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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Remarks on a nonlinear nonlocal operator in Orlicz spaces

Ernesto Correa / Arturo de Pablo
Published Online: 2019-05-22 | DOI: https://doi.org/10.1515/anona-2020-0002

Abstract

We study integral operators Lu(χ)=ψ(u(x)u(y))J(xy)dyof the type of the fractional p-Laplacian operator, and the properties of the corresponding Orlicz and Sobolev-Orlicz spaces. In particular we show a Poincaré inequality and a Sobolev inequality, depending on the singularity at the origin of the kernel J considered, which may be very weak. Both inequalities lead to compact inclusions. We then use those properties to study the associated elliptic problem Lu=fin a bounded domain Ω,and boundary condition u ≡ 0 on Ωc;both cases f = f(x) and f = f(u) are considred, including the generalized eigenvalue problem f(u)=λψ(u).

Keywords: Nonlocal equations; integral operators; p–fractional Laplacian; Orlicz spaces

MSC 2010: 45P05; 46E35; 45G10

1 Introduction

The aim of this paper is to study the properties of the nonlinear nonlocal operator

Lu(χ)=LJ,ψu(χ)ψ(u(χ)u(y))J(χy)dy,(1.1)

where ψ:is a nondecreasing, continuous, unbounded odd function, and J:N+is a measurable function satisfying

{J(z)>0,J(z)=J(z),JL1(B1),Nmin(1,|z|q0)J(z)dz<,forsomeq0>0.(H0)

Br denotes the ball Br={zN:|z|<r}.This set of hypotheses is assumed throughout the paper without further mention. We also denote g* = inf {q0 > 0 : (H0) holds }, which measures in some sense the differential character of the operator.

The power case ψ(s)=k|s|p2sfor some p>1,J(z)=c|z|Nσp/2for some 0<σ<2,is known as the σ– fractional p–Laplacian operator. We want to consider here general functions ψand J more than just powers, so we are led to study some Orlicz and Sobolev-Orlicz spaces, see below, which makes the study nontrivial. On the other hand, we are also interested in the limit case of integrability, which in our context means q* = 0, that is, the singularity of the kernel can be weaker than that of any fractional Laplacian or p– Laplacian. Some of the results also hold for more general kernels, J = J(x,y), satisfying only a lower estimate J(x,y)J0(xy),with J0 in the above hypotheses, but we prefer to keep the proofs in a simpler way.

For problems including operators like (1.1), in particular the fractional p–Laplacian, and the motivations for their study we refer to [9].

1.1 The associated Orlicz spaces

Formula (1.1) makes sense pointwise for regular functions with some extra restriction on the nonlinearity ψ and the kernel J, see Section 2. In order to define the operator Lin weak sense we consider the nonlocal nonlinear interaction energy (linear in the second variable)

ε(u;φ)=122Nψ(u(χ)u(y))(φ(χ)φ(y))J(xy)dχdy,(1.2)

and we put

Lu,φ=ε(u;φ).

Clearly, by the symmetry properties of ψ and J we have ε(u;φ)=NLuφfor regular functions. But the above allows to define Lalso for functions in a Sobolev type space. To this end we define the functionals

F(u)=NΨ(u(χ))dχ,(1.3)

E(u)=122NΨ(u(χ)u(y))J(xy)dχdy,(1.4)

with ψ' = ψ. The properties of ψ imply that ψ is an strict Young function, so we can consider the Orlicz spaces

LΨ(N)={u:N,F(u)<},(1.5)

WJ,Ψ(N)={uLΨ(N),E(u)<}.(1.6)

Observe that in general ε(u;u)cE(u)for any constant c > 0, the equality being true only in the power case ψ(u)=k|u|p2u,and then c = p. What we have is that ε is the Euler-Lagrange operator associated to the functional E, that is,

E(u),φ=ε(u;φ)

for every u,φWJ,Ψ(N).

The above spaces do not have good properties unless we impose some conditions on the nonlinearity ψ. The simplest case is when

c1sp1Ψ(s)c2sp1,s>0,p>1,(1.7)

so that the space LΨ(N)coincides with Lp(N),and the Sobolev space WJ,Ψ(N)is denoted by WJ,p(N).But we are interested in more general functions. Thus we consider the set, for some pq>1,

Γp,q={Ψ:+,convex,symmetric,satisfyingΨ(0)=0,Ψ(1)=1,qsΨ(s)Ψ(s)ps0}.(1.8)

The condition ψ(1) = 1 is for normalization purposes and simplifies some expression. We thus deal with functions that lie between two powers, for instance a sum of powers, but we also allow for perturbation of powers like Ψ(s)=c|s|p|log(1+s)|r,min{p,p+r}>1.The first property deduced from (1.8) is the relation between the interaction energy ε and the functional E,

qE(u)ε(u;u)pE(u).(1.9)

Our main interest lies in studying the properties of the spaces (1.3) and (1.4) for nonlinearities ψ in the class Γp,q.In particular we have that LΨ(N)andWJ,Ψ(N)are reflexive Banach spaces, with norms defined, for instance, in (2.1) and (2.3). On the other hand, if q > q*, see (H0), then the functional E(u) is well defined and finite for functions satisfying F(u)<,see Proposition 2.3. This means the inclusion W1,Ψ(RN)WJ,Ψ(RN) the former being the standard Orlicz-Sobolev space of functions in LΨ(N)with gradient also in LΨ(N).

When dealing with problems defined in bounded domains ΩN,since the Dirichlet conditions must be prescribed in the complement ΩcN\Ω,instead of just on the boundary, precisely by the nonlocal character of the operator, it is convenient to consider the space

W0J,Ψ(Ω)={uWJ,Ψ(N),u0inΩc}.

Without imposing any singularity condition on the kernel J at the origin, besides of course being not integrable, that is q* may be zero, we show that a Poincaré inequality holds, so that we have the embedding

W0J,Ψ(Ω)LΨ(Ω).(1.10)

Observe that if J were integrable then W0J,Ψ(Ω)LΨ(Ω).

Assuming now q* > 0 (and some monotonicity near the origin, (3.5)) we obtain a better result, namely a Sobolev embedding

W0J,Ψ(Ω)LΨr(Ω),1r<r{NNqifq<N,ifqN,(1.11)

which is compact. The borderline r = r* when q* < N produces also an embedding W0J,Ψ(Ω)LΨr(Ω),provided J(z)c|z|Nqnear the origin, but without compactness. In the limit case q* = 0, which would give r* = 1 in (1.11), we do obtain compactness of the inclusion (1.10) by assuming a minimum of singularity on the kernel, the extra condition lim|z|0+|z|NJ(z)=.See Theorems 3.2–3.4.

1.2 Elliptic problems

With this machinery we next study the problem

{Lu=f,inΩ,u=0,inΩc.(1.12)

This problem must be considered in weak sense with the aid of the interaction energy ε, that is, any solution u satisfies

ε(u;φ)=Ωfφ,φW0J,Ψ(Ω).(1.13)

We study first the case f = f(x) in an appropriate space. We obtain existence and uniqueness of a solution, see Theorem 4.1. We also show some integrability properties of the solution in terms of the data f when ψ is restricted to the power-like case (1.7), see Theorems 4.3 and 4.4. In particular the solution is bounded provided fLm(Ω)withm>N/qifq>0,see Theorem 4.5. For the corresponding results in the case of the fractional p–Laplacian see [4]. We do not address regularity issues in this paper. For Hölder regularity results in the case of the fractional p–Laplacian we refer to [12], [13], [22], [24].

We then pass to study the case f = f(u) in problem (1.12). In Theorem 5.1 we prove existence of a nonnegative nontrivial solution in the lower range, which roughly speaking in the power-like case ψ(s) ~ sp–1, fit) ~ ctm–1, means m < p. The intermediate range p<m<m=NpNq,below the Sobolev exponent, is studied in Theorem 5.2 using the Mountain Pass Theorem. We also use a Pohozaev inequality in order to show nonexistence, in the exact power case, for supercritical powers m>m=NpNδ,δ>0being a constant depending on the kernel J, see Corollary 5.4. We must remark that all the conditions on the reaction f are very involved in terms of ψ, and are not as clean as suggested by the above, see the precise conditions (5.2) and (5.4). We refer to [20] for the study of nonlinear problems like the above, even with more general reactions, for the fractional p–Laplacian case.

We finally are interested in the limit case m = p, which corresponds to the so called generalized eigenvalue problem

Lu=λψ(u),in Ω,u=0,in Ωc.

We prove that there exists a first positive eigenvalue and a first positive eigenfunction, which is bounded provided q* > 0, Theorem 6.1. The fractional p–eigenvalues have been studied in [22] and [17].

1.3 Organization of the paper

We begin with a preliminary Section 2 where we study the properties of the Orlicz spaces LΨ(N)and WJ,Ψ(N)by means of some inequalities satisfied by the nonlinearity ψ and the functionals F and E. Section 3 shows the Sobolev inclusions of the spaces W0J,Ψ(Ω).Finally Sections 46 are devoted to the study of problem (1.12) for the different reactions commented upon before. In what follows the letters c or ci will denote some constants that do not depend on the relevant quantities, and may change from line to line.

After the writing of this paper was completed, we learned that, at the very same time and independently of us, Fernández-Bonder and Salort proved in [15] some properties of fractional Sobolev-Orlicz spaces defined in a way related to ours.

2 Preliminaries

In this section we study in detail the properties of the Orlicz spaces LΨ(N)andWJ,Ψ(N)defined in (1.5) and (1.6), and the corresponding spaces in a bounded domain . We refer to [25] for instance for the general theory of Orlicz spaces. We begin by studying the Young functions in the set Γ p,q. First observe that ΨΓp,q,pq0,implies

min{|s|p,|s|q}Ψ(s)max{|s|p,|s|q}.

Associated to any given positive function g:++we consider its characteristic functions, for s > 0,

yg¯(s)=infχ>0g(sχ)g(χ),yg+(s)=supχ>0g(sχ)g(χ).

These are nondecreasing functions that satisfy

Lemma 2.1

For any ΨΓp,q,pq>0,

min{sp,sq}yΨ(s)yΨ+(s)max{sp,sq}qpyΨ(s)syΨ(s)yΨ+(s)pqyΨ+(s)s.

Proof. If s > 1 we have that

log(Ψ(sχ)Ψ(χ))=χsχΨ(t)Ψ(t)dtpχsχ1tdt=plogs,

and thus Ψ(sχ)spΨ(χ).The other estimates for Ψare analogous. The inequalities for Ψ' are deduced from the definition of Γp,q.

The complementary function Φ of a Young function Ψ is defined such that (Φ)1=Ψ.If we normalize it to satisfy Φ(1)=1we have, for every pq>1([25, Corollary 1.1.3])

ΨΓp,qΦΓq,p,p=pq1,q=qq1.

These two functions satisfy the Young inequality

abΨ(a)+Φ(b),a,b,

and equality holds only if b=Ψ(|a|)signa.From this point on we always assume q > 1.

Let Ψ ∈ Γp,q be fixed and consider the corresponding Orlicz space LΨ(N).It is a linear space that satisfies

Lp(N)Lp(N)LΨ(N)Lp(N)+Lp(N),

and in the case of bounded domains

Lp(Ω)LΨ(Ω)Lq(Ω).

Also it is a Banach space with norm, called Luxemburg norm,

uLΨ=inf{k>0:F(u/k)1}.(2.1)

We recall that other equivalent norms are also used in the literature. The following result allows us to use F(u) instead of uLΨin most calculations.

Lemma 2.2

yΨ(uLΨ)F(u)yΨ+(uLΨ).(2.2)

Proof. Let a=uLΨ.We clearly have F(u/a)1.Then

F(u)=NΨ(u(χ))dχyΨ+(a)N(u(χ)a)dχyΨ+(a).

On the other hand, for every ε > 0 we have so that

F(u)yΨ(a+ε)NΨ(u(χ)a+ε)dχyΨ(a+ε).

The dual space of LΨ(N)isLΦ(N),where Φ is the complementary function, and thus they are both reflexive Banach spaces.

We also consider the Sobolev type space WJ,Ψ(N).In the same way as before it is a Banach space with norm

uWJ,Ψ=uLΨ+[u]WJ,ΨuLΨ+inf{k>0:E(u/k)1}.(2.3)

The second term is a kind of Gagliardo seminorm in the context of Young functions. For this seminorm an analogous property as that of Lemma 2.2 also holds,

yΨ([u]WJ,Ψ)E(u)yΨ+([u]WJ,Ψ).(2.4)

In order to show that this space is reflexive as well we consider the weighted space

LΨ(2N,J)={w:2N,2NΨ(w(χ,y))J(xy)dχdy<}

and put M=LΨ(N)×LΨ(2N,J).Clearly, the product space M is reflexive. The operator T:WJ,Ψ(N)Mdefined by Tu = [u,w], where w(x,y) = u(x) – u(y),is an isometry. Since WJ,Ψ(N)is a Banach space, T(WJ,Ψ(N))is a closed subspace of M. It follows that T(WJ,Ψ(N))is reflexive (see [7, Proposition 3.20]), and consequently WJ,Ψ(N)is also reflexive.

We now take a look at the properties of the space WJ,Ψ(N)in terms of the properties of the kernel J, in particular its singularity at the origin, which is reflected in the exponent q*, see (H0).

Proposition 2.3

If ΨΓp,qwith p ≥ q > q* then

W1,Ψ(N)WJ,Ψ(N)

and moreover

E(u)c(F(u)+F(u)).(2.5)

Proof. We decompose the integral

E(u)=12N(|z|<1Ψ(u(χ)u(χ+z))J(z)dz+|z|>1Ψ(u(χ)u(χ+z))J(z)dz)dχ=12(I1+I2).

The far away integral is easy to estimate

I22NΨ(u(χ))dχ|z|>1J(z)dz=cF(u).

As to the inner integral, we have

I1N|z|<1Ψ(u(χ)u(χ+z)|z|)y+(|z|)J(z)dzdχN|z|<1Ψ(01|u(χ+tz)|dt)y+(|z|)J(z)dzdχN|z|<101Ψ(|u(χ+tz)|)dty+(|z|)J(z)dzdχ|z|<101NΨ(|u(χ+tz)|)dχdty+(|z|)J(z)dz=NΨ(|u(χ)|)dχ|z|<1y+(|z|)J(z)dz=cF(|u|),

since y+(|z|)=yΨ+(|z|)|z|qin the set {|z|<1},and using hypothesis (H0).

If the kernel J behaves like that of the fractional Laplacian

c1|z|NαJ(z)c2|z|Nα,(2.6)

then we also have the following interpolation estimate

Proposition 2.4

If J satisfies (2.6) for some α > 0, and Ψ ∈ Γ p,q with p ≥ q > α then

E(u)cF(u)max{(F(u)F(u))α/p,(F(u)F(u))α/q}.(2.7)

Proof. We apply inequality (2.5) to the rescaled function uλ(x) = u(λx). We first observe that by (2.6),

E(uλ)=122NΨ(u(λχ)u(λy))J(χy)dχdy=122Nλ2NΨ(u(χ)u(y))J(λ1(xy))dχdyc122NλN+αΨ(u(χ)u(y))J(χy)dχdy=cλN+αE(u).

Thus

E(u)cλNαE(uλ)λNαc(F(uλ)+F(uλ))cλNα(λNF(u)+yΨ+(λ)λNF(u))cλα(a+bmax{λp,λq})cg(λ),

a=F(u),b=F(u).Minimizing the right-hand side in A we get

min{g(λ)}={a(b/a)α/qifa/b<q/α1,a(b/a)α/qifa/b>q/α1,a(1+b/a)ifq/α1<a/b<p/α1.

From this we easily deduce (2.7).

In the power-like case we obtain from the above the well-known interpolation result.

Corollary 2.5

If J satisfies (2.6) for some α > 0, and ψ satisfies (1.7) with p > α then

E(u)cF1α/p(u)Fα/p(u),

or which is the same

uWα/2,pcup1α/pupα/p.

We now turn our attention to the operator L.The pointwise expression (1.1) does not always have a meaning.

Let us look at some easy situations where Luis well defined.

We may take, for instance, Ψ" nondecreasing and uC02(N).Another less trivial example is q > q* + 1 and uCα(N)L(N)withqq1<α<1,so that

|Lu(χ)|u|χy|>1J(χy)dy+|χy|<1|χy|(q1)αJ(χy)dy<.

We now show some useful inequalities. The first one is a Kato type inequality, that is, the result of applying the operator Lto a convex function of u. We refer to [21] and [10], respectively, for the well-known inequalities

Δ|u|sign(u)(Δ)u,(Δ)σ/2(u2)2u(Δ)σ/2u.

Proposition 2.6

If A is a positive convex function and Luis well defined, then L(A(u))is also well defined and

L(A(u))yψ+(A(u))Lu.

Proof. We just observe that since A is convex and ψ is nondecreasing, we have

ψ(A(u(χ))A(u(y)))ψ(A(u(χ))(u(χ)u(y)))yψ+(A(u(χ)))ψ(u(χ)(y)).

Now integrate with respect to J(x - y) dy to get the result.

As a Corollary we obtain an integral version of the Kato inequality, useful in the applications.

Corollary 2.7

Assume Gyψ+(A)A.Then

ε(u,G(u))qE(A(u)).

Of later use are also the following two inequalities

ε(u,u+)ε(u+,u+),E(u)E(|u|),(2.8)

whose proof is immediate just looking at the signs of the corresponding functions.

Related to those inequalities is the well known Stroock-Varopoulos inequality, see [29] for the linear case Ψ(s)=|s|2andJ(z)=|z|Nσfor some 0<σ<2,and [5] for general Lévy kernels J. It is of the type of the integral Kato inequality, but the functions for which it holds is different. In the case of powers they coincide but for the coefficient, which is always better in the Stroock-Varopoulos inequality. We show here a generalized Stroock-Varopoulos inequality.

Proposition 2.8

Assume δ=infs>0ψ(s)yψ+(s)>0andletuWJ,Ψ(N)such that G(u),A(u)WJ,Ψ(N),where A and G satisfy G|Ψ(A)|.Then

ε(u;G(u))δqpE(A(u)).(2.9)

Proof. The proof follows from a calculus estimate. For any d > c we have that

Ψ(|A(d)A(c)|)Ψ(cd|A(s)|ds)yΨ+(dc)Ψ(1dccd|A(s)|ds)yΨ+(dc)dccdΨ(|A(s)|)dspqyψ+(dc)|G(d)G(c)|pδqψ(dc)|G(d)G(c)|.

The same inequality is obtained for d ≤ c. We now deduce (2.9) by choosing d = u(x), c = u(y) and integrate with respect to J(x – y) dxdy.

For instance in the case of a sum of powers, Ψ(s)=i1MkiSpi,p1<p2<pM,we have yψ+(s)=max{sp11,spM1}and δ=min{k1p1,kMpM}.

All the above inequalities hold also, with different constants, for nonlinearities that behave like a power, i.e., when they satisfy (1.7) instead of (1.8). In particular in that case the integral Kato inequality and the Stroock-Varopoulos inequality coincide, but for the coefficient, both giving

ε(u;|u|r1u)cE(|u|r+p1p).(2.10)

We also obtain some calculus inequalities needed in proving uniqueness results in the last sections. We borrow ideas from [23] and [18] that deal with the exact power case.

Lemma 2.9

Let ψ be a nonnegative, nondecreasing, continuous odd function and let ψ = ψ'.

  • i)

    If ψ satisfies

sψ(s)ψ(s)1foreverys0,(2.11)

then

(ψ(a)ψ(b))(ab)4Ψ(ab2).(2.12)

  • ii)

    If ψ is concave in (0, ∞) then

(ψ(a)ψ(b))(ab)ψ(|a|+|b|)(ab)2.(2.13)

  • iii)

    If ψ satisfies

c1|s|p2ψ(s)c2|s|p2forsome1<p<2andeverys0,(2.14)

then

(ψ(a)ψ(b))(ab)c(Ψ(ab))2p(Ψ(a)+Ψ(b))2pp.(2.15)

Proof. i) We begin by proving a Clarkson inequality. Condition (2.11) implies that the function g(s)=Ψ(|s|)is convex. Therefore

Ψ(a+b2)+Ψ(ab2)=g((a+b2)2)+g((a+b2)2)g((a+b2)2+(ab2)2)=g(a2+b22)12(g(a2)+g(b2))=12(Ψ(a)+Ψ(b)).

Now the convexity of Ψ implies

Ψ(a)Ψ(b)+Ψ(b)(ab),

and also

Ψ(a+b2)Ψ(b)+12Ψ(b)(ab),

so that

Ψ(a)+Ψ(b)2Ψ(a+b2)+2Ψ(ab2)2Ψ(b)+Ψ(b)(ab)+2Ψ(ab2).

This gives

Ψ(a)Ψ(b)+Ψ(b)(ab)+2Ψ(ab2),

and reversing the roles of a and b,

Ψ(b)Ψ(a)+Ψ(a)(ba)+2Ψ(ab2).

Adding these two inequalities we get (2.12).

  • ii)

    Developing the function Ψ around the point s = a we get

Ψ(b)=Ψ(a)+Ψ(a)(ba)+(ba)201(1s)Ψ(a+s(ba))dsΨ(a)+Ψ(a)(ba)+(ba)2Ψ(a+b)01(1s)ds.

We have used that |a+s(ba)||a|+|b|andΨis nonincreasing in (0, ∞). Observe that though Ψ" is singular at zero, the integral is convergent. We conclude as before.

  • iii)

    As ii), using (2.14) in the last step.

To end this section devoted to the preliminary properties of E and F, we point out a result on symmetrization that says that the energy E(u) decreases when we replace u by its symmetric rearrangement (the radially deacreasing function with the same distribution function as u).

Theorem 2.10

If uWJ,Ψ(N)and u* is its decreasing rearrangement, then

E(u)E(u).

This property is well known for the norm in W0σ/2,p(Ω),0<σ2,p>1,see [1], and is proved in [11] for general kernels when p = 2. The same proof can be used to get the result in our situation, so we omit the details. See also [3].

3 Sobolev inclusions

In this section we consider a nonlinearity ΨΓp,q,pq>max{q*, 1} fixed. As to the kernel J, besides condition (H0) we also consider, for some results, the singularity condition at the origin

J(z)c|z|Nαfor0<|z|<1,α>0.(3.1)

Clearly it must be αq.In fact in the fractional p–Laplacian case it is α=σp/2.Other kernels could also be considered, for instance J(z)=|z|Nμ|log(|z|/2)|β,for0<|z|<1,with μ0,(andβ1ifμ=0).In that case it is q=μ.Ifμ>0then J satisfies (3.1) with α = μ if β ≥ 0, but if β < 0 it satisfies (3.1) only with 0<α<μ.A more intricate example can be constructed by the following piecewise definition of J,

J(z)={|z|Nif22k1<|z|22k,|z|Nμif22k<|z|22k+1,

k1,μ>0.Here we have q* = μ while condition (3.1) does not hold for any a > 0.

Assume now that u has support contained in Ω¯.Then

E(u)=122NΨ(u(χ)u(y))J(χy)dχdy=12ΩΩΨ(u(χ)u(y))J(χy)dχdy+ΩΩΨ(u(χ))J(χy)dχdyΩΨ(u(χ))Λ(Ω;χ)dχ,

where

Λ(Ω;x)=ΩcJ(xy)dy.

If

μ=min{J(z):|z|R}>0,R>δ=supχΩdist(χ,Ωc),

then

Λ(Ω;χ)μ|{δ<|z|<R}|=A>0foreveryχΩ.

This gives the Poincaré inequality

E(u)AF(u),(3.2)

and the inclusion W0J,Ψ(Ω)LΨ(Ω).We remark that in the case of integrable kernel J we immediately would get E(u)cJ1F(u),and thus W0J,Ψ(Ω)LΨ(Ω).

In order to obtain better energy estimates in the case q* > 0, which would yield better space embeddings, we need a better estimate of the function Λ(Ω; •) in terms of the kernel J. The following result is essentially contained in [28, Lemma A.1].

Proposition 3.1

Λ(Ω;χ)P((|Ω|ωN)1/N)foreveryχΩ,

where P(s)=|z|>sJ(z)dz.In particular, if J satisfies (3.1) then

Λ(Ω;χ)c(Ω)|Ω|α/NforeveryΩΩ.

This estimate allows us to prove, assuming condition (3.1), the Sobolev embedding W0J,Ψ(Ω)LΨr(Ω)forevery 1rrNNα,ifα<N,for every 1r<ifαN.The proof uses ideas of [27] and [14]. If a ≥ N we obtain the result substituting a by any number below N and close to N, since (3.1) still holds for that exponent.

Theorem 3.2

Assume J satisfies condition (3.1) with 0 < α < N. Then there exists a positive constant C = C(N, p, q, α, ) such that, for any function uW0J,Ψ(Ω)we have uLΨr(Ω)for every 1rrNNαand

Ψ(u)rCE(u).(3.3)

Proof. We prove the inequality for r = r*, and then the result for r < r* follows by Hölder inequality. We can assume, without loss of generality, that u is radially deacreasing and Ω=BR,since substituting u by its symmetric decreasing rearrangement u*, we have by Theorem 2.10,

Ψ(u)r=Ψ(u)rCE(u)CE(u).

We may also consider the case of u bounded, since if not, taking the sequence uT = min{u, T}, and thanks to the Dominated Convergence Theorem, we would get the result in the limit T ⟶ ∞. We now define

Ak:={χN:u(χ)>2k},ak=|Ak|,Dk:=Ak\Ak+1,dk=|Dk|.

We have Ak = BRk, with Rk+1RkR.Also ak = dk = 0 for all large k, say for k > M. Now we compute,

Ψ(u)r=(k=MDkΨr(u(χ))dχ)1/rk=MΨ(2k+1)dk1/rck=MΨ(2k)ak1/r,

since r > 1. On the other hand, if χDiand yDj,with ji2,then

|u(χ)u(y)|2i2j+12i1.

Thus

i=Mj=i2DiDjΨ(u(χ)u(y))J(χy)dydχi=MΨ(2i1)Diji2DjJ(χy)dydχi=MΨ(2i1)DiAi1cJ(χy)dydχci=MΨ(2i)ai1α/Ndi=ci=MΨ(2i)ai1α/N(aik=i+1Mdk)=c(AB).

The second term can be estimated as

B=i=Mk=i+1MΨ(2i)ai1α/Ndk=dk=k=Mi=MΨ(2i)ai1α/Ndkk=Mi=MΨ(2i)ai1α/Ndkk=MΨ(2k)ak1α/Ndki=k1y+(2ik)=k=MΨ(2k)ak1α/Ndkm=1y+(2m)=ck=MΨ(2k)ak1α/Ndk=c(AB).

We deduce the estimate

E(u)i=Mj=i2DiDjΨ(u(χ)u(y))J(χy)dydχCA=Ci=MΨ(2i)ai1α/Nai.

We conclude, using [27, Lemma 5], since 1r=1αN,

E(u)Ci=MΨ(2i)ai1α/NCΨ(u)r.

We also prove that the above embedding is compact provided r < r*. To this end we first show the compactness of the inclusion for r = 1 and then interpolate with the continuity for r = r*. It is important to remark that the inclusion W0J,Ψ(Ω)LΨ(Ω)is compact even when q* = 0, which implies r * =1, provided the following conditions on the kernel at the origin hold

lim|z|0+|z|NJ(z)=,(3.4)

J(z1)cJ(z2)forevery0<|z1||z2|1,andsomec>0.(3.5)

This implies some kind of minimal singularity and some monotonicity near the origin. In particular this allows to consider for instance a kernel of the form J(z)=|z|N|log||z|β,β>0,for|z|0.See [11] for the caseΨ(s)=|s|2.

Theorem 3.3

Assume J satisfies (3.4) and (3.5). Then the embedding W0J,Ψ(Ω)LΨ(Ω)is compact.

Proof. The idea of the proof goes back to the Riesz-Fréchet-Kolmogorov work. We follow the adaptation to the fractional Laplacian framework performed in [14].

Let AW0J,Ψ(Ω)be a bounded set. We show that A is totally bounded in LΨ(Ω),i.e.,foranyϵ(0,1)there exist β1,βMLΨ(B1)such that for any u ∈ A there exists j ∈ { 1 , . . . , M} such that

F(uβj)є.(3.6)

We take a collection of disjoints cubes Q1,...QM' of side ρ < 1 such that Ωj=1MQj.For any x ∈ we define j(x) as the unique integer in {1,..., M' } for which χQj(χ).Also, for any u ∈ A, let

Q(u)(χ):=1|Qj(χ)|Qj(χ)u(y)dy.

Notice that

Q(u+v)=Q(u)+Q(u)foranyu,vA,

and that Q(u) is constant, say equal to qj (u), in any Qj, for; ∈ {1,..., M'}. Therefore, we can define

S(u):=ρN(Ψ(q1(u)),,Ψ(qM(u)))M,

and consider the spatial 1-norm in Mas

v1:=j=1M|yj|,foranyv=(y1,yM)M.

We observe that

F(Q(u))=j=1MQjΨ(Q(u)(χ))dχρNj=1MΨ(qj(u))=S(u)1,(3.7)

and also, by Jensen inequality and (3.2),

S(u)1=j=1MρN|Ψ(qj(u))|=ρNj=1MΨ(1ρNQju(y)dy)j=1MQjΨ(u(y))dy=ΩΨ(u(y))dyc.(3.8)

In the same way,

F(Q(u)a)=F(Q(u)Q(a))S(u)S(a))1

for every constant a. In particular from (3.8) we obtain that the set S(A) is bounded in Mand so, since it is finite dimensional, it is totally bounded. Therefore, there exist b1,bKMsuch that

S(A)i=1KBη(bi),(3.9)

where Bη(bi) are the 1–balls of radius η centered at bi. For any i{1,,K},we write the coordinates of bi asbi=(bi,1,,bi,M)M.ForanyχΩweset

βi(χ)=Ψ1(ρNbi,j(χ)),

where j(x) is as above. Notice that βi is constant on Qj i.e. if χQjthen

Q(βi)(χ)=Ψ1(ρNbi,j(χ))=Ψ1(ρNbi,j(χ))=βi(χ)(3.10)

and so qj(βi)=Ψ1(ρNbi,j);thus S(βi)=bi.Furthermore, again by Jensen inequality,

F(uQ(u))=j=1MQjΨ(u(χ)Q(u)(χ))dχ=j=1MQjΨ(1ρNQj(u(χ)u(y))dy)dχ1ρNj=1MQjQjΨ(u(χ)u(y))dydχ1l(ρ)j=1MQjQjΨ(u(χ)u(y))J(χy)dydχcl(ρ),

where is some function satisfying ℓ(z) ≤ |z|N J(z). Using (3.4), (3.5) we can take ℓ radial satisfying ℓ(0+) = ∞.

Consequently, for any j{1,,K},recalling (3.7) and (3.10)

F(uβj)F(uQ(u))+F(Q(u)Q(βj))+F(Q(βj)βj)c(1l(ρ)+S(u)S(βj))1).

Now recalling (3.9) we take; ∈ {1,...,K} such that S(u) ∈ Bη(bj), that is

S(u)S(βj))1=S(u)bj1<η.

We conclude by choosing ρ and η small so as to have c 1(ρ)+η<ϵ.

As a corollary we obtain the full compactness result in the fractional case.

Theorem 3.4

Assume J satisfies (3.1) and (3.5). Then the embedding W0J,ΨΩLΨr(Ω)is compact for every 1r<rifα<N,forevery1r<ifαN.

Proof. As before if we obtain the result substituting α by any number below N. By classical interpolation

Ψ(u)rΨ(u)1λΨ(u)r1λcF(u)λ,

where 1r=λ+1λr.Therefore we can obtain, instead of (3.6), the estimate

ΩΨr(u(χ)βj)dχcϵλr,

and we are done.

4 The problem with reaction f = f(x)

We start with this section the study of some elliptic type problems associated to our nonlinear nonlocal operator

Here we consider the problem

{Lu=f(χ),inΩ,u=0,inΩc.(4.1)

Given any fϵ(W0J,Ψ(Ω)),the dual space, we say that uϵW0J,Ψ(Ω)is a weak solution to (4.1) if (1.13) holds.

By Poincaré inequality (3.2) we have that fϵ(W0J,Ψ(Ω))for instance provided fϵLΦ(Ω),where Φ is the complementary function of Ψ.

We next show that problem (4.1) has a weak solution. We do not know if this solution is a strong solution, that is if Luis defined pointwise and the equality in (4.1) holds almost everywhere. On the other hand, we are able to show uniqueness assuming some extra conditions on the function Ψ. In the exact power case Ψ(s)=|s|pthese extra conditions cover the full range p > 1.

Theorem 4.1

For any fϵ(W0J,Ψ(Ω))there exists a solution uϵW0J,Ψ(Ω)to problem (4.1). If ψ satisfies either condition (2.11) or (2.14) then the solution is unique.

Proof. Existence follows by minimizing in W0J,Ψ(Ω)the functional

I(v)=E(v)Ωfv.

Clearly it is well defined, lower semicontinuous and Fréchet differentiable with

I(v),φ=ε(v;φ)Ωfφ

for every v,φϵW0J,Ψ(Ω).To see that it is coercive we first observe that vWJ,Ψimplies E(v).

Actually, by (2.2), (2.4) and Poincaré inequality,

vWJ,Ψ=vLΨ+[v]WJ,ΨC((yΨ)1(F(v))+(yΨ)1(E(v)))cmax{(E(v))1/p,(E(v))1/p}.

Now use Hölder inequality in Orlicz spaces,

|Ωfv|cvW0J,Ψc=supWW0J,Ψ=1|Ωfw|.

The last quantity is known as the Orlicz norm of f in (W0J,Ψ(Ω)),an is equivalent to the Luxemburg norm, see [25]. We thus get

I(v)E(v)c(E(v))1/q

as vWJ,Ψ.Therefore there exists a minimum of I, attained by compactness for some function uW0J,Ψ(Ω) which is a weak solution to our problem.

We now show uniqueness. Suppose by contradiction that there exist two functions u1,u2W0J,Ψ(Ω)such that

ε(u1;φ)=ε(u2;φ)φW0J,Ψ(Ω).(4.2)

Assume first that (2.11) holds. We have, denoting a = u1(x) – u1(y), b = u2(x) – u2(y), and using (2.12),

E(u1u2)=122NΨ(ab)J(xy)dxdyc2N(ψ(a)ψ(b))(ab)J(xy)dxdy=c(ε(u1;u1u2)ε(u2;u1u2))=0

by (4.2). This implies u1=u2.

Assume now condition (2.14). We calculate, using Hölder inequality and (2.15),

E(u1u2)=122NΨ(ab)J(xy)dxdyc(2N(Ψ(ab))2/p(Ψ(a)+Ψ(b))2ppJ(xy)dxdy)p2(2N(Ψ(a)+Ψ(b))J(xy)dxdy)1p2c(2N(Ψ(a)Ψ(b))(ab)J(xy)dxdy)p2(2N(Ψ(a)+Ψ(b))J(xy)dxdy)1p2=c(ε(u1;u1u2)ε(u2;u1u2))p2(E(u1)+E(u2))1p2=0.

A maximum principle is easy to obtain.

Proposition 4.2. If uWJ,Ψ(N)then

ε(u,φ)φWJ,Ψ(N),φ0u0inΩc}u0inΩ.

Proof. Since u0anduWJ,Ψ(N),we have, by (2.8),

0ε(u,u)ε(u,u)0.

Hence u=0.

We now study the integrability properties of the solution in terms of the integrability of the datum in the power-like case (1.7). In the exact power case of the fractional p–Laplacian these integrability properties have been obtained in [4]. Our proofs in the more general case treated in this paper differ from theirs in that we are using Stroock-Varopoulos inequality instead of Kato inequality, and that we allow for the limit case q* = 0, which does not make sense in the fractional p–Laplacian. All the proofs are based on the well known Moser iteration technique for the standard Laplacian case, see for example the book [19].

The first result uses no singularity condition on the kernel J, besides being nonintegrable.

Theorem 4.3

Assume condition (1.7). If u is a weak solution to problem (4.1) with fLm(Ω)then u ∈ Lm(p1)(Ω).

Of course this result is not trivial only if m>pp1,since u being a weak solution it belongs toW0J,Ψ(Ω)Lp(Ω).

Proof. Without loss of generality we may assume u0,and this simplifies notation; the general case is obtained in a similar way. We define for β1andK>0the function

H(s)={sβ,sK,linear,s>K.

We choose as test function φ=G(u)=0uΨ(H(s))ds.It is easy to check that φW0J,Ψ(Ω).In fact

E(φ)yΨ+(Ψ(βKβ1))E(u)<.

We obtain on one hand, using the Stroock-Varopoulos inequality (2.9) and the Poincaré inequality (3.2),

ε(u;G(u))cE(H(u))cF(H(u)),(4.3)

and on the other hand, using Hölder inequality,

ΩfG(u)fmG(u)m.(4.4)

Letting Kin the definition of H, the inequalities (4.3) and (4.4) give

upβpβcfmu((β1)p+1)m(β1)p+1.(4.5)

Choosing now β=m(p1)p,we get

um(p1)cfm1p1.

The same proof allows to gain more integrability when condition (3.1) holds.

Theorem 4.4

Assume conditions (1.7) and (3.1) and let u be a weak solution to problem (4.1). where fLm(Ω),. m<N/α.ThenuLm(p1)NNmα(Ω).

Again this result is not trivial only if m>NpNpN+α,since then m(p1)NNmα>NpNα.

Proof. In the previous proof, using Sobolev inequality (3.3) instead of Poincaré inequality, we obtain in (4.5)

upβpββfmu((β1)p+1)m(β1)p+1,

r=NNα.Choosing now β=m(p1)p(rm),we get

um(p1)NNmαcfm1p1.

Even more, assuming a better integrability condition on f we get that the solution is bounded. This is a well known result for the standard Laplacian or the fractional Laplacian.

Theorem 4.5

Assume conditions (1.7) and (3.1). If u is a weak solution to problem (4.1), where fLm(Ω)with m>N/α,thenuL(Ω).

Proof. We change here slightly the test function used in the previous two proofs. We define for β1andKk(k to be chosen later) a C1([k,))function H, as follows:

H(s)={sβkβ,s[k,K],linear,s>K.

Let us also define v = u + k, and choose as test function φ=G(v)=kvΨ(H(s))ds.We obtain on one hand, using the Stroock-Varopoulos inequality (2.9) and the Sobolev inequality (3.3),

ε(u;G(v))cE(H(v))cΨ(H(v))r,(4.6)

and on the other hand, using Hölder inequality,

ΩfG(v)ΩfvΨ(H(v))ΩfvpΨ(H(v))ckp1fmvH(v)pmp,(4.7)

since vk.Inequality (4.6) together with (4.7), and the properties of Ψ, lead to

H(v)rp(cfmkp1)1/pvH(v)pm.(4.8)

We choose k=(cfm)1p1and letKin the definition of H, so that the inequality (4.8) becomes

ur*pββupmβ.

Hence for all β1the inclusion uLpmβ(Ω)implies the stronger inclusion uLrpβ(Ω),sincer=NNα>m=mm1providedm>Nα Observe that u being a weak solution it belongs to W0J,Ψ(Ω),and thus uLNpNα(Ω).The result follows now iterating the estimate starting with β=N(m1)(Nα)m>1,see for example [19, Theorem 8.15] for the details in the standard Laplacian case. This gives uL(Ω).In fact we get the estimate

uc(E(u)1p+fm1p1).

5 The problem with reaction f = f(u)

We study in this section the nonlinear elliptic type problem

{Lu=f(u),inΩ,u0,u/0,inΩ,u=0,inΩc.(5.1)

We first show existence in the lower case, i.e., when f:[0,)is a continuous function satisfying

0<μ<q1p:|f(t)|c1+c2Ψμ(t),liminft0+f(t)Ψμ(t)c3>0.(5.2)

In the power-like case (1.7) with f(t) = tm–1 this means 0 < m < p. See [8] for the classical sublinear problem for L=Δand [11] for general Lwithq0,both in the case Ψ(s)=|s|2.

Theorem 5.1

Under the assumption (5.2) problem (5.1) has a solution uW0J,Ψ(Ω)

Proof. We define the energy functional I:W0J,Ψ(Ω)defined by

I(v)=E(v)ΩG(v),

where G(u)=0uf(s)ds.This functional is easily seen to be weakly lower semicontinuous, and is well defined since

|ΩG(v)|c1|Ω|+c2|Ω|1μ(F(v))μ<.(5.3)

On the other hand, this same estimate also gives coercivity since μ < 1, and then

I(v)E(v)c(E(v))μasvWJ,Ψ.

Let now {vn}W0J,Ψ(Ω)be a minimizing sequence for I,

liminfnI(vn)=v=infuW0J,Ψ(Ω)I(u).

This sequence is bounded in W0J,Ψ(Ω),and therefore we can assume that there is a subsequence, still denoted {v n}, such that vnuinW0J,Ψ(Ω).Therefore vnuinLΨ(Ω).We thus deduce by (5.3)

ΩG(vn)ΩG(u),

so that

vI(u)liminfn(E(vn)ΩG(vn))=liminfnI(vn)=v.

This shows that I (u) = v and u is a global minimum for I, hence a solution to (5.1). It is easy to see that we can replace u by |u|since I(|u|)I(u).In order to show that u is nontrivial let us check that I(u)<0.In fact, given any vW0J,Ψ(Ω)we have

I(εv)yΨ+(ε)E(v)ε(yΨ(ε))μΩG(u)εq(v)ε1+pμΩG(u)<0,

for small ε > 0,since q > 1 + pμ. We deduce that v < 0 and u/0.

Unfortunately we are only able to prove uniqueness in the exact power case Ψ(s)=|s|p.In fact uniqueness follows in that case using a standard argument by means of a Picone inequality proved in [16], see [8] and [11]. Though a Picone inequality could be obtained also assuming that (1.7) is satisfied, it is not sharp enough to prove uniqueness. In the more general case of ΨΓp,qsuch Picone type inequality is not even known to hold.

We now assume condition (3.1) and consider nonlinear functions f in the intermediate range, that is above the power p – 1 but subcritical in the sense of Sobolev, see Theorem 3.2. The precise conditions on f are

ρ>q:tf(t)ρG(t)t>0;1<r<r,t0>0:tf(t)cψr(t)t>t0;λ0>0:f(λt)λρf(t)t>0,λ>λ0,(5.4)

where G' = f. When f(t)=tm1these condition hold with ρ = m provided p<m<NqNα.

Theorem 5.2

Assume J satisfies (3.1), ψ satisfies either (2.11) or (2.14), and f is a nondecreasing function satisfying (5.4). Then problem (5.1) has a solution

Proof. As before we consider the functional

I(v)=E(v)ΩG(v),

whose critical points are the solutions to our problem. This functional is well defined in W0J,Ψ(Ω)thanks to the Sobolev embedding and the second condition in (5.4). We therefore apply the standard variational technique based on the Mountain Pass Theorem [2]. We only have to prove that the functional satisfies the Palais-Smale condition and has the appropriate geometry.

We first prove that any Palais-Smale sequence has a convergent subsequence. Let {vn} be a sequence satisfying

I(vn)v,I(vn),φ0φ(W0J,Ψ(Ω)).

By the first condition in (5.4), and using (1.9), we have

I(vn),vn=ε(vn;vn)Ωvnf(vn)pE(vn)ρΩG(vn).

On the other hand, for all large n such that I(vn)1we have

|I(vn),vn|vnWJ,Ψ.

Therefore

v+1I(vn)=I(vn)1ρI(vn),vn+1ρI(vn),vn(1pρ)E(vn)1ρvnWJ,Ψ(1pρ)min{vnWJ,Ψp,vnWJ,Ψq}1ρvnWJ,Ψ.

This implies vnWJ,Ψkfor every n, so that there exists a subsequence, still denoted {vn},converging weakly to some uW0J,Ψ(Ω),and by Theorem 3.4 it is vnvstrongly in Lψγ(Ω)for every1r<r*.The second condition in (5.4) implies vnf(vn)vf(v)inL1(Ω).Now write,

ε(vn;vnv)ε(v;vnv)=I(vn),vnvI(v),vnv+Ω(f(vn)f(v))(vnv)0.

Using inequalities (2.12) or (2.15) as in the proof of uniqueness in Theorem 4.1, we obtain

E(vnv)0,

that is vnvinW0J,Ψ(Ω),and Palais-Smale condition holds.

Let us now look at the behaviour of I close to the origin and far from it. First I (0) = 0. Also, given any vW0J,ψ(Ω),we have by Poincaré inequality and the second condition in (5.4)

I(v)=E(v)ΩG(v)c1Ωψ(v)c2Ωψr(v)c1F(v)c3Fr(v)>0

for every F(v) small. But ||v||WJ,Ψsmall implies F(v) small. We have obtained

ε,δ>0:I(v)>I(0)+δvW0J,Ψ(Ω),||v||WJ,Ψ=ε.

On the other hand, if A > 0 is large, using the third condition in (5.4), we get

I(λv)λpE(v)λρΩG(v)<0,

since p < ρ. Thus

v¯W0J,Ψ(Ω),||v¯||WJ,Ψ>ε:I(v¯)<I(0).

This ends the proof by an application of the Mountain Pass Theorem. Actually, if we define

Θ={hC([0,1];W0J,Ψ(Ω)):h(0)=0,h(1)=v¯},

then

η=infhΘmaxt[0,1]I(h(t))

is a critical value with I(u) = η for some uW0J,Ψ(Ω),which is a solution to our problem.

The exponent r* in (5.4) is sharp in the fractional p–Laplacian case. In fact, in the fractional Laplacian case p = 2 this has been proved in [26] by means of a Pohozaev identity when Ω is star-shaped. Their proof was adapted in [11] for more general kernels again with Ψ(s)=|s|2,obtaining an exponent which depends onthe kernel and is presumed not to be optimal. The proof of this last result works verbatim for general powers Ψ(s)=|s|p,but not for other functions, since homogeneity is crucial in the argument.

Let, for A > 1,

μ(λ)=λNsupzNz0J(z/λ)J(z),(5.5)

and assume μ(λ) < ∞ for λ close to 1.

Theorem 5.3

If u is a bounded solution to problem (5.1) with Ψ(s)=|s|pand Ω is star-shaped, then

Ωuf(u)NpNδΩG(u),

where δ=μ(1+)andG=f.

Corollary 5.4

Problem (5.1) with f(u)=um1,Ψ(s)=|s|pandΩstar-shaped has no bounded solutions for any exponent m>m*=NpNδ.

We observe that this nonexistence result depends not only on the behaviour of the kernel at the origin, but on its global behaviour, see (5.5). In fact when the kernel is

J(z)={|z|Nα1if|z|<1,|z|Nα2if|z|>1,

α1 < p, α2 > 0, we get σ = max{α1, α2}. It will be interesting to know if only the singularity of J at the origin determines by its own the existence or nonexistence of solution. If this is the case we would get, in the critical singularity exponent q* = 0 in (H0), that there is no solution for any m > p. This, together with the existence result for m < p of Theorem 5.1, leaves only the case m = p to be studied. We dedicate next section to this task.

6 The generalized eigenvalue problem

In this last section we study the parametric problem

{Lu=λψ(u),inΩ,u=0inΩc.(6.1)

Though the problem is not homogeneous due to the presence of the non homogeneous function ψ, since it is precisely this same function ψ that defines both, the operator and the reaction, it can also be called generalized eigenvalue problem, as is usual for the p–Laplacian or the fractional p–Laplacian, see [22], [23]. The first (generalized) eigenvalue and eigenfunction are obtained minimizing

I(v)=E(v)F(v),vW0J,Ψ(Ω)\{0}.

In fact, if u is a minimum, the function g(t) = I (u + ), for any admisible function φsatisfies g(0)=E(u)F(u)=λ1,g'(0) = 0, that is,

E(u),φ=λ1F(u),φ,

which is the associated Euler-Lagrange equation, the weak formulation (1.13).

Theorem 6.1

Define

λ1=infvW0J,Ψ(Ω)\{0}I(v).

Then λ1 is positive and is achieved by some uW0J,Ψ(Ω)\{0}.Moreover, the function u is a weak solution to problem (6.1). The solution does not change sign, and it is moreover bounded if (1.7) and (3.1) holds for some α >0.

Proof. The inequality (3.2) immediately gives λ1 > 0. Consider M={vW0J,Ψ(Ω):F(v)=1}.Let{vn}be a minimizing sequence for I in M, that is

limnI(vn)=λ1=infvMI(v)>0.

Then {vn}is bounded in W0J,Ψ(Ω),so there exists a subsequence, still denoted by {vn},such that vnuinW0J,Ψ(Ω).As usual, by Theorem 3.4 there exists a subsequence converging to u in LΨ(Ω), so F(u) = 1 and uM.This gives

λ1I(u)=E(u)limnE(vn)=limnI(vn)=λ1,

and then I(u) = λ1. The functionals E and T are differentiable, and so is I, and we have

0=I(u),φ=1F(φ)(E(u),φI(u)F(u),φ).

Therefore

ε(u;φ)=E(u),φ=I(u)F(u),φ=λ1Ωψ(u)φ,

for every φW0J,Ψ(Ω).The fact that the eigenfunction is nonnegative or nonpositive follows by (2.8) which impliesI(±|u|)I(u).The boundedness of u assuming (3.1) is easily proved again by the Moser iterative scheme as performed in [6]. The key point is the use of the Stroock-Varopoulos inequality (2.10) and condition (1.7), and finally apply Theorem 4.5. See also [17].

Acknowledgement

Work supported by the Spanish project MTM2014-53037-P.

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About the article

Received: 2018-08-03

Accepted: 2018-11-24

Published Online: 2019-05-22

Published in Print: 2019-03-01


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 305–326, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2020-0002.

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© 2020 E. Correa and A. de Pablo, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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