

(Rn) is what one expectsin the nonlocal framework considered here. This is one of the novelty withrespect to the analog of the definition of supersolutions in the local case, saywhen s = 1, and it is necessary since here one has to use in a precise way thedefinition in (10) to deal with the long-range interactions; see Remark 6 below,and also, the regularity estimates in the aforementioned papers [9, 10, 20, 23].It is also worth noticing that in Definition 3 it makes no di↵erence to assumeu 2 Lp�1

sp

(Rn) instead of u� 2 Lp�1

sp

(Rn), as the next lemma implies.

Lemma 1 Let u be a weak supersolution in B2r

(x0

). Then, for c ⌘ c(n, p, s),

Tail(u;x0

, r)

c⇣

rsp�1�n

p�1 [u]W

h,p�1(Br(x0))

+ r�n

p�1 kukL

p�1(Br(x0))

+Tail(u�;x0

, r)⌘

with

h = max

⇢

0,sp� 1

p� 1

�

< s.

In particular, if u is a weak supersolution in an open set ⌦, then u 2 Lp�1

sp

(Rn).

Proof Firstly, we write the weak formulation, for nonnegative � 2 C10

(Br/2

(x0

))such that � ⌘ 1 in B

r/4

(x0

), with 0 � 1 and |r�| 8/r. We have

0 Z

Br(x0)

Z

Br(x0)

|u(x)� u(y)|p�2

�

u(x)� u(y)��

�(x)� �(y)�

K(x, y) dxdy

+

Z

Rn\Br(x0)

Z

Br/2(x0)

|u(x)� u(y)|p�2

�

u(x)� u(y)�

�(x)K(x, y) dxdy

= I1

+ I2

.

The first term is easily estimated using |�(x)� �(y)| 8|x� y|/r as

I1

c

rmin{sp,1} [u]p�1

W

h,p�1(Br(x0))

In order to estimate the second term, we have

|u(x)� u(y)|p�2

�

u(x)� u(y)�

2p�1

�

up�1

+

(x) + up�1

� (y)�

� up�1

+

(y),



and thus

I2

c r�sp kukp�1

L

p�1(Br(x0))

+c rn�spTail(u�;x0

, r)p�1� rn�sp

cTail(u;x

0

, r)p�1.

By combining the preceding displays we get the desired estimates. The secondstatement plainly follows by an application of Holder’s Inequality.

Remark 6 The left-hand side of the inequality in (11) is finite for every u 2W s,p

loc

(⌦) \ Lp�1

sp

(Rn) and for every ⌘ 2 C10

(⌦). Indeed, for an open set D

such that supp ⌘ ⇢ D b ⌦, we have by Holder’s Inequality�

�

�

�

Z

Rn

Z

Rn

|u(x)� u(y)|p�2

�

u(x)� u(y)��

⌘(x)� ⌘(y)�

K(x, y) dxdy

�

�

�

�

c

Z

D

Z

D

|u(x)� u(y)|p�1|⌘(x)� ⌘(y)| dxdy

|x� y|n+sp

+ c

Z

Rn\D

Z

supp ⌘

�

|u(x)|p�1 + |u(y)|p�1

�

|⌘(x)||z � y|�n�sp dxdy

c [u]p�1

W

s,p(D)

[⌘]W

s,p(D)

+ c kukp�1

L

p(D)

k⌘kL

p(D)

+ cTail(u; z, r)p�1k⌘kL

1(D)

,

where r := dist(supp ⌘, @D) > 0, z 2 supp ⌘, and c ⌘ c(n, p, s,⇤, r,D). Wenotice that all the terms in the right-hand side are finite since u, ⌘ 2 W s,p(D)and Tail(u; z, r) < 1.

2.1 Algebraic inequalities

We next collect some elementary algebraic inequalities. In order to simplifythe notation in the weak formulation (11), from now on we denote by

L(a, b) := |a� b|p�2(a� b), a, b 2 R. (12)

Notice that L(a, b) is increasing with respect to a and decreasing with respectto b.

Lemma 2 ([20, Lemma 2.1-2.2]). Let 1 < p 2 and a, b, a0, b0 2 R. Then

|L(a, b)� L(a0, b0)| 4|a� a0 � b+ b0|p�1. (13)

Let p � 2 and a, b, a0, b0 2 R. Then

|L(a, b)� L(a0, b)| c |a� a0|p�1 + c |a� a0||a� b|p�2,

and

|L(a, b)� L(a, b0)| c |b� b0|p�1 + c |b� b0||a� b|p�2,

where c depends only on p.



Remark 7 Finally, we would like to make the following observation. In the restof the paper, we often use the fact that there is a constant c > 0 dependingonly on p such that

1

c

�

|a|p�2a� |b|p�2b�

(a� b)

(|a|+ |b|)p�2(a� b)2 c,

when a, b 2 R, a 6= b. In particular,�

|a|p�2a� |b|p�2b�

(a� b) � 0, a, b 2 R. (14)

2.2 Some recent results on nonlocal fractional operators

In this section, we recall some recent results for fractional weak sub- andsupersolutions, which we adapted to our framework for the sake of the reader;see [9, 10, 20] for the related proofs. Notice that the proofs of Theorems 3and 4 below make sense even if we assume u 2 W s,p

loc

(⌦) \ Lp�1

sp

(Rn) insteadof u 2 W s,p(Rn).

Firstly, we state a general inequality which shows that the natural ex-tension of the Caccioppoli inequality to the nonlocal framework has to takeinto account a suitable tail. For other fractional Caccioppoli-type inequalities,though not taking into account the tail contribution, see [32,33], and also [13].

Theorem 3 (Caccioppoli estimate with tail) ( [10, Theorem 1.4]). Letu be a weak supersoltion to (5). Then, for any B

r

⌘ Br

(z) ⇢ ⌦ and any

nonnegative ' 2 C10

(Br

), the following estimate holds trueZ

Br

Z

Br

K(x, y)|w�(x)'(x)� w�(y)'(y)|p dxdy

c

Z

Br

Z

Br

K(x, y)�

max{w�(x), w�(y)}�

p|'(x)� '(y)|p dxdy (15)

+c

Z

Br

w�(x)'p(x) dx

supy 2 supp'

Z

Rn\Br

K(x, y)wp�1

� (x) dx

!

,

where w� := (u�k)� for any k 2 R, K is any measurable kernel satisfying (6),and c depends only on p.

Remark 8 We underline that the estimate in (15) holds by replacing w� withw

+

:= (u� k)+

in the case when u is a fractional weak subsolution.

A first natural consequence is the local boundedness of fractional weaksubsolutions, as stated in the following



Theorem 4 (Local boundedness) ([10, Theorem 1.1 and Remark 4.2]). Letu be a weak supersolution to (5) and let B

r

⌘ Br

(z) ⇢ ⌦. Then the following

estimate holds true

ess supBr/2

u �Tail(u+

;x0

, r/2) + c ��✓

Z

Br

up

+

dx

◆

1p

, (16)

where Tail(·) is defined in (10), � = (p� 1)n/sp2, the real parameter � 2 (0, 1],and the constant c depends only on n, p, s, and ⇤.

It is worth noticing that the parameter � in (16) allows a precise interpola-tion between the local and nonlocal terms. Combining Theorem 3 togetherwith a nonlocal Logarithmic-Lemma (see [10, Lemma 1.3]), one can prove thatboth the p-minimizers and weak solutions enjoy oscillation estimates, whichnaturally yield Holder continuity (see Theorem 5) and some natural Harnackestimates with tail, as the nonlocal weak Harnack estimate presented in The-orem 6 below.

Theorem 5 (Holder continuity) ([10, Theorem 1.2]). Let u be a weak solu-

tion to (5). Then u is locally Holder continuous in ⌦. In particular, there are

positive constants ↵, ↵ < sp/(p� 1), and c, both depending only on n, p, s,⇤,

such that if B2r

(x0

) ⇢ ⌦, then

oscB%(x0)

u c⇣%

r

⌘

↵

"

Tail(u;x0

, r) +

✓

Z

B2r(x0)

|u|p dx◆

1p

#

holds whenever % 2 (0, r], where Tail(·) is defined in (10).

Theorem 6 (Nonlocal weak Harnack inequality) ([9, Theorem 1.2]). bea weak supersolution to (5) such that u � 0 in B

R

⌘ BR

(x0

) ⇢ ⌦. Let

t :=

(

(p�1)n

n�sp

, 1 < p < n

s

,

+1, p � n

s

.(17)

Then the following estimate holds for any Br

⌘ Br

(x0

) ⇢ BR/2

(x0

) and for

any t < t

✓

Z

Br

ut dx

◆

1t

c ess infB2r

u+ c⇣ r

R

⌘

spp�1

Tail(u�;x0

, R),

where Tail(·) is defined in (10), and the constant c depends only on n, p, s,

and ⇤.

To be precise, the case p � n

s

was not treated in the proof of the weak Harnackwith tail in [9], but one may deduce the result in this case by straightforwardmodifications.



As expected, the contribution given by the nonlocal tail has again to beconsidered and the result is analogous to the local case if u is nonnegative inthe whole Rn.

We finally conclude this section by recalling three results for the solutionto the obstacle problem in the fractional nonlinear framework we are dealingin. First, we consider the following set of functions,

Kg,h

(⌦,⌦0) =n

u 2 W s,p(⌦0) : u � h a. e. in ⌦, u = g a. e. on Rn \⌦o

,

where ⌦ b ⌦0 are open bounded subsets of Rn, h : Rn ! [�1,1) is theobstacle, and g 2 W s,p(⌦0) \ Lp�1

sp

(Rn) determines the boundary values. Thesolution u 2 K

g,h

(⌦,⌦0) to the obstacle problem satisfies

hA(u), v � ui � 0 for all v 2 Kg,h

(⌦,⌦0),

where the functional A(u) is defined, for all w 2 Kg,h

(⌦,⌦0) \W s,p

0

(⌦), as

hA(u), wi :=Z

Rn

Z

Rn

L(u(x), u(y))�

w(x)� w(y)�

K(x, y) dxdy.

The results needed here are the uniqueness, the fact that such a solution is aweak supersolution and/or a weak solution to (5), and the continuity of thesolution up to the boundary under precise assumptions on the functions g, hand the set ⌦.

Theorem 7 (Solution to the nonlocal obstacle problem) ([20, Theorem1]). There exists a unique solution to the obstacle problem in K

g,h

(⌦,⌦0).Moreover, the solution to the obstacle problem is a weak supersolution to (5)in ⌦.

Corollary 1 ([20, Corollary 1]). Let u be the solution to the obstacle problem

in Kg,h

(⌦,⌦0). If Br

⇢ ⌦ is such that

ess infBr

(u� h) > 0,

then u is a weak solution to (5) in Br

. In particular, if u is lower semicontin-

uous and h is upper semicontinuous in ⌦, then u is a weak solution to (5) in⌦

+

:=�

x 2 ⌦ : u(x) > h(x)

.

Theorem 8 ([20, Theorem 9]) Suppose that every x0

2 @⌦ satisfies

inf0<r<r0

|(Rn \⌦) \Br

(x0

)||B

r

(x0

)| � �⌦

(18)

for some r0

> 0 and �⌦

2 (0, 1), and suppose that g 2 Kg,h

(⌦,⌦0). Let u solve

the obstacle problem in Kg,h

(⌦,⌦0). If g is continuous in ⌦0 and h is either

continuous in ⌦ or h ⌘ �1, then u is continuous in ⌦0.

Notice that if D and ⌦ are open sets such that D b ⌦, we always find anopen set U such that D b U b ⌦ and Rn \ U satisfies the measure densitycondition (18).



3 Properties of the fractional weak supersolutions

In order to prove all the main results in the present manuscript and to de-velop the basis for the fractional nonlinear Potential Theory, we need to per-form careful computations on the strongly nonlocal form of the operators Lin (5). Hence, it was important for us to understand how to modify the clas-sical techniques in order to deal with nonlocal integro-di↵erential energies, inparticular to manage the contributions coming from far. Therefore, in thissection we state and prove some general and independent results for fractionalweak supersolutions, to be applied here in the rest of the paper. We providethe boundedness from below and some precise control from above of the frac-tional energy of weak supersolutions, which could have their own interest inthe analysis of equations involving the (nonlinear) fractional Laplacian and re-lated nonlinear integro-di↵erential operators. Next, we devote our attention tothe essential properties of the weak fractional supersolutions, by investigatingnatural comparison principles, and lower semicontinuity. We then discuss thepointwise convergence of sequences of supersolutions and other related results.Our results aim at constituting the fractional counterpart of the basis of theclassical nonlinear Potential Theory.

3.1 A priori bounds for weak supersolutions

The next result states that weak supersolutions are locally essentially boundedfrom below.

Lemma 3 Let v be a weak supersolution in ⌦, let h 2 Lp�1

sp

(Rn) and assume

that h v 0 almost everywhere in Rn. Then, for all D b ⌦ there is a

constant C ⌘ C(n, p, s,⇤,⌦, D, h) such that

ess infD

v � �C.

Proof Let B2r

(x0

) ⇢ ⌦. Let 1 �0 < � 2 and ⇢ = (� � �0)r/2. ThenB

2⇢

(z) ⇢ B�r

(x0

) ⇢ ⌦ for a point z 2 B�

0r

(x0

). Thus, using the fact thatv� = �v � 0 is a weak subsolution, we can apply the estimate in Theorem 4choosing the interpolation parameter � = 1 there. We have

ess supB⇢(z)

v� Tail(v�; z, ⇢) + c

Z

B2⇢(z)

vp�(x) dx

!

1p

.

Since h v, the tail term can be estimated as follows

Tail(v�; z, ⇢) c

⇢spZ

B�r(x0)\B⇢(z)

hp�1

� (x)|x� z|�n�sp dx

!

1p�1



+ c

⇢spZ

Rn\B�r(x0)

hp�1

� (x)|x� z|�n�sp dx

!

1p�1

c

⇢spZ

B�r(x0)

hp�1

� (x)⇢�n�sp dx

!

1p�1

+ c

⇢spZ

Rn\B�r(x0)

hp�1

� (x)⇣ ⇢

�r|x� x

0

|⌘�n�sp

dx

!

1p�1

c (� � �0)� n

p�1

"

Z

B�r(x0)

hp�1

� (x) dx

!

1p�1

+Tail(h�;x0

,�r)

#

c (� � �0)� n

p�1

"

Z

B2r(x0)

hp�1

� (x) dx

!

1p�1

+Tail(h�;x0

, r)

#

.

For the average term, in turn,

|B�r

(x0

)||B

2⇢

(z)| =

✓

�r

2⇢

◆

n

=

✓

�

� � �0

◆

n

,

and thus by Young’s Inequality, we obtain

Z

B2⇢(z)

vp�(x) dx

!

1p

c (� � �0)�n

p

Z

B�r(x0)

vp�(x) dx

!

1p

c

ess supB�r(x0)

v�

!

1p

(� � �0)�n

Z

B2r(x0)

hp�1

� (x) dx

!

1p

1

2ess supB�r(x0)

v� + c (� � �0)� n

p�1

Z

B2r(x0)

hp�1

� (x) dx

!

1p�1

.

Since the estimates above hold for every z 2 B�

0r

(x0

), we have after combiningthe estimates for tail and average terms

ess supB�0r

v�

1

2ess sup

B�r

v� + c (� � �0)� n

p�1

"

✓

Z

B2r

hp�1

� dx

◆

1p�1

+Tail(h�;x0

, r)

#

.

Now, a standard iteration argument yields

ess supBr(x0)

v� c

"

Z

B2r(x0)

hp�1

� dx

!

1p�1

+Tail(h�;x0

, r)

#

.

which is bounded by a constant independent of v since h� 2 Lp�1

sp

(Rn).



To finish the proof, let D b ⌦. We can cover D by finitely many ballsB

ri(xi

), i = 1, . . . , N , with B2ri(xi

) ⇢ ⌦, and the claim follows since

ess infD

v � � max1iN

ess supBri (xi)

v� � �C.

From Theorem 3 we can deduce a Caccioppoli-type estimate as in thefollowing

Lemma 4 Let M > 0. Suppose that u is a weak supersolution in B2r

⌘ B2r

(z)such that u M in B

3r/2

. Then, for a positive constant c ⌘ c(n, p, s,⇤), itholds

Z

Br

Z

Br

|u(x)� u(y)|p

|x� y|n+sp

dxdy c r�spHp, (19)

where

H := M +

✓

Z

B3r/2

up

�(x) dx

◆

1p

+Tail(u�; z, 3r/2).

Proof Let � 2 C10

(B4r/3

) such that 0 � 1, � = 1 in Br

, and |D�| c/r.Setting w := 2H � u, we get

0 1

|Br

|

Z

Rn

Z

Rn

L(u(x), u(y))�

w(x)�p(x)� w(y)�p(y)�

K(x, y) dxdy

= � 1

|Br

|

Z

B3r/2

Z

B3r/2

L(w(x), w(y))�

w(x)�p(x)� w(y)�p(y)�

K(x, y) dxdy

+2

|Br

|

Z

Rn\B3r/2

Z

B3r/2

L(u(x), u(y))w(x)�p(x)K(x, y) dxdy

=: �I1

+ 2I2

. (20)

Following the proof of Theorem 3, by assuming w(x) � w(y), we can deducethat

I1

� 1

c

Z

B3r/2

Z

B3r/2

|u(x)� u(y)|p

|x� y|n+sp

�

max�

�(x), �(y) �

p

dxdy

� c

Z

B3r/2

Z

B3r/2

�

2H � u(x)�

p

|�(x)� �(y)|p

|x� y|n+sp

dxdy

� 1

c

Z

Br

Z

Br

|u(x)� u(y)|p

|x� y|n+sp

dxdy � c r�spHp. (21)

Furthermore,

I2

c

Z

Rn\B3r/2

Z

B4r/3

�

u(x)� u(y)�

p�1

+

�

2H � u(x)�

|x� y|�n�sp dxdy

c

Z

Rn\B3r/2

Z

B4r/3

�

Hp�1 + up�1

� (y)��

2H + u�(x)�

|y � z|�n�sp dxdy



c r�spHp + cH

Z

Rn\B3r/2

up�1

� (y)|y � z|�n�sp dy

c r�spHp, (22)

where, in particular, we used Jensen’s Inequality to estimateZ

B4r/3

u�(x) dx ✓

Z

B4r/3

up

�(x) dx

◆

1p

cH.

By combining (20) with (21) and (22), we plainly obtain the estimate in (19).

Using the previous result we may prove a uniform bound in W s,p.

Lemma 5 Let M > 0 and let h 2 Lp�1

sp

(Rn) with h M . Let u be a weak

supersolution in ⌦ such that u � h almost everywhere in Rn and u M

almost everywhere in ⌦. Then, for all D b ⌦ there is a constant C ⌘C(n, p, s,⇤,⌦, D,M, h) such that

Z

D

Z

D

|u(x)� u(y)|p

|x� y|n+sp

dxdy C. (23)

Proof Let D b ⌦ and denote d := dist(D, @⌦) > 0. We can cover the diagonalD :=

�

(x, y) 2 D ⇥D : |x� y| < d

4

of D ⇥D with finitely many sets of theform B

d/2

(zi

) ⇥ Bd/2

(zi

), i = 1, . . . , N , such that Bd

(zi

) ⇢ ⌦. By Lemma 3we can assume that u is essentially bounded in D by a constant independentof u. Since u M is a weak supersolution in B

d

(zi

) and u � h 2 Lp�1

sp

(Rn),we have by Lemma 4 that

Z

Bd/2(zi)

Z

Bd/2(zi)

|u(x)� u(y)|p

|x� y|n+sp

dxdy C 0

for every i = 1, . . . , N , where C 0 ⌘ C 0(n, p, s,⇤, d,M, h). Thus, we can splitthe integral in (23) as follows

Z

D

Z

D

|u(x)� u(y)|p

|x� y|n+sp

dxdy N

X

i=1

Z

Bd/2(zi)

Z

Bd/2(zi)

|u(x)� u(y)|p

|x� y|n+sp

dxdy

+

ZZ

(D⇥D)\D

|u(x)� u(y)|p

|x� y|n+sp

dxdy.

Now, notice that the first term in the right-hand side of the preceding inequal-ity is bounded from above by

N

X

i=1

Z

Bd/2(zi)

Z

Bd/2(zi)

|u(x)� u(y)|p

|x� y|n+sp

dxdy NC 0;

and the second term byZ

D

Z

D

|u(x)� u(y)|p

(d/4)n+sp

dxdy C 00|⌦0|2

with C 00 independent of u. Combining last three displays yields (23).



3.2 Comparison principle for weak solutions

We next prove a comparison principle for weak sub- and supersolution, whichtypically constitutes a powerful tool, playing a fundamental role in the wholePDE theory.

Lemma 6 (Comparison Principle) Let ⌦ b ⌦0 be bounded open subsets

of Rn. Let u 2 W s,p(⌦0) be a weak supersolution to (5) in ⌦, and let v 2W s,p(⌦0) be a weak subsolution to (5) in ⌦ such that u � v almost everywhere

in Rn \⌦. Then u � v almost everywhere in ⌦ as well.

Proof Consider the function ⌘ := (u � v)�. Notice that ⌘ is a nonnegativefunction in W s,p

0

(⌦). For this, we can use it as a test function in (11) for bothu, v 2 W s,p(⌦0) and, by summing up, we get

0 Z

Rn

Z

Rn

|u(x)� u(y)|p�2

�

u(x)� u(y)��

⌘(x)� ⌘(y)�

K(x, y) dxdy (24)

�Z

Rn

Z

Rn

|v(x)� v(y)|p�2

�

v(x)� v(y)��

⌘(x)� ⌘(y)�

K(x, y) dxdy.

It is now convenient to split the integrals above by partitioning the whole Rn

into separate sets comparing the values of u with those of v, so that, from (24)we get

0 Z

{u<v}

Z

{u<v}

�

L(u(x), u(y))� L(v(x), v(y))��

⌘(x)� ⌘(y)�

K(x, y) dxdy

(25)

+

Z

{u�v}

Z

{u<v}

�

L(u(x), u(y))� L(v(x), v(y))�

⌘(x)K(x, y) dxdy

�Z

{u<v}

Z

{u�v}

�

L(u(x), u(y))� L(v(x), v(y))�

⌘(y)K(x, y) dxdy.

The goal is now to prove that the right-hand side of the inequality above isnonpositive. In view of the very definition of ⌘ and the inequalities in Remark 7(see, in particular, (14) there), we can estimate the three terms in (25) asfollows

[...] �Z

{u<v}

Z

{u<v}

�

L(u(x), u(y))� L(v(x), v(y))�

⇥�

u(x)� u(y)� v(x) + v(y)�

K(x, y) dxdy

+

Z

{u�v}

Z

{u<v}

�

L(v(x), v(y))� L(v(x), v(y))�

⌘(x)K(x, y) dxdy

�Z

{u<v}

Z

{u�v}

�

L(v(x), v(y))� L(v(x), v(y))�

⌘(y)K(x, y) dxdy

0. (26)



By combining (26) with (25), we deduce that all the terms in (25) have to beequal to 0, which implies ⌘ = 0 almost everywhere in {u < v}, in turn givingthe desired result.

In particular, since the weak sub- and supersolutions belongs locally toW s,p, we get the following comparison principle.

Corollary 2 Let D b ⌦. Let u be a weak supersolution to (5) in ⌦, and let

v be a weak subsolution to (5) in ⌦ such that u � v almost everywhere in

Rn \D. Then u � v almost everywhere in D.

3.3 Lower semicontinuity of weak supersolutions

Now, we give an expected lower semicontinuity result for the weak supersolu-tions, which, as in the classic local setting, is a fundamental object to provideother important topological tools in order to develop the entire nonlinear Po-tential Theory. As we can see in the proof below, we will be able to obtain sucha property essentially via the supremum estimates given by Theorem 4 per-forming here a careful choice of the interpolation parameter � in (16) betweenthe local contributions and the nonlocal ones. This is a relevant di↵erence withrespect to the classical nonlinear Potential Theory, where on the contrary thelower semicontinuity is a straight consequence of weak Harnack estimates (see,e. g., [16, Theorem 3.51 and 3.63]).

Theorem 9 (Lower semicontinuity of supersolutions) Let u be a weak

supersolution in ⌦. Then

u(x) = ess lim infy!x

u(y) for a. e. x 2 ⌦.

In particular, u has a lower semicontinuous representative.

Proof Let ⌦0 b ⌦ and

E :=

⇢

x 2 ⌦0 : limr!0

Z

Br(x)

|u(x)� u(y)| dy = 0, |u(x)| < 1�

.

Then, in particular, |⌦0 \E| = 0 by Lebesgue’s Theorem. Fix z 2 E and r > 0.We may assume B

2r

(z) b ⌦. Since v := u(z) � u is a weak subsolution, wehave by Theorem 4 that

ess supBr(z)

v �Tail(v+

; z, r) + c ��

Z

B2r(z)

vp+

dx

!

1/p

(27)



whenever r r and � 2 (0, 1], where Tail is defined in (10) and positiveconstants � and c are both independent of u, r, z and �. Firstly, by the triangleinequality v

+

|u(z)|+ u� so that we immediately have

supr2(0,r)

Tail(v+

; z, r) c |u(z)|+ c supr2(0,r)

Tail(u�; z, r).

Also, for some constant c independent of u, r and z, we can write

supr2(0,r)

Tail(v+

; z, r) c |u(z)|+ c

rspZ

Rn\Br(z)

|u�(x)|p�1|x� z|�n�sp dx

!

1p�1

+ c supr2(0,r)

rspZ

Br(z)\Br(z)

|u�(x)|p�1|x� z|�n�sp dx

!

1p�1

c |u(z)|+ cTail(u�; z, r) + c ess supBr(z)

u� =: M,

where M is finite. Indeed, one can use the fact that z 2 E, that u belongs tothe tail space Lp�1

sp

(Rn), and that u is locally essentially bounded from belowin view of Lemma 3.

Now, a key-point in the present proof does consist in taking advantage ofthe ductility of the estimate in (16), which permits us to suitably choosing theparameter � there in order to interpolate the contribution given by the localand nonlocal terms. For this, given " > 0 we choose � < "/2M and thus weget

�Tail(v+

; z, r) <"

2(28)

whenever r 2 (0, r).Then we estimate the term with an integral average. Since z 2 E and u is

locally essentially bounded from below,Z

B2r(z)

�

u(z)�u(x)�

p

+

dx ess supx2B2r(z)

�

u(z)�u(x)�

p�1

+

Z

B2r(z)

|u(z)�u(x)| dx ! 0

as r ! 0. Thus, we can choose a sequence {r"

}">0

, r"

2 (0, r), such that

c ��

Z

B2r" (z)

�

u(z)� u(x)�

p

+

dx

!

1/p

<"

2. (29)

Clearly r"

! 0 as " ! 0. Combining the estimates (27), (28), and (29), itfollows

ess supBr" (z)

�

u(z)� u�

",

and consequently

u(z) lim"!0

⇣

ess infBr" (z)

u+ "⌘

= ess lim infy!z

u(y).



The reverse inequality will follow because z is a Lebesgue point:

u(z) = limr!0

Z

Br(z)

u(x) dx � limr!0

ess infBr(z)

u = ess lim infy!z

u(y),

and thus the claim holds for z 2 E. Finally, almost every x 2 ⌦ can bechosen as the role of z above when choosing ⌦0 such that x 2 ⌦0. The proofis complete.

3.4 Convergence results for weak supersolutions

We begin with an elementary result showing that a truncation of a weaksupersolution is still a weak supersolution.

Lemma 7 Suppose that u is a weak supersolution in ⌦. Then, for k 2 R,min{u, k} is a weak supersolution in ⌦ as well.

Proof Clearly min{u, k} 2 W s,p

loc

(⌦) \ Lp�1

sp

(Rn). Thus we only need to checkthat it satisfies the weak formulation. To this end, take a nonnegative testfunction � 2 C1

0

(⌦). For any " > 0 we consider the marker function ✓"

defined by

✓"

:= 1�min

⇢

1,(u� k)

+

"

�

.

We choose ⌘ = ✓"

� as a test function in the weak formulation of u. Then weget

0 Z

Rn

Z

Rn

L(u(x), u(y))�

✓"

(x)�(x)� ✓"

(y)�(y)�

K(x, y) dxdy,

where we denoted by L the function defined in (12). To estimate the integrand,we decompose Rn ⇥Rn as a union of

E1

:= {(x, y) 2 Rn ⇥Rn : u(x) k , u(y) k} ,E

2,"

:= {(x, y) 2 Rn ⇥Rn : u(x) � k + " , u(y) � k + "} ,E

3,"

:= {(x, y) 2 Rn ⇥Rn : u(x) � k + " , u(y) < k + "} ,E

4,"

:= {(x, y) 2 Rn ⇥Rn : u(x) < k + " , u(y) � k + "} ,E

5,"

:= {(x, y) 2 Rn ⇥Rn : k < u(x) < k + " , u(y) k} ,E

6,"

:= {(x, y) 2 Rn ⇥Rn : u(x) k , k < u(y) < k + "} ,E

7,"

:= {(x, y) 2 Rn ⇥Rn : k < u(x) < k + " , k < u(y) < k + "} .

Note that on E1

we have u = min{u, k} and ✓"

= 1, whereas on E2,"

the testfunction vanishes since ✓

"

(x) = ✓"

(y) = 0. On the other hand, on E3,"

we have



that ✓"

(x) = 0 and L(u(x), u(y)) > 0. Thus, using ✓"

(y) � �{uk}(y) and�(x) � 0, we get

ZZ

E3,"

L(u(x), u(y))�

✓"

(x)�(x)� ✓"

(y)�(y)�

K(x, y) dxdy

�Z

{uk}

Z

{u�k+"}L(k, u(y))�(y)K(x, y) dxdy

"!0�! �Z

{uk}

Z

{u�k}L(k, u(y))�(y)K(x, y) dxdy

Z

{uk}

Z

{u�k}L(k, u(y))

�

�(x)� �(y)�

K(x, y) dxdy.

The convergence follows by the monotone convergence theorem, and the lastinequality follows since � is nonnegative. Similar reasoning holds on E

4,"

byexchanging the roles of x and y. On E

5,"

we have L(u(x), u(y)) > 0, ✓"

(y) = 1,and ✓

"

(x) = 1� (u(x)� k)/", giving the estimate

L(u(x), u(y))�

✓"

(x)�(x)� ✓"

(y)�(y)�

= L(u(x), u(y))�

�(x)� �(y)�

� L(u(x), u(y))u(x)� k

"�(x)

|u(x)� u(y)|p�1|�(x)� �(y)|.

Thus,ZZ

E5,"

L(u(x), u(y))�

✓"

(x)�(x)� ✓"

(y)�(y)�

K(x, y) dxdy

ZZ

E5,"

|u(x)� u(y)|p�1|�(x)� �(y)|K(x, y) dxdy ! 0

as " ! 0 by the dominated convergence theorem since �{k<u<k+"} ! 0pointwise as " ! 0. The uniform upper bound follows from the fact thatu 2 W s,p

loc

(⌦)\Lp�1

sp

(Rn) and � 2 C10

(⌦). Similar reasoning holds on E6,"

byexchanging the roles of x and y.

Finally, on E7,"

we have ✓"

= 1� (u� k)/", implying

L(u(x), u(y))�

✓"

(x)�(x)� ✓(y)�(y)�

= �"p�1L(✓"

(x), ✓"

(y))�

✓"

(x)�(x)� ✓"

(y)�(x) + ✓"

(y)�(x)� ✓"

(y)�(y)�

= �"p�1|✓"

(x)� ✓"

(y)|p�(x) + L(u(x), u(y))✓"

(y)�

�(x)� �(y)�

|u(x)� u(y)|p�1|�(x)� �(y)|

since 0 ✓"

1. Consequently,ZZ

E7,"

L(u(x), u(y))�

✓"

(x)�(x)� ✓"

(y)�(y)�

K(x, y) dxdy

ZZ

E7,"

|u(x)� u(y)|p�1|�(x)� �(y)|K(x, y) dxdy ! 0



as "! 0 by the dominated convergence theorem. Indeed, we have that |u(x)�u(y)|�

E7," ! 0 almost everywhere as " ! 0, and the uniform upper boundfollows as in the case of E

5,"

. Collecting all the cases gives the desired non-negativeness of the weak formulation for min{u, k} finishing the proof.

Remark 9 We could also prove that the pointwise minimum of two weak su-persolutions is a weak supersolution. However, we do not state the proof heresince it will immediately follow from our results for (s, p)-superharmonic func-tions in Section 4.

Finally, we state and prove a very general fact which assures that (point-wise) limit functions of suitably bounded sequences of weak supersolutions aresupersolutions as well.

Theorem 10 (Convergence of sequences of supersolutions) Let g 2Lp�1

sp

(Rn) and h 2 Lp�1

sp

(Rn) be such that h g in Rn. Let {uj

} be a sequence

of weak supersolutions in ⌦ such that h uj

g almost everywhere in Rn

and uj

is uniformly locally essentially bounded from above in ⌦. Suppose that

uj

converges to a function u pointwise almost everywhere as j ! 1. Then u

is a weak supersolution in ⌦ as well.

Proof Fix a nonnegative � 2 C10

(⌦) and let D1

be an open set such thatsupp� ⇢ D

1

b ⌦. Furthermore, let D2

be an open set such that D1

b D2

b ⌦

and take large enough M > 0 satisfying uj

M almost everywhere in D2

.First, from Lemma 5 we deduce that

Z

D1

Z

D1

|uj

(x)� uj

(y)|p

|x� y|n+sp

dxdy C < 1

uniformly in j. Therefore, Fatou’s Lemma yields that u 2 W s,p(D1

). Moreover,the pointwise convergence implies that h u g a. e. in Rn. Accordingly, wemay rewrite as

0 Z

Rn

Z

Rn

L(uj

(x), uj

(y))�

�(x)� �(y)�

K(x, y) dxdy

=

Z

Rn

Z

Rn

L(u(x), u(y))�

�(x)� �(y)�

K(x, y) dxdy

+

Z

Rn

Z

Rn

�

L(uj

(x), uj

(y))� L(u(x), u(y))��

�(x)� �(y)�

K(x, y) dxdy.

We further split the second term on the right-hand side in the display aboveinto the following two terms, by using the fact that supp� ⇢ D

1

will assurethe needed separation to write the contribution on D

1

⇥D1

,Z

Rn

Z

Rn

�

L(uj

(x), uj

(y))� L(u(x), u(y))��

�(x)� �(y)�

K(x, y) dxdy



=

Z

D1

Z

D1

�

L(uj

(x), uj

(y))� L(u(x), u(y))��

�(x)� �(y)�

K(x, y) dxdy

+ 2

Z

Rn\D1

Z

D1

�

L(uj

(x), uj

(y))� L(u(x), u(y))�

�(x)K(x, y) dxdy

=: E1,j

+ 2E2,j

.

Our goal is now to show that

limj!1

(E1,j

+ 2E2,j

) = 0,

which then proves that u is a weak supersolution in ⌦, as desired.Considering first E

2,j

, we have the pointwise upper bound�

�L(uj

(x), uj

(y))� L(u(x), u(y))�

�

c�

gp�1

+

(x) + gp�1

+

(y) + hp�1

� (x) + hp�1

� (y)�

,

and therefore, by the dominated convergence theorem,

limj!1

E2,j

= limj!1

Z

Rn\D1

Z

D1

�

L(uj

(x), uj

(y))� L(u(x), u(y))�

�(x)K(x, y) dxdy

= 0.

Therefore, it remains to show that limj!1 E

1,j

= 0. To this end, denotein short

j

(x, y) :=�

L(uj

(x), uj

(y))� L(u(x), u(y))��

�(x)� �(y)�

K(x, y),

and rewriteZ

D1

Z

D1

j

(x, y) dxdy

=

Z

Aj,✓

Z

Aj,✓

j

(x, y) dxdy +

ZZ

(D1⇥D1)\(Aj,✓⇥Aj,✓)

j

(x, y) dxdy,

where we have set

Aj,✓

:=n

x 2 D1

: |uj

(x)� u(x)| < ✓o

.

On the one hand, by Holder’s Inequality we get that

ZZ

E

j

(x, y) dxdy c

✓

ZZ

E

|uj

(x)� uj

(y)|p

|x� y|n+sp

+|u(x)� u(y)|p

|x� y|n+sp

dxdy

◆

p�1p

⇥✓

ZZ

E

|�(x)� �(y)|p

|x� y|n+sp

dxdy

◆

1p



whenever E is a Borel set of D1

⇥ D1

. The first integral in the right-handside of the inequality above is uniformly bounded in j, since the sequence u

j

is equibounded in W s,p(D1

) as seen in the beginning of the proof. Also, sincethe function � : Rn ⇥Rn ! R, defined by

�(x, y) :=|�(x)� �(y)|p

|x� y|n+sp

,

belongs to L1(Rn ⇥Rn), we deduce that

limj!1

ZZ

(D1⇥D1)\(Aj,✓⇥Aj,✓)

�(x, y) dxdy = 0,

because�

�(D1

⇥ D1

) \ (Aj,✓

⇥ Aj,✓

)�

� ! 0 as j ! 1 for any ✓ > 0 by thepointwise convergence of u

j

to u.On the other hand,

| j

(x, y)| c|�(x)� �(y)||x� y|n+sp

�

�uj

(x)� u(x)� uj

(y) + u(y)�

�

⇥Z

1

0

�

�t�

uj

(x)� uj

(y)�

+ (1� t)�

u(x)� u(y)�

�

�

p�2

dt.

Now, we have to distinguish two cases depending on the summability ex-ponent p. In the case when p � 2, we obtain in A

j,✓

⇥Aj,✓

that

| j

(x, y)| c ✓��

|uj

(x)� uj

(y)|+ |u(x)� u(y)|�

p�1��

|x� y|s(p�1��)|�(x)� �(y)||x� y|n+s(1+�)

,

for any small �, and thus by Holder’s Inequality we obtain

Z

Aj,✓

Z

Aj,✓

j

(x, y) dxdy c ✓�C

✓

Z

D1

Z

D1

|�(x)� �(y)|q

|x� y|n+s(1+�)q

dxdy

◆

1q

,

where q := [p/(p�1��)]0 = p/(1+�) and C is independent of j and ✓. Taking

� = min

⇢

1� s

2s,p� 1

2

�

,

we finally get thatZ

Aj,✓

Z

Aj,✓

j

(x, y) dxdy eC✓�, (30)

where eC is independent of j and ✓.On the other hand, in the case when 1 < p < 2, we obtain by (13)

| j

(x, y)| c|u

j

(x)� u(x)� uj

(y) + u(y)|p�1

|x� y|s(p�1)

|�(x)� �(y)||x� y|n+s



c ✓�|u

j

(x)� u(x)� uj

(y) + u(y)|p�1��

|x� y|s(p�1��)|�(x)� �(y)||x� y|n+s(1+�)

in Aj,✓

⇥Aj,✓

, and now it su�ces to act as in the case p � 2 above in order toprove the estimate in (30) also in such a sublinear case.

Finally, it su�ces to collect all the estimates above in order to concludethat actually

limj!1

E1,j

= 0

holds since ✓ can be chosen arbitrarily small. This finishes the proof.

If the sequence is increasing, we do not have to assume any boundednessfrom above.

Corollary 3 Let {uj

} be an increasing sequence of weak supersolutions in ⌦

such that uj

converges to a function u 2 W s,p

loc

(⌦)\Lp�1

sp

(Rn) pointwise almost

everywhere in Rn as j ! 1. Then u is a weak supersolution in ⌦ as well.

Proof For any M > 0, denote by uM

:= min{u,M} and uM,j

:= min{uj

,M},which is a weak supersolution by Lemma 7. Then {u

M,j

}j

is a sequence satis-fying the assumptions of Theorem 10 converging pointwise almost everywhereto u

M

, and consequently uM

is a weak supersolution in ⌦. Let ⌘ 2 C10

(⌦) bea nonnegative test function. Since

|L(uM

(x), uM

(y))| |u(x)� u(y)|p�1

for every M > 0 and every x, y 2 Rn, where u 2 W s,p

loc

(⌦)\Lp�1

sp

(Rn), we canlet M ! 1 to obtain by the dominated convergence theorem that

Z

Rn

Z

Rn

L(u(x), u(y))�

⌘(x)� ⌘(y)�

K(x, y) dxdy � 0.

We conclude that u is a weak supersolution in ⌦.

A similar result as Theorem 10 holds also for sequences of weak solutions.

Corollary 4 Let h, g 2 Lp�1

sp

(Rn) and let {uj

} be a sequence of weak solutions

in ⌦ such that h uj

g and uj

! u pointwise almost everywhere in Rn as

j ! 1. Then u is a weak solution in ⌦.

Proof Since both uj

and �uj

are weak supersolutions in ⌦, we have that |uj

|is uniformly locally essentially bounded in ⌦ by Lemma 3. Then u is a weaksolution in ⌦ since both u and �u are weak supersolutions by Theorem 10.

We conclude the section with a crucial convergence result concerning con-tinuous weak solutions.



Corollary 5 Let h, g 2 Lp�1

sp

(Rn) and let {uj

} be a sequence of continuous

weak solutions in ⌦ such that h uj

g and that limj!1 u

j

exists almost

everywhere in Rn. Then u := limj!1 u

j

exists at every point of ⌦ and u is a

continuous weak solution in ⌦.

Proof According to Corollary 4, u is a weak solution in ⌦. Therefore onlycontinuity of u in ⌦ and pointwise convergence need to be checked. LettingB

3r

(x0

) be a ball in ⌦, we have by Lemma 3 and the uniform Tail spacebounds that

supj

supB2r(x0)

|uj

|+Tail(uj

;x0

, r)

!

C,

where C is independent of uj

and u. Using now the Holder continuity estimatein Theorem 5, we see that

oscB⇢(x0)

uj

c⇣⇢

r

⌘

↵

supB2r(x0)

|uj

|+Tail(uj

;x0

, r)

!

c⇣⇢

r

⌘

↵

C,

where ⇢ 2 (0, r) and ↵ ⌘ ↵(n, p, s,⇤) 2 (0, 1). Therefore the sequence {uj

}is equicontinuous on compact subsets of ⌦, and thus the continuity of u andpointwise convergence in ⌦ follow from the Arzela–Ascoli theorem. This fin-ishes the proof.

4 (s, p)-superharmonic functions

In this section, we study the nonlocal superharmonic functions for the non-linear integro-di↵erential equations in (2), which we have defined in the in-troduction; recall Definition 1. As well-known, the superharmonic functionsconstitute an important class of functions which have been extensively usedin PDE and in classical Potential Theory, as well as in Complex Analysis.Their fractional counterpart has to take into account the nonlocality of theoperators in (5) and thus it has to incorporate the summability assumptionsof the negative part of the functions in the tail space Lp�1

sp

defined in (4).

4.1 Bounded (s, p)-superharmonic functions

We first move towards proving Theorem 1(iv). We begin with an elementaryapproximation result for lower semicontinuous functions. The proof is standardand goes via infimal convolution. However, due to the nonlocal framework weneed a suitable pointwise control of approximations over Rn, and hence wepresent the details.



Lemma 8 Let u be an (s, p)-superharmonic function in ⌦ and let D b ⌦.

Then there is an increasing sequence of smooth functions { j

} such that

limj!1

j

(x) = u(x) for all x 2 D.

Proof Define the increasing sequence of continuous functions { e j

} as follows

e j

(x) := miny2D

n

min�

j, u(y)

+ j2|x� y|o

� 1

j.

Notice that, by the very definition, e j

(x) u(x) � 1/j < u(x) in D. Since u

is locally bounded from below, u(y) � �M in D for some M < 1. Also, bythe lower semicontinuity, the minimum is attained at some y

j

2 D, and thuswe have

j � 1

j� e

j

(x) � �M + j2|x� yj

|� 1

j,

which yields

|x� yj

| j +M

j2=: r

j

< 1,

where rj

! 0 as j ! 1. Since u is lower semicontinuous, we have that in D

u(x) limj!1

infy2Brj (x)

min�

j, u(y)

� 1

j

!

limj!1

infy2Brj (x)

n

min�

j, u(y)

+ j2|x� y|o

� 1

j

!

= limj!1

e j

(x).

Hence, limj!1 e

j

(x) = u(x) for all x 2 D. Finally, since { e j

} is an increasingsequence of continuous functions in D and

e j+1

� e j

� 1

j� 1

j + 1> 0 in D,

we can find smooth functions j

such that e j

j

< e j+1

in D. Now { j

}is the desired sequence of functions.

Using the previous approximation lemma, we can show that the (s, p)-superharmonic functions can be also approximated by continuous weak super-solutions in regular sets.

Lemma 9 Let u be an (s, p)-superharmonic function in ⌦ and let D b ⌦

be an open set such that Rn \D satisfies the measure density condition (18).Then there is an increasing sequence {u

j

}, uj

2 C(D), of weak supersolutions

in D converging to u pointwise in Rn.



Proof Let U be an open set satisfying D b U b ⌦, which is possible byUrysohn’s Lemma. By Lemma 8, there is an increasing sequence of smoothfunctions {

j

}, j

2 C1(U), converging to u pointwise in U . For each j,define

gj

(x) :=

(

j

(x), x 2 U,

min{j, u(x)}, x 2 Rn \ U.

Clearly gj

2 W s,p(U) \ Lp�1

sp

(Rn) by smoothness of j

and the fact thatu� 2 Lp�1

sp

(Rn). Now we can solve the obstacle problem using the functions gj

as obstacles to obtain solutions uj

2 Kgj ,gj (D,U), j = 1, 2, . . . , so that u

j

iscontinuous inD by Theorem 8 and a weak supersolution inD by Theorem 7. Tosee that {u

j

} is an increasing sequence, denote by Aj

:= D \ {uj

> gj

}. Sinceuj

is a weak solution in Aj

by Corollary 1 and clearly uj+1

� uj

in Rn \ Aj

,the comparison principle (Lemma 6) implies that u

j+1

� uj

. Similarly, uj

u

by Definition 1(iii). Since gj

converges pointwise to u, we must also have that

limj!1

uj

(x) = u(x) for all x 2 Rn.

This finishes the proof.

Below we will show that, as expected, an (s, p)-superharmonic functionbounded from above is a weak supersolution to (5). This proves the first state-ment of Theorem 1(iv).

Theorem 11 Let u 2 Lp�1

sp

(Rn) be an (s, p)-superharmonic function in ⌦

that is locally bounded from above in ⌦. Then u is a weak supersolution in ⌦.

Proof Let D b ⌦ be an open set such that Rn\D satisfies the measure densitycondition (18). Then by Lemma 9 there is an increasing sequence {u

j

} ofweak supersolutions in D converging to u pointwise in Rn such that each u

j

iscontinuous inD. Since each u

j

satisfies u1

uj

u with u1

, u 2 Lp�1

sp

(Rn) andu is bounded from above in D, u is a weak supersolution in D by Theorem 10.Finally, because of the arbitrariness of the set D b ⌦, we can deduce that thefunction u is a weak supersolution in ⌦, as desired.

If an (s, p)-superharmonic function is a fractional Sobolev function, it is aweak supersolution as well. This gives the second statement of Theorem 1(iv).

Corollary 6 Let u 2 W s,p

loc

(⌦) \ Lp�1

sp

(Rn) be an (s, p)-superharmonic func-

tion in ⌦. Then u is a weak supersolution in ⌦.

Proof For anyM > 0, denote by uM

:= min{u,M}, which is (s, p)-superharmonicin ⌦ as a pointwise minimum of two (s, p)-superharmonic functions. By The-orem 11 u

M

is a weak supersolution in ⌦. Consequently, Corollary 3 yieldsthat u is a weak supersolution in ⌦.



On the other hand, lower semicontinuous representatives of weak superso-lutions are (s, p)-superharmonic.

Theorem 12 Let u be a lower semicontinuous weak supersolution in ⌦ sat-

isfying

u(x) = ess lim infy!x

u(y) for every x 2 ⌦. (31)

Then u is an (s, p)-superharmonic function in ⌦.

Proof According to the definition of u, by Lemma 3, together with (31), wehave that (i–ii) and (iv) of Definition 1 hold. Thus it remains to check that usatisfies the comparison given in Definition 1(iii). For this, take D b ⌦ anda weak solution v in D such that v 2 C(D), v u almost everywhere inRn \D and v u on @D. For any " > 0 define v

"

:= v � " and consider theset K

"

=�

v"

� u

\ D. Notice that by construction the set K"

is compactand K

"

\ @D = ;. Thus, it su�ces to prove that K"

= ;. This is now a plainconsequence of the comparison principle proven in Section 3. Indeed, one canfind an open set D

1

such that K"

⇢ D1

b D. Moreover, v"

u in Rn \ D1

almost everywhere and thus Corollary 2 yields u � v"

almost everywhere inD

1

. In particular, u � v � " almost everywhere in D. To obtain an inequalitythat holds everywhere in D, fix x 2 D. Then there exists r > 0 such thatB

r

(x) ⇢ D and

u(x) � ess infBr(x)

u� " � infBr(x)

v � 2 " � v(x)� 3 ",

by (31) and continuity of v. Since " > 0 and x 2 D were arbitrary, we haveu � v in D. This finishes the proof.

From Theorem 11 and Theorem 12 we see that a function is a continuousweak solution in ⌦ if and only if it is both (s, p)-superharmonic and (s, p)-subharmonic in ⌦.

4.2 Pointwise behavior

We next investigate the pointwise behavior of (s, p)-superharmonic functionsin ⌦ and start with the following lemma.

Lemma 10 Let u be (s, p)-superharmonic in ⌦ such that u = 0 almost every-

where in ⌦. Then u = 0 in ⌦.

Proof Since u is lower semicontinuous, we have u 0 in ⌦. Furthermore, wecan assume that u 0 in the wholeRn by considering the (s, p)-superharmonicfunction min{u, 0} instead of u. Let z 2 ⌦ and take R > 0 such that B

R

(z) b⌦. By Lemma 9 there is an increasing sequence {u

j

} of weak supersolutions



in BR

(z) converging to u pointwise in Rn such that each uj

is continuous inB

R

(z). Then it holds, in particular, that uj

(z) u(z). Thus, it su�ces toshow that for every " > 0 there exists j such that u

j

(z) � �". To this end, let" > 0. Since �u

j

is a weak subsolution in B2r

(z) for any r R/2, applyingTheorem 4 with � = 1 we have that

supBr(z)

(�uj

) c

✓

Z

B2r

(�uj

)p+

dx

◆

1p

+Tail((�uj

)+

; z, r)

c

✓

Z

B2r

|uj

|p dx◆

1p

+ c

rspZ

BR\Br

|uj

(y)|p�1|z � y|�n�sp dy

!

1p�1

+ c

rspZ

Rn\BR

|uj


!

1p�1

c

✓

Z

B2r

|uj

|p dx◆

1p

+ c

rspZ

BR\Br

|uj


!

1p�1

(32)

+ c⇣ r

R

⌘

spp�1

Tail(u1

; z,R).

Now, we first choose r to be so small that the last term on the right-handside of (32) is smaller than "/3. Then we can choose j so large that each ofthe two first terms on the right-hand side of (32) is smaller than "/3. This ispossible according to the dominated convergence theorem since u

j

! 0 almosteverywhere in B

R

(z) as j ! 1 and |uj

| |u1

| for every j. Consequently,uj

(z) � �" and the proof is complete.

An (s, p)-superharmonic function has to coincide with its inferior limitsin ⌦. In particular, the function cannot have isolated smaller values in singlepoints. This gives Theorem 1(i).

Theorem 13 Let u be (s, p)-superharmonic in ⌦. Then

u(x) = lim infy!x

u(y) = ess lim infy!x

u(y) for every x 2 ⌦.

In particular, infD

u = ess infD

u for any open set D b ⌦.

Proof Fix x 2 ⌦ and denote by � := ess lim infy!x

u(y). Then

� � lim infy!x

u(y) � u(x)

by the lower semicontinuity of u. To prove the reverse inequality, pick t < �.Then there exists r > 0 such that B

r

(x) ⇢ ⌦ and u � t almost everywhere inB

r

(x). By Lemma 10 the (s, p)-superharmonic function

v := min{u, t}� t



is identically 0 in Br

(x). In particular, u(x) � t and the claim follows byarbitrariness of t < �.

4.3 Summability of (s, p)-superharmonic functions

We recall a basic result from [23, Lemma 7.3], which is in turn based on theCaccioppoli inequality and the weak Harnack estimates for weak supersolu-tions presented in [9]. In [23] it is given for equations involving nonnegativesource terms, but the proof is identical in the case of weak supersolutions. Theneeded information is that the weak supersolution belongs locally to W s,p.

Lemma 11 Let u be a nonnegative weak supersolution in B4r

⌘ B4r

(x0

) ⇢ ⌦.

Let h 2 (0, s), q 2 (0, q), where

q := min

⇢

n(p� 1)

n� s, p

�

. (33)

Then there exists a constant c ⌘ c(n, p, s,⇤, s� h, q � q) such that

✓

Z

B2r

Z

B2r

|u(x)� u(y)|q

|x� y|n+hq

dxdy

◆

1q

c

rh

✓

ess infBr

u+Tail(u�;x0

, 4r)

◆

holds.

The next theorem tells that the positive part of an (s, p)-superharmonicfunction also belongs to the Tail space and describes summability propertiesof solutions, giving Theorem 1(ii).

Theorem 14 Suppose that u is an (s, p)-superharmonic function in B2r

(x0

).Then u 2 Lp�1

sp

(Rn). Moreover, defining the quantity

M := supz2Br(x0)

infBr/8(z)

u+

+Tail(u�; z, r/2) + supB3r/2(x0)

u�

!

,

then M is finite and for h 2 (0, s), q 2 (0, q) and t 2 (0, t), where q is as

in (33) and t as in (17), there is a positive finite constant C ⌘ C(n, p, s,⇤, s�h, q � q, t� t) such that

rh [u]W

h,q(Br(x0))

+ kukL

t(Br(x0))

CM. (34)

Proof First, M is finite due to assumptions (i) and (iv) of Definition 1. Since uis locally bounded from below, we may assume, without loss of generality, thatu is nonnegative in B

3r/2

(x0

). Let uk

:= min{u, k}, k 2 N. By Theorem 11we have that u

k

is a lower semicontinuous weak supersolution in B2r

(x0

). Let



z 2 Br

(x0

). The weak Harnack estimate (Theorem 6) for uk

together withFatou’s Lemma, after letting k ! 1, then imply that

rnt kuk

L

t(Br/4(z))

c infBr/2(z)

u+ cTail(u�; z, r/2) (35)

for any t 2 (0, t). Similarly, Lemma 11 applies for uk

, and we deduce from it,by Fatou’s Lemma that

rh+nq [u]

W

h,q(Br/4(z))

c infBr/8(z)

u+ cTail(u�; z, r/2) (36)

for any h 2 (0, s) and q 2 (0, q). Now (34) follows from (35) and (36) aftera covering argument. Finally, Lemma 1 implies that u 2 Lp�1

sp

(Rn) from theboundedness of [u]

W

h,q(Br(x0))

and kukL

t(Br(x0))

when taking t = q = p� 1.

4.4 Convergence properties

We next collect some convergence results related to (s, p)-superharmonic func-tions. The first one is that the limit of an increasing sequence of (s, p)-super-harmonic functions in an open set ⌦ is either identically +1 or (s, p)-super-harmonic in ⌦. Observe that ⌦ does not need to be a connected set which isin strict contrast with respect to the local setting.

Lemma 12 Let {uk

} be an increasing sequence of (s, p)-superharmonic func-

tions in an open set ⌦ converging pointwise to a function u as k ! 1. Then

either u ⌘ +1 in ⌦ or u is (s, p)-superharmonic in ⌦.

Proof Observe that since u � u1

and (u1

)� 2 Lp�1

sp

(Rn) by Definition 1(iv),we also have that u� 2 Lp�1

sp

(Rn).

Step 1. Assume first that there is an open set D ⇢ ⌦ such that u is finitealmost everywhere in D. Then we clearly have that u satisfies (i–ii), (iv) ofDefinition 1 in D. Thus we have to check Definition 1(iii). Let D

4

b D and letv be as in Definition 1(iii) (with D ⌘ D

4

), i. e., v 2 C(D4

) is a weak solutionin D

4

such that v+

2 L1(Rn) and v u on @D4

and almost everywhere onRn \D

4

. For any " > 0, by the lower semicontinuity of u� v, there are opensets D

1

, D2

, D3

such that D1

b D2

b D3

b D4

, Rn \D2

satisfies the measuredensity condition (18), and {u v� "}\D

4

⇢ D1

. In particular, u > v� " onD

4

\D1

and almost everywhere on Rn \D4

. Since D3

b D4

we have by thecompactness that there is large enough k

"

such that D3

\D2

b {uk

> v � "}for k > k

"

. Indeed, since

{u > v � "} \D4

=[

k

{uk

> v � "} \D4

,



we have that�

{uk

> v�"}\D4

k

is an open cover for the compact setD3

\D2

.Defining eu

k

= v� " on D3

\D2

and euk

= min{uk

, v� "} on Rn \D3

, we haveby Lemma 13 below (applied with ⌦ ⌘ D

3

, D ⌘ D2

, uk

⌘ euk

) that there isa sequence of weak solutions {v

k

} in D2

such that vk

2 C(D2

), vk

! v � "

in D2

and almost everywhere in Rn \ D2

, and that vk

uk

on @D2

andalmost everywhere on Rn\D

2

whenever k > k"

. Therefore, by Definition 1(iii),uk

� vk

in D2

as well. Since the convergence of vk

! v � " is uniform in D1

by Arzela–Ascoli Theorem as k ! 1, we obtain that u � v � 2" in D1

, andtherefore also in the whole D

4

. This shows that u is (s, p)-superharmonic inD.

Step 2. Let us next assume that u is not finite on a Borel subset of E of ⌦having positive measure. Using inner regularity of the Lebesgue measure wefind a compact set K ⇢ ⌦ with positive measure such that u = +1 on K.Then there has to be a ball B

r

(x0

) such that |K \Br

(x0

)| > 0 and B2r

(x0

) b⌦. In particular, for nonnegative (s, p)-superharmonic functions defined asw

k

:= uk

�infB2r(x0)

u1

, k 2 N, we have by the monotone convergence theoremthat kw

k

kL

p�1(Br(x0))

! +1 as k ! 1. Then Theorem 14 implies thatinf

B⇢(z)w

k

! +1 as k ! 1 for some smaller ball B⇢

(z) and that u ⌘ +1in B

⇢

(z). This also implies that u /2 Lp�1

sp

(Rn).

Step 3. Conclusion. If there is any non-empty open setD b ⌦ such that u isfinite almost everywhere inD, then Step 1 yields that u is (s, p)-superharmonicin D. Therefore Theorem 14 implies that in fact u 2 Lp�1

sp

(Rn). By Step 2this excludes the possibility of having a Borel subset E of ⌦ with positivemeasure such that u is not finite on E. Therefore, the only possibility that thesituation in Step 2 can occur is that every ball B

r

(z) such that B2r

(z) b ⌦

contains a Borel set Ez,r

with positive measure such that u is not finite onE

z,r

. Step 2 then implies that infBr(z)

u = +1, and hence either u ⌘ +1 in ⌦or u is finite almost everywhere in ⌦, implying that u is (s, p)-superharmonicin ⌦ by Step 1.

In the proof above we appealed to the following stability result.

Lemma 13 Suppose that v is a continuous weak solution in ⌦ and let D b ⌦

be an open set such that Rn \D satisfies the measure density condition (18).Assume further that there are h, g 2 Lp�1

sp

(Rn) and a sequence {uk

} such that

h uk

g and uk

! v almost everywhere in Rn \⌦ as k ! 1. Then there is

a sequence of weak solutions {vk

} in D such that vk

2 C(D), vk

= v on ⌦ \D,

vk

= uk

on Rn \ ⌦, and vk

! v everywhere in D and almost everywhere on

Rn \D as k ! 1.

Proof Let U be such that D b U b ⌦ and v 2 W s,p(U). Setting gk

:= v

on ⌦ and gk

= uk

on Rn \ ⌦, we find by Corollary 1 functions {vk

}, vk

2K

gk,�1(D,U), as in the statement. Indeed, gk

2 W s,p(U) \ Lp�1

sp

(Rn). We



may test the weak formulation of vk

with �k

:= (vk

� v)�U

2 W s,p

0

(D) andobtain after straightforward manipulations (see e.g. proof of [20, Lemma 3])that, for a universal constant C,

kvk

kW

s,p(U)

C kvkW

s,p(U)

+ C

Z

Rn\D

(|h(x)|+ |g(x)|)p�1

(1 + |x|)n+sp

dx

!

1p�1

. (37)

Therefore the sequence {vk

} is uniformly bounded in W s,p(U), and the pre-compactness of W s,p(U), as shown for instance in [11, Theorem 7.1], guar-antees that there is a subsequence {v

kj}j converging almost everywhere to ev

as j ! 1. By Corollary 5 the convergence is pointwise in D and ev is (s, p)-harmonic inD. We will show that actually ev = v inD. Since every subsequenceof {v

k

} has such a subsequence, we have that limk!1 v

k

= v pointwise in D.To see that v = ev in D, we test the weak formulation with ⌘

k

:= (v �vk

)�U

2 W s,p

0

(D), relabeling the subsequence. Notice that ⌘k

is a feasible testfunction since v, v

k

2 W s,p(U) and vk

= v in U \D. The weak formulation forv and v

k

gives

0 =

Z

Rn

Z

Rn

�

L(v(x), v(y))� L(vk

(x), vk

(y))��

⌘(x)� ⌘(y)�

K(x, y) dxdy

=

Z

U

Z

U

�


(x), vk

(y))�

⇥�

v(x)� v(y)� vk

(x) + vk

(y)�

K(x, y) dxdy

+ 2

Z

Rn\U

Z

U

�


(x), vk

(y))��

v(x)� vk

(x)�

K(x, y) dxdy

=: I1,k

+ 2I2,k

.

We claim that limk!1 I

2,k

= 0. Indeed, noticing that since vk

(x) = v(x) forx 2 U \D, we may rewrite

I2,k

=

Z

Rn\U

Z

D

�


(x), vk

(y))��

v(x)� vk

(x)�

K(x, y) dxdy.

The involved measure K(x, y) dxdy is finite on D ⇥Rn \ U and thus we haveby the dominated convergence theorem, using the uniform bounds h u

k

g,the estimate in (37), and the fact that ev = v almost everywhere on Rn \ D,that

limk!1

I2,k

=

Z

Rn\U

Z

D

�

L(v(x), v(y))�L(ev(x), v(y))��

v(x)�ev(x)�

K(x, y) dxdy.

Therefore limk!1 I

2,k

� 0 by the monotonicity of t 7! L(t, v(y)). Thus, Fa-tou’s Lemma implies that

0 � lim infk!1

I1,k

�Z

U

Z

U

�

L(v(x), v(y))� L(ev(x), ev(y))�



⇥�

v(x)� v(y)� ev(x) + ev(y)�

K(x, y) dxdy,

proving by the monotonicity of L that ev = v almost everywhere. This finishesthe proof.

We also get a fundamental convergence result for increasing sequences of(s, p)-harmonic functions, improving Corollary 5.

Theorem 15 (Harnack’s convergence theorem) Let {uk

} be an increas-

ing sequence of (s, p)-harmonic functions in ⌦ converging pointwise to a func-

tion u as k ! 1. Then either u ⌘ +1 in ⌦ or u is (s, p)-harmonic in ⌦.

Proof By Lemma 12 either u ⌘ +1 or u is (s, p)-superharmonic in ⌦. In thelatter case, Theorem 14 implies that u 2 Lp�1

sp

(Rn), and thus by Corollary 5,u is (s, p)-harmonic in ⌦.

4.5 Unbounded comparison

In Definition 1(iii) we demanded that the comparison functions are globallybounded from above. A reasonable question is then that how would the defi-nition change if one removes this assumption. In other words, if the solutionis allowed to have too wild nonlocal contributions, would this be able to breakthe comparison? The answer is negative. Indeed, the next lemma tells thatone can remove the boundedness assumption v

+

2 L1(Rn) in the definitionof (s, p)-superharmonic functions and still get the same class of functions. Thisis Theorem 1(iii).

Lemma 14 Let u be an (s, p)-superharmonic function in ⌦. Then it satisfies

the following unbounded comparison statement:

(iii’) u satisfies the comparison in ⌦ against solutions, that is, if D b ⌦ is an

open set and v 2 C(D) is a weak solution in D such that u � v on @D and

almost everywhere on Rn \D, then u � v in D.

Proof Let u be an (s, p)-superharmonic function in ⌦. We will show thatthen it also satisfies (iii’). To this end, take D b ⌦ and v as in (iii’). Let" > 0. Due to lower semicontinuity of u � v and the boundary condition, theset K

"

:= {u v � "} \ D is a compact set of D. Therefore we find opensets D

1

, D2

such that K"

⇢ D1

b D2

b D and Rn \D2

satisfies the measuredensity condition (18). Truncate v as u

k

:= min{v�", k}. Applying Lemma 13(with ⌦ ⌘ D and D ⌘ D

2

) we find a sequence of continuous weak solutions{v

k

} in D2

such that vk

! v � " in D2

. The convergence is uniform in D1

.Therefore, there is large enough k such that |v

k

� v| 2" on D1

. Moreover,by the comparison principle (Lemma 6), v

k

v in Rn. Since u > vk

� " on



@D1

and almost everywhere in Rn \ D1

by the definition of K"

, we have byDefinition 1(iii) that u � v

k

� " � v � 3" in D1

, and thus we also have thatu � v�3" in the whole D, because in D\K

"

we have u > v�". Since this holdsfor an arbitrary positive ", we have that (iii’) holds, completing the proof.

We conclude the section by a more general version of the comparison prin-ciple.

Theorem 16 (Comparison principle) Let u be (s, p)-superharmonic in ⌦

and let v be (s, p)-subharmonic in ⌦. If u � v almost everywhere in Rn \ ⌦and

lim inf⌦3y!x

u(y) � lim sup⌦3y!x

v(y) for all x 2 @⌦

such that both sides are not simultaneously +1 or �1, then u � v in ⌦.

Proof Suppose that u and v satisfy the assumptions of the theorem. Let " > 0.Then there exists an open set D b ⌦ such that u � v � " in ⌦ \ D by theboundary condition for u and v. We may also assume that Rn \D satisfies themeasure density condition (18). Let U be an open set such that D b U b ⌦,and let {

j

}, j

2 C1(U), be an increasing sequence converging pointwiseto u in U . Such a sequence exists according to Lemma 8. Then

j

� v � 2"in U \D whenever j is large enough by compactness of U \D together withupper semicontinuity of v � 2". For such j, let g :=

j

�U

+ u�Rn\U , whichis in W s,p(U) by smoothness of

j

and in Lp�1

sp

(Rn) since u 2 Lp�1

sp

(Rn)by Theorem 14. Letting now h 2 K

g,�1(D,U) solve the related Dirichletproblem, h 2 C(D) is a weak solution in D by Corollary 1 and Theorem 8.Since u � h � v � 2" in @D and almost everywhere in Rn \ D, we haveaccording to Lemma 14 that u � h � v � 2" in D as well. Also u � v � " in⌦ \D by the choice of D in the beginning, and consequently u � v� 2" in ⌦.The claim follows by letting "! 0.

5 The Perron method

We now turn our focus on Dirichlet boundary value problems. Collecting someof the tools so far, it is rather straightforward to prove existence results outsideof the natural energy classes. For instance we record the following existenceand regularity result, which often in practice turns out to be very useful.

Theorem 17 Let ⌦ b ⌦0 be bounded open sets, and assume that Rn \⌦ sat-

isfies the measure density condition (18). Suppose that g 2 C(⌦0)\Lp�1

sp

(Rn).Then there is a weak solution in ⌦, which is continuous in ⌦0 and has bound-

ary values g on Rn \⌦. Such a solution is unique.



Proof By Lemma 8 there is an increasing sequence { j

}, j

2 C1(⌦0) \Lp�1

sp

(Rn), such that j

! g pointwise in Rn as j ! 1. Solving the Dirichletboundary value problem we find weak solutions u

j

2 K j ,�1(⌦,⌦0) \C(⌦0),

j = 1, 2, . . . , and the sequence is increasing. By [20, Theorem 5] we have thatuj

is uniformly bounded from above in ⌦ and hence Theorem 15 gives that uj

converges to an (s, p)-harmonic function u in ⌦. Theorem 8 gives a uniform (inj) modulus of continuity for u

j

’s on compact subsets of ⌦0, and thus u 2 C(⌦0)is a weak solution as in the statement.

The uniqueness follows easily, since if u1

, u2

are two solutions as in thestatement, then {u

1

� u2

+"} is compact set of ⌦ for all " > 0, and comparisonthen yields that u

1

u2

+" in ⌦. Since this holds for arbitrarily small positive", and we may interchange the roles of u

1

and u2

, we deduce that u1

⌘ u2

.

However, our tools provide a much more general setup for Dirichlet prob-lems, given by the Perron method. Indeed, we conclude this paper by intro-ducing a natural nonlocal counterpart of the celebrated Perron method in non-linear Potential Theory, as mentioned in the introduction; recall Definition 2there.

5.1 Poisson modification

We start by defining the nonlocal Poisson modification.

Theorem 18 Let D b ⌦ be open sets such that Rn \D satisfies the measure

density condition (18). Let u be (s, p)-superharmonic in ⌦. Then there is a

continuous weak solution w in D such that the function Pu,D

, defined as

Pu,D

(x) :=

(

w(x), x 2 D,

u(x), x 2 Rn \D,

is an (s, p)-superharmonic function in ⌦ satisfying Pu,D

u everywhere in

Rn. The function Pu,D

is called Poisson modification of u in D.

Proof Let U be an open set such that D b U b ⌦ and Rn \ U satisfiesthe measure density condition (18). Then by Lemma 9 there is an inreasingsequence {u

k

}, uk

2 W s,p(U) \ C(U), such that uk

= u outside of U anduk

converges to u pointwise in Rn. Further, we find weak solutions wk

2K

uk,�1(D,U) \ C(U) in D by Corollary 1 and Theorem 8. Moreover, {wk

}is an increasing sequence by the comparison principle, and we may definew := lim

k!1 wk

. Since wk

u by the comparison property, also w u.In addition, w is a continuous weak solution in D by Theorem 15, and

lower semicontinuous in U as a limit of an increasing sequence of continuousfunctions, and thus also in ⌦. Furthermore, according to its definition, w =



u everywhere on Rn \ D. It is then clear that it satisfies (i–ii) and (iv) ofDefinition 1.

We next check that w satisfies also Definition 1(iii). Let E b ⌦ be openand let v 2 C(E) be a weak solution in E bounded from above such that v w

on @E and almost everywhere on Rn \ E. Since w u in Rn, we have thatv u on @E and almost everywhere on Rn \ E, and thus by Definition 1(iii)that v u in E as well. This implies that, since w = u on @D, we have thatv w on @(E \ D) and almost everywhere on Rn \ (E \ D). Now, for anyx 2 @(E \D) we have

lim infE\D3y!x

w(y) � w(x) � v(x) = lim supE\D3y!x

v(y)

by the lower semicontinuity of w and continuity of v up to the boundary.This shows by the comparion principle, Theorem 16, that w satisfies Defini-tion 1(iii), and hence w is (s, p)-superharmonic in ⌦. This finishes the proofsince P

u,D

⌘ w.

The next two lemmas show that there is a natural ordering for the Poissonmodifications.

Lemma 15 Let D b ⌦ be open sets such that Rn \ D satisfies the measure

density condition (18). Let u and v be (s, p)-superharmonic functions in ⌦ such

that u v. Then Pu,D

Pv,D

in ⌦. In particular, the Poisson modification

of u in D is unique.

Proof By the proof of Theorem 18, there is an increasing sequence {wk

} con-verging pointwise to P

u,D

such that wk

2 C(D) is a weak solution in D.Since P

v,D

� Pu,D

� wk

in Rn \D and Pv,D

is (s, p)-superharmonic in ⌦ byTheorem 18, Lemma 14 yields P

v,D

� wk

in D. Letting k ! 1 finishes theproof.

Lemma 16 Let D b U b ⌦ be open sets such that both Rn \D and Rn \ Usatisfy the measure density condition (18). Let u be (s, p)-superharmonic in ⌦.

Then Pu,D

� Pu,U

in ⌦.

Proof Since Pu,D

= u inRn\D, we have Pu,D

� Pu,U

inRn\D by Theorem 18.Moreover, according to Theorem 18 P

u,D

is (s, p)-superharmonic in⌦ and Pu,U

is a continuous weak solution in U � D. Thus, Lemma 14 implies Pu,D

� Pu,U

in D.

5.2 Perron solutions

We conclude this paper by considering the Perron solutions we defined inDefinition 2. The first property is that upper and lower Perron solutions arein order.



Lemma 17 The Perron solutions Hg

and Hg

satisfy Hg

� Hg

in Rn.

Proof If Ug

or Lg

is empty, there is nothing to prove since Hg

⌘ +1 orH

g

⌘ �1, respectively. Assume then that the classes are non-empty and takeu 2 U

g

and v 2 Lg

. Then

lim inf⌦3y!x

u(y) � ess lim supRn\⌦3y!x

g(y) � ess lim infRn\⌦3y!x

g(y) � lim sup⌦3y!x

v(y)

for every x 2 @⌦ by Definition 2(iii). Both sides of the inequality above cannotbe simultaneously �1 or +1 according to Definition 2(ii). Moreover, sinceu = g = v almost everywhere in Rn\⌦, we have u � v in ⌦ by the comparisonprinciple, Theorem 16. Finally, taking the infimum over {u 2 U

g

} and thesupremum over {v 2 L

g

} finishes the proof.

The second straightforward observation is that for bounded boundary val-ues the Perron classes are non-empty.

Lemma 18 If g 2 Lp�1

sp

(Rn) is bounded from above, then the class Ug

is

nonempty.

Proof Let supRn g M < 1 and take u := M�⌦

+ g�Rn\⌦ . Then clearly u

satisfies the properties (ii-iv) of Definition 2. To obtain the property (i), wefirst have that u 2 W s,p

loc

(⌦) \ Lp�1

sp

(Rn), and testing against a nonnegativetest function ⌘ 2 C1

0

(⌦) givesZ

Rn

Z

Rn

L(u(x), u(y))�

⌘(x)� ⌘(y)�

K(x, y) dxdy

= 2

Z

Rn\⌦

Z

⌦

L(M, g(y))⌘(x)K(x, y) dxdy � 0.

Thus u is a weak supersolution in ⌦, and further (s, p)-superharmonic in ⌦

by Theorem 12.

Now, we are in a position to prove the main theorem in this section, i. e.,Theorem 2 stated in the introduction, which gives the expected alternativeresult, saying that the Perron solution has to be identically +1 or �1, or(s, p)-harmonic.

Proof (Proof of Theorem 2) Let us denote by Hg

:= Hg

, the case of Hg

:=H

g

being completely analogous. We may assume that Ug

is non-empty sinceotherwise H

g

⌘ 1 in ⌦. Since Ug

is non-empty, we must have that (Hg

)+

2Lp�1

sp

(Rn). According to Choquet’s Topological Lemma (see, e. g. [16, Lemma8.3]), there exists a decreasing sequence {u

j

} of functions in Ug

converging toa function u such that bu = bH

g

. Here the lower semicontinuous regularizationof a function f : Rn ! [�1,1] is defined by

bf(x) := limr!0

infBr(x)

f.



In particular, bf f and bf is lower semicontinuous. Let D b ⌦ be an open setsuch that Rn\D satisfies the measure density condition (18). Then P

uj ,D 2 Ug

and it is (s, p)-harmonic in D for all j by Lemma 18. Moreover, {Puj ,D}

j

is adecreasing sequence by Lemma 15. Let v

D

:= limj!1 P

uj ,D. By Harnack’sconvergence theorem (Theorem 15) either v

D

⌘ �1 in D or v is (s, p)-harmonic in D. Furthermore, we may take any larger U containing D suchthat Rn \ U satisfies the measure density condition (18). Since P

uj ,U Puj ,D

by Lemma 16, we have that limj!1 P

uj ,U ⌘ �1 in U if vD

⌘ �1 in D.Thus, H

g

⌘ �1 in ⌦ if vD

⌘ �1 in any regular open component D.

Suppose now that Hg

6⌘ �1 in ⌦. Therefore vD

is (s, p)-harmonic inD whenever the complement of D b ⌦ satisfies the measure density condi-tion (18). Let D be such a set. Theorem 18 yields H

g

Puj ,D u

j

, and takingthe limit as j ! 1 and, furthermore, lower semicontinuous regularizations,we obtain

Hg

vD

u and bHg

bvD

bu = bHg

, (38)

respectively. Consequently,

vD

= bvD

= bHg

Hg

vD

in D,

and thus Hg

= vD

in D.

To obtain the (s, p)-harmonicity for Hg

in ⌦, let {Dk

}, k = 1, 2, . . . , be anexhaustion of ⌦ by open regular subsets such that D

k

b Dk+1

. Proceedingas above, we obtain functions v

k

:= limj!1 P

uj ,Dk that are (s, p)-harmonicin D

k

and vk

= Hg

in Dk

. In particular, we have that Hg

is continuous in⌦. Since P

uj ,Dk+1 Puj ,Dk for every j by Lemma 16, we have v

k+1

vk

.Let us denote by v := lim

k!1 vk

. Then for any U b ⌦, v is (s, p)-harmonicin U by Theorem 15 since v

k

is (s, p)-harmonic in U when k is large enough.The possibility that v ⌘ �1 in U is excluded since v = H

g

6⌘ �1 in D1

.Consequently, v is (s, p)-harmonic in the whole ⌦. Now

Hg

= bHg

= bvk

� bv = v in ⌦

by continuity of Hg

and v in ⌦ together with (38). The reverse inequalityholds by (38), and thus H

g

= v in ⌦. Moreover, since Hg

= g = v almosteverywhere in Rn \⌦, we conclude that H

g

is (s, p)-harmonic in ⌦.

Remark 10 Lemma 18 and Theorem 2 do also hold for the more general Perronsolutions mentioned in Remark 5.

We conclude this paper with the lemma below, which assures that if thereis a solution to the Dirichlet problem, then it is necessarily the Perron solution.In particular, this is the case under the natural hypothesis of Theorem 17.



Lemma 19 Assume that h 2 C(⌦) is a weak solution in ⌦ such that

lim⌦3y!x

h(y) = g(x) for every x 2 @⌦ and h = g a. e. in Rn \⌦

for some g 2 C(⌦0) \ Lp�1

sp

(Rn) with ⌦0 c ⌦. Then Hg

= h = Hg

.

Proof The situation is symmetric, so we only need to prove the result for Hg

.We have h � H

g

since h 2 Ug

. To obtain the reverse inequality, let u 2 Ug

.Then for every " > 0 there exists an open set D b ⌦ such that u + " > h inRn \D. Consequently, u+ " � h in D since u+ " is (s, p)-superharmonic in ⌦.Letting " ! 0 we obtain that u � h, and taking the infimum over U

g

yieldsH

g

� h.

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Fractional superharmonic functions and the perron method