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Mechanisms and applications of the anti-inflammatory effects of photobiomodulation

  • Photobiomodulation (PBM) also known as low-level level laser therapy is the use of red and near-infrared light to stimulate healing, relieve pain, and reduce inflammation. The primary chromophores have been identified as cytochrome c oxidase in mitochondria, and calcium ion channels (possibly mediated by light absorption by opsins). Secondary effects of photon absorption include increases in ATP, a brief burst of reactive oxygen species, an increase in nitric oxide, and modulation of calcium levels. Tertiary effects include activation of a wide range of transcription factors leading to improved cell survival, increased proliferation and migration, and new protein synthesis. There is a pronounced biphasic dose response whereby low levels of light have stimulating effects, while high levels of light have inhibitory effects. It has been found that PBM can produce ROS in normal cells, but when used in oxidatively stressed cells or in animal models of disease, ROS levels are lowered. PBM is able to up-regulate anti-oxidant defenses and reduce oxidative stress. It was shown that PBM can activate NF-kB in normal quiescent cells, however in activated inflammatory cells, inflammatory markers were decreased. One of the most reproducible effects of PBM is an overall reduction in inflammation, which is particularly important for disorders of the joints, traumatic injuries, lung disorders, and in the brain. PBM has been shown to reduce markers of M1 phenotype in activated macrophages. Many reports have shown reductions in reactive nitrogen species and prostaglandins in various animal models. PBM can reduce inflammation in the brain, abdominal fat, wounds, lungs, spinal cord.

    Citation: Michael R Hamblin. Mechanisms and applications of the anti-inflammatory effects of photobiomodulation[J]. AIMS Biophysics, 2017, 4(3): 337-361. doi: 10.3934/biophy.2017.3.337

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  • Photobiomodulation (PBM) also known as low-level level laser therapy is the use of red and near-infrared light to stimulate healing, relieve pain, and reduce inflammation. The primary chromophores have been identified as cytochrome c oxidase in mitochondria, and calcium ion channels (possibly mediated by light absorption by opsins). Secondary effects of photon absorption include increases in ATP, a brief burst of reactive oxygen species, an increase in nitric oxide, and modulation of calcium levels. Tertiary effects include activation of a wide range of transcription factors leading to improved cell survival, increased proliferation and migration, and new protein synthesis. There is a pronounced biphasic dose response whereby low levels of light have stimulating effects, while high levels of light have inhibitory effects. It has been found that PBM can produce ROS in normal cells, but when used in oxidatively stressed cells or in animal models of disease, ROS levels are lowered. PBM is able to up-regulate anti-oxidant defenses and reduce oxidative stress. It was shown that PBM can activate NF-kB in normal quiescent cells, however in activated inflammatory cells, inflammatory markers were decreased. One of the most reproducible effects of PBM is an overall reduction in inflammation, which is particularly important for disorders of the joints, traumatic injuries, lung disorders, and in the brain. PBM has been shown to reduce markers of M1 phenotype in activated macrophages. Many reports have shown reductions in reactive nitrogen species and prostaglandins in various animal models. PBM can reduce inflammation in the brain, abdominal fat, wounds, lungs, spinal cord.


    This work is dedicated to Ireneo in memoriam, for his teachings, friendship and enthusiasm.

    This work deals with the following problem:

    {(Δ)su=λu|x|2s+(F(u)(x))p+ρf in Ω,u>0 in Ω,u=0 in (RNΩ), (1)

    where λ>0, ρ>0, s(0,1), 2s<N, 1<p<, ΩRN is a bounded regular domain containing the origin and f is a measurable non-negative function satisfying suitable hypotheses.

    By (Δ)s we denote the fractional Laplacian of order 2s introduced by M. Riesz in [39], that is,

    (Δ)su(x):=aN,s P.V. RNu(x)u(y)|xy|N+2sdy,s(0,1),

    where

    aN,s=22s1πN2Γ(N+2s2)|Γ(s)|,

    is the normalizing constant that gives the Fourier multiplier identity

    F((Δ)su)(ξ)=|ξ|2sF(u)(ξ), for uS(RN).

    See [26] for details. Our goal is to find natural conditions on p and f (related to the value of λ), in order to get the existence of positive solutions.

    If λ=0, the problem (1) can be seen as a Kardar-Parisi-Zhang stationary equation with fractional diffusion and nonlocal gradient term. We refer to [30] for the main model and additional properties of the local case.

    The nonlocal case s(0,1), but still with the local gradient term, was used recently in order to describe the growing surface in the presence of self-similar hopping surface diffusion. We refer the reader to the papers [29,32,33,35] for a physical rigorous justification.

    Existence results for the corresponding problem were obtained in [23] and [11] under suitable hypotheses on f and p. As it was shown in [11], if p>11s, then the corresponding problem does not have positive solutions with global regularity of the gradient, even in the case of regular datum f. Existence of a solution, in the viscosity sense, is proved in [9,16,17] for some particular cases.

    The case λ=0, under the presence of a nonlocal gradient term, was analyzed recently in [7]. Without any limitation on the value of p and under suitable hypothesis of f, the author proved the existence of a solution using a priori estimates and fixed point arguments.

    The case λ>0 with a local gradient term was considered in [10] and [12]. Here the authors showed the existence of a critical exponent related to the existence of solutions. Our work can be seen as the non-local counterpart of [12]. However, the non-local gradient term makes the problem more difficult and fine analysis is needed to determine the existence or non-existence scheme.

    Notice that for λ>0, problem (1) is related to the Hardy inequality proved in [28], (see also [18] and [38] for equivalent forms.) Namely, for ϕC0(IRN), we have

    IRN|ξ|2s|ˆϕ|2dξΛN,sIRN|x|2sϕ2dx, (2)

    where

    ΛN,s:=22sΓ2(N+2s4)Γ2(N2s4) (3)

    is optimal and not attained.

    It is clear that

    lims1ΛN,s=(N22)2,

    the Hardy constant in the local case.

    Inequality (2) can be also formulated in the following way

    aN,s2IRNIRN|ϕ(x)ϕ(y)|2|xy|N+2sdxdyΛN,sIRNϕ2|x|2sdx,ϕC0(IRN).

    If λ>ΛN,s, then we can prove that problem (1) has no positive supersolution. Hence, we assume throughout this paper that 0<λ<ΛN,s.

    The presence of the Hardy potential forces the solution to enjoy a singular behavior near the singular point zero and then a loss of regularity is generated.

    The paper is organized as follows. In Section 2 we present the functional setting used in order to study our problem. More precisely we describe some related spaces, as the Bessel potential space, and their relationship with the fractional Sobolev space. We introduce also the different forms of the fractional gradient that will be used throughout the paper. In Subsection 2.1 we recall the global regularity results for the Poisson fractional problem proved in [6]. This will be the key in order to show the fractional regularity in our problem.

    The analysis of the problem under the presence of the Hardy potential, without the nonlocal gradient term, is considered in Section 3. More precisely, we will consider the semilinear problem

    {(Δ)su=λu|x|2s+f in Ω,u=0 in RNΩ,

    where fLm(Ω) with m1. Some partial regularity results are known in the case where λ<Js,mΛN,s4N(m1)(N2ms)m2(N2s)2.

    However for Js,mλΛN,s, using a different approach based on weighted spaces, we are able to complete the full picture of regularity. As a consequence, we get a complete classification of the fractional regularity of the solution to the above problem.

    The first analysis of the KPZ problem (1) is done in Section 4. We begin by considering the case where F(u)(x)=|(Δ)s2u(x)|. Using suitable radial computations in the whole space, we derive the existence of a critical exponent p+(λ,s) such that if p>p+(λ,s), then for all ρ>0, the problem (1) has no positive solution in a weak sense. Some other non existence results are proved for ρ large under technical condition on p.

    The case p<p+(λ,s) is analyzed in Subsection 4.2. Under the hypothesis that f is bounded, we are able to show the existence of a supersolution for ρ small. Moreover, for p<NNs, and for all fL1(Ω) that satisfy a suitable integral condition near the origin, we are able to show the existence of a weak solution for ρ<ρ.

    In Section 5 we treat the KPZ problem, namely equation (1), under the presence of another version of the non local gradient.

    More precisely, we consider the case where F(u)(x)=(aN,s2IRN|u(x)u(y)|2|xy|N+2sdy)12. Then, also in this case, we are able to show the existence of a critical exponent p+(λ,s) such that non existence holds if p>p+(λ). The proof of the non existence in this case is more technical and need some additional estimates.

    Finally, at the end of the section we formulate some interesting open problems that may describe a full picture for the existence in our problem.

    The goal of this section is to establish some useful tools and definitions that will play an important role in what follows.

    Definition 2.1. Let ΩIRN be a bounded domain and s(0,1). For p[1,), the fractional Sobolev space Ws,p(Ω) is defined by

    Ws,p(Ω):={uLp(Ω):Ω×Ω|u(x)u(y)|p|xy|N+spdxdy<}.

    Ws,p(Ω) is a Banach space endowed with the norm

    uWs,p(Ω):=(upLp(Ω)+Ω×Ω|u(x)u(y)|p|xy|N+spdxdy)1p.

    The space Ws,p0(Ω) is defined as follows:

    Ws,p0(Ω):={uWs,p(IRN):u=0 in IRNΩ}.

    This is a Banach space endowed with the norm

    uWs,p0(Ω):=(DΩ|u(x)u(y)|p|xy|N+spdxdy)1/p,

    where

    DΩ:=(IRN×IRN)(CΩ×CΩ)=(Ω×IRN)(CΩ×Ω).

    Now, for s(0,1) and 1p<+ we define the Bessel potential space by setting

    Ls,p(RN) := ¯{uC0(RN)}||.||Ls,p(RN),

    where

    ||u||Ls,p(RN)=(1Δ)s2uLp(RN) and (1Δ)s2u=F1((1+||2)s2Fu), uCc(RN).

    Let us stress that, in the case where s(0,1) and 1<p<+,

    uLs,p(RN):=uLp(RN)+(Δ)s2uLp(RN)

    is an equivalent norm for Ls,p(RN) (see e.g., [1,page 5] for a precise explanation of this fact). Let us as well recall that, for all 0<ϵ<s<1 and all 1<p<+, by [13,Theorem 7.63, (g)], we have

    Ls+ϵ,p(RN)Ws,p(RN)Lsϵ,p(RN).

    For ϕC0(IRN) we define the fractional gradient of order s of ϕ by

    sϕ(x):=IRNϕ(x)ϕ(y)|xy|sxy|xy|dy|xy|N, xIRN. (4)

    Notice that, as it was proved in [46,Theorem 2] and [42,Theorem 1.7], we have

    Ls,p(IRN):={uLp(IRN) such that |su|Lp(IRN)}={uLp(IRN) such that |(Δ)s2u|Lp(IRN)}

    with the equivalent norms

    |u|Ls,p(IRN):=uLp(IRN)+suLp(IRN)uLp(IRN)+(Δ)s2uLp(IRN).

    Another type of "nonlocal gradient" can be defined also by

    Ds(u)(x)=(aN,s2IRN|u(x)u(y)|2|xy|N+2sdy)12. (5)

    We refer to [20] and [36] for some motivation of this non local version of the gradient.

    In this case one has

    lims1(1s)D2s(u(x))=|u(x)|2, uC0(RN). (6)

    If p>2NN+2s, it was proved in [46] that the Bessel potential space Ls,p(IRN) can be defined also as the set of functions uLp(IRN) such that Ds(u)Lp(IRN). The space Ls,p(IRN) can be equipped with the equivalent norms

    |||u|||Ls,p(IRN)=uLp(IRN)+Ds(u)Lp(IRN).

    The next Sobolev inequality in Ls,p(IRN) is proved in [13], see also [25].

    Theorem 2.2. Let 1<p< and s(0,1) be such that sp<N. Then there exist two positive constants S1:=S2(N,p,s) and S2:=S1(N,p,s) such that for all uLs,p(IRN), we have

    S1||u||Lps(IRN)suLp(IRN),

    and

    S2||u||Lps(IRN)(Δ)s2uLp(IRN),

    with ps=pNNps.

    If ΩIRN, we define the space Ls,p0(Ω) as the set of functions uLs,p(IRN) with u=0 in IRNΩ.

    From Lemma 1 in [46], if p>2NN+2s and Ω is a bounded domain, then there exist C1:=C1(Ω,N,p,s) and C2:=C2(Ω,N,p,s), two positive constants, such that for all uLs,p0(Ω)

    C1|u|Ls,p(IRN)Ds(u)Lp(IRN)C2|u|Ls,p(IRN).

    Notice that if Ω is a bounded domain, we can endow Ls,p0(Ω) with the equivalent norms suLp(IRN) or (Δ)s2uLp(IRN). In the same way, by assuming in addition that p>2NN+2s, then we can equip Ls,p0(Ω) also with the equivalent norms Ds(u)Lp(IRN). We refer to [47] for more details about the properties of the Bessel potential space and its relation with the fractional Sobolev space.

    The next Hardy inequality will be useful in order to prove the non existence result above the critical exponent. See [7] for the proof.

    Proposition 2.3. Let ΩRN be a regular domain with 0Ω and 0<s<1. Suppose that p>2NN+2s with ps<N and define

    L(Ω):=inf{IRN(Ds(ϕ)(x))pdxΩ|ϕ(x)|p|x|psdx:ϕC0(Ω){0}}. (7)

    Then L(Ω)>0 and L(Ω)=L does not depends on Ω. Moreover, the weight |x|ps is optimal in the sense that, for all ε>0 we have

    inf{IRN(Ds(ϕ)(x))pdxΩ|ϕ(x)|p|x|ps+εdx:ϕC0(Ω){0}}=0.

    Finally, we recall the next standard result from harmonic analysis. See for instance [45,Theorem I, Section 1.2, Chapter V].

    Theorem 2.4. Let 0<ν<N and 1p<< be such that 1+1=1p+νN. For gLp(IRN), we define

    Jν(g)(x)=IRNg(y)|xy|νdy.

    Then, it follows that:

    a) Jν is well defined in the sense that the integralconverges absolutely for almost all xRN.

    b) If p>1, then Jν(g)L(IRN)cp,lgLp(IRN). c) If p=1, then |{xRN|Jν(g)(x)>σ}|(AgL1(IRN)σ).

    The goal of this section is to state some well known results about the regularity of the Poisson equation

    {(Δ)su=g in Ω,u=0 in RNΩ, (8)

    where Ω is a bounded regular domain of IRN and gLm(Ω) with m1. We begin by the sense for which solutions are defined.

    Definition 2.5. We define the class of test functions

    T(Ω)={ϕ|(Δ)s(ϕ)=ψ in Ω,ϕ=0 in RNΩ,ψC0(Ω)}. (9)

    Notice that if vT(Ω) then, using the results in [34], vHs0(Ω)L(Ω). Moreover, according to the regularity theory developed in [43], if Ω is smooth enough, there exists a constant β>0 (that depends only on the structural constants) such that vCβ(Ω) (see also [31]).

    Definition 2.6. We say that uL1(Ω) is a weak solution to (8) if for gL1(Ω) we have that

    Ωuψdx=Ωgϕdx,

    for any ϕT(Ω) with ψC0(Ω).

    Recall also the definition of the truncation operator Tk,

    Tk(σ)=max{k;min{k,σ}} and Gk(σ)=σTk(σ). (10)

    From [2,22,34] we have the next existence result.

    Theorem 2.7. Suppose that gL1(Ω), then problem (8) has a unique weak solution u obtained as the limit of {un}nN, the sequence of unique solutions to the approximating problems

    {(Δ)sun=gn(x) in Ω,un=0 in IRNΩ, (11)

    with gn=Tn(g). Moreover,

    Tk(un)Tk(u) strongly in Hs0(Ω),k>0, (12)
    uLq(Ω), q[1,NN2s) (13)

    and

    |(Δ)s2u|Lr(Ω), r[1,NNs). (14)

    In addition, if s>12, then uW1,q0(Ω) for all 1q<NN(2s1) and unu strongly in W1,q0(Ω).

    In what follows we denote Gs the Green function associated to the fractional laplacian (Δ)s.

    Notice that Gs(x,y) solves the problem

    {(Δ)syGs(x,y)=δx(y) if yΩ,Gs(x,y)=0 if yIRNΩ, (15)

    where xΩ is fixed and δx is Dirac's delta function.

    It is clear that if u is the unique weak solution to problem (8), then

    u(x)=ΩGs(x,y)g(y)dy.

    We collect in the next Proposition some useful properties of the Green function Gs (See [21] and [19] for the proof).

    Proposition 2.8. Assume that s(0,1). Then, for almost every x, y Ω, we have

    Gs(x,y)1|xy|N2s(δs(x)|xy|s1)(δs(y)|xy|s1)1|xy|N2s(δs(x)δs(y)|xy|2s1). (16)

    In particular, we have

    Gs(x,y)C1min{1|xy|N2s,δs(x)|xy|Ns,δs(y)|xy|Ns} for a.e. x,yΩ. (17)

    In the case where gLm(Ω), we can improve the regularity results of Theorem 2.7. More precisely from [11], we have the next theorem.

    Theorem 2.9. Assume that gLm(Ω) with m>1 and let u be the uniquesolution to problem (8), then there exists a positiveconstant C:=C(N,s,m,Ω) (that can change from a line to anotherone), such that

    1) If 1<m<N2s, then uLmNN2ms(Ω), uδsLmNNms(Ω) and

    ||u||LmNN2ms(Ω)+||uδs||LmNNms(Ω)C||g||Lm(Ω).

    2) If m=N2s, then uLr(Ω) for all r<, uδsLmNNms(Ω) and

    ||u||Lr(Ω)+||uδs||LmNNms(Ω)C||g||Lm(Ω).

    3) If N2s<m<Ns, then uL(Ω), uδsLmNNms(Ω) and

    ||u||L(Ω)+||uδs||LmNNms(Ω)C||g||Lm(Ω).

    4) If m=Ns, then uL(Ω), uδsLp(Ω) for all p< and

    ||u||L(Ω)+||uδs||Lp(Ω)C||g||Lm(Ω).

    5) If m>Ns, then uL(Ω), uδsL(Ω) and

    ||u||L(Ω)+||uδs||L(Ω)C||g||Lm(Ω).

    Related to the fractional regularity of the solution to problem (8), a global fractional Calderon-Zygmund regularity result was obtained recently in [6].

    Theorem 2.10. Let s(0,1) and consider u to be the (unique) weak solution to problem (8) with fLm(Ω). Then we have

    1) If mNs, then for all 1p<, there exists a positive constant C=C(N,s,p,m,Ω) such that

    (Δ)s2uLp(RN)CgLm(Ω).

    Moreover uLs,p(RN) for all 1p< and

    uLs,p(RN)CgLm(Ω).

    2) 1m<Ns, then, for all 1p<mNNms, there exists a positive constant C=C(N,s,p,m,Ω) such that

    (Δ)s2uLp(RN)CgLm(Ω).

    Hence uLs,p(RN) for all 1p<mNNmsand

    uLs,p(RN)CgLm(Ω).

    As a direct consequence of the relation between the fractional Sobolev space Ws,p(RN) and the Bessel potential space Ls,p(RN), we get the next result.

    Corollary 2.11. Let s(0,1). Consider u to be the unique solution of problem (8) with gLm(Ω). Then

    1) If 1m<Ns,

    we have, for all 1<p<mNNms, that there exists C=C(N,s,m,p,Ω) such that

    uWs,p(RN)CgLm(Ω).

    2) If mNs then, for all 1<p<, there exists C=C(N,s,m,p,Ω) such that

    uWs,p(RN)CgLm(Ω).

    Let us recall that another version of the nonlocal gradient is given by

    Ds(u)(x)=(aN,s2IRN|u(x)u(y)|2|xy|N+2sdy)12.

    Taking into consideration the result of [46], we get the following corollary.

    Corollary 2.12. Assume that the conditions of Theorem 2.10 hold. Then we have

    1) If m>Ns, then for all 2NN+2s<p<, there exists C=C(N,s,m,p,Ω) such that

    Ds(u)Lp(RN)CgLm(Ω).

    2) If 2NN+4s<mNs, then for all 2NN+2s<p<mNNms, there exists C=C(N,s,m,p,Ω) such that

    Ds(u)Lp(RN)CgLm(Ω).

    In this subsection we analyze the question of regularity of the solution to the problem

    {(Δ)su=λu|x|2s+f in Ω,u=0 in RNΩ, (18)

    in Lebesgue spaces and fractional Sobolev spaces according to the regularity of the datum f. Here ΩIRN is a bounded regular domain containing the origin and s(0,1). We will suppose that fLm(Ω) with m1 and 0<λ<ΛN,s.

    If f=0, we define the radial potential v±αλ(x)=|x|N2s2±αλ with αλ given by

    λ=λ(αλ)=λ(αλ)=22sΓ(N+2s+2αλ4)Γ(N+2s2αλ4)Γ(N2s+2αλ4)Γ(N2s2αλ4). (19)

    From [8], we obtain that v±αλ solves the homogeneous equation

    (Δ)su=λu|x|2s in RN{0}. (20)

    It is clear that λ(α)=λ(α)=mαλmαλ, with mαλ=2αλ+sΓ(N+2s+2αλ4)Γ(N2s2αλ4).

    Notice that

    0<λ(αλ)=λ(αλ)ΛN,s if and only if 0αλ<N2s2.

    Define

    μ(λ)=N2s2αλ and ˉμ(λ)=N2s2+αλ. (21)

    For 0<λ<ΛN,s, then 0<μ(λ)<N2s2<ˉμ<(N2s). Since N2μ(λ)2s=2αλ>0 and N2ˉμ(λ)2s=2αλ<0, then (Δ)s/2(|x|μ(λ))L2(Ω), but (Δ)s/2(|x|ˉμ(λ)) does not.

    As it was proved in [8], if fL1(Ω), then the existence of a solution to problem (18) is guaranteed under the necessary and sufficient condition Br(0)f|x|μ(λ)dx<. Hence, throughout this section this condition will be assumed.

    The first result concerning the behavior in the neighborhood of zero is given by the next Proposition proved in [8].

    Proposition 3.1. Let uL1loc(IRN) be such that u0 in IRN and (Δ)suL1loc(Ω). Assume that

    (Δ)suλu|x|2s in Ω,0<λ<ΛN,s.

    Then, there exists r>0 and a positive constant CC(r,N,λ) such that

    u(x)C|x|μ(λ)=C|x|N2s2+αλ in Br(0)⊂⊂Ω.

    We are now in position to prove the first regularity results, in fractional Sobolev space, to the solution of problem (18).

    Theorem 3.2. Assume that fLm(Ω) with m>1 satisfying the condition Br(0)f|x|μ(λ)dx<. Let uL1(Ω) tobe the unique weak solution to (18) with λ<ΛN,s. Then there exists a positive constant C=C(N,m,p,s,Ω) such that

    1) If mN2s, then u|x|2sLσ(Ω) for all 1σ<Nμ(λ)+2s and |(Δ)s2u|Lp(IRN) for all 1p<Nμ(λ)+s. Moreover we have

    ||(Δ)s2u||Lp(IRN)C||f||Lm(Ω).

    2) If 1<m<N2s and λ<Js,mΛN,s4N(m1)(N2ms)m2(N2s)2, then |(Δ)s2u|Lp(IRN) forall 1p<NmNms. Moreover we have

    ||(Δ)s2u||Lp(RN)C||f||Lm(Ω).

    Proof. We begin by analyzing the first case. Assume that fLm(Ω) with m>N2s. From Theorem 4.1 in [8], we obtain that u(x)C|x|μ(λ)χΩ. Hence u|x|2sC|x|μ(λ)2sχΩ. As a consequence, we deduce that u|x|2sLσ(Ω) for all 1σ<Nμ(λ)+2s.

    Setting gu|x|2s+f, it follows that gLσ(Ω) for all σ<Nμ(λ)+2s. Using the regularity result in Theorem 2.9, we conclude that uLt(Ω) for all t<Nμ(λ). Now by Theorem 2.10, it holds that |(Δ)s2u|Lp(IRN) for all 1p<Nμ(λ)+s and

    ||(Δ)s2u||Lp(IRN)C||f||Lm(Ω).

    Hence we conclude.

    We treat now the case 1<m<N2s and 0<λ<Js,mΛN,s4N(m1)(N2ms)m2(N2s)2.

    Recall that u solves problem (18). Then by Theorem 4.2 of [8], we get the existence of positive constant C(N,s,m) such that

    ||u||Lms(Ω)C||f||Lm(Ω) where ms=mNN2sm. (22)

    Since p<NmNms, then we get the existence of m1<m such that p<Nm1Nm1s. Fixed m1<m, using Hölder inequality we deduce that

    Ωum1|x|2sm1dxC.

    Since m1<m, it follows that gλu|x|2s+fLm1(Ω).

    On the other hand m1<m<N2s<Ns, therefore using the regularity result in Theorem 2.10, we deduce that

    ||(Δ)s2u||Lp(IRN)C||g||Lm1(Ω) for all p<Nm1Nm1s.

    Thus

    ||(Δ)s2u||Lp(IRN)C||f||Lm(Ω) for all p<NmNms,

    and the result follows in this case.

    In order to treat the general case Js,mΛN,s4N(m1)(N2ms)m2(N2s)2λ<ΛN,s, we need to develop a new approach.

    Let u be the unique weak solution to problem (18). Setting v(x):=|x|μ(λ)u(x), it follows that v solves the problem

    {Lμ(λ)v=|x|μ(λ)f(x)=:˜f(x) in Ω,v=0 in RNΩ, (23)

    with

    Lγv:=aN,s P.V. RNv(x)v(y)|xy|N+2sdy|x|γ|y|γ. (24)

    Since Br(0)f|x|μ(λ)dx<, then ˜fL1(Ω). Thus v can be seen as the unique entropy solution to problem (23) as defined in [2]. Following closely the argument used in [4], we get the next general regularity result.

    Theorem 3.3. Let s(0,1) and 0<λ<ΛN,s. Assume that ˜fLq(Ω,|x|β(q1)dx) with q>1 and 2Nμ(λ)N2sβ2(μ(λ)+s). Let us denoteby C:=C(N,β,λ,s,q,Ω) a positive constant that may change from line to other.

    Then if v solves problem (23), we have

    1) If β<2(μ(λ)+s) and q>(Nβ)2(μ(λ)+s)β, then vL(Ω). Moreover,

    vL(Ω)C˜fLq(Ω,|x|β(q1)dx).

    2) If β<2(μ(λ)+s) and q=(Nβ)2(μ(λ)+s)β, then vLr(Ω,|x|βdx), for all 1r<+. Moreover

    (Ω|v|r |x|βdx)1rC˜fLq(Ω,|x|β(q1)dx).

    3) If either β=2(μ(λ)+s) or β<2(μ(λ)+s) and 1<q<Nβ2(μ(λ)+s)β, then |v|rL1(Ω,|x|βdx), for all 1rr=(Nβ)qNβq(2(μ(λ)+s)β). Moreover

    (Ω|v|r |x|βdx)1rC˜fLq(Ω,|x|β(q1)dx).

    Before proving the previous Theorem, we recall the following weighted fractional Caffarelli-Khon-Nirenberg inequality, whose proof can be found in [3,15,37].

    Theorem 3.4. Assume that s(0,1) and 2s<γ<N2s2. Let θ[γ,γ+s], then there exists a positive constant C:=C(N, s, γ, θ), such that for all ϕC0(IRN), we have

    C(IRN|ϕ|ˆσ|x|ˆσθdx)2ˆσIRNIRN|ϕ(x)ϕ(y)|2|xy|N+2s|x|γ|y|γdxdy,

    with ˆσ=2NN2s+2(θγ).

    Setting β=ˆσθ, we obtain that 2NγN2sβ2(γ+s) and

    C(IRN|ϕ|ˆσ|x|βdx)2ˆσIRNIRN|ϕ(x)ϕ(y)|2|xy|N+2s|x|γ|y|γdxdy. (25)

    Notice that by substituting the value of θ in the formula of ˆσ, we reach that ˆσ=2(Nβ)N2(γ+s).

    Proof of Theorem 3.3. Notice that, using the notation of Theorem 3.4, then, in our case, we have γ=μ(λ)(0,N2s2).

    The main idea of the proof is to use a suitable test function and an approximation argument. To make the paper self contained as possible, we include here all the details.

    Without loss of generality we can assume that q>1 and ˜f0. Thus v0 in IRN.

    Consider the following approximating problem

    {Lμ(λ)vn=˜fn(x) in Ω,vn=0 in RNΩ, (26)

    where ˜fn(x)=Tn(˜f(x)) is the truncation of ˜fn as defined in (10).

    Since v is the unique solution to problem (23), at least in the entropy sense, then

    vnv a.e. in IRN and vnv strongly in L1(IRN).

    In the rest of the proof, we denote by C any positive constant that depends only on N,s,q,r,Ω, and is independent of n,˜f, v, that may change from line to other.

    It is not difficult to show that vn is bounded. Thus, for α>0 fixed, to be chosen later, using vαn as a test function in (26), it holds that

    12DΩ(vn(x)vn(y))(vαn(x)vαn(y))|xy|N+2s|x|μ(λ)|y|μ(λ)dxdy=Ω˜fnvαn(x)dx.

    By the algebraic inequality

    (ab)(aαbα)C(aα+12bα+12)2,

    we reach that

    CDΩ(vα+12n(x)vα+12n(y))2|xy|N+2s|x|μ(λ)|y|μ(λ)dxdyΩ˜fn(x)vαn(x)dx.

    Using the weighted fractional Caffarelli-Khon-Nirenberg inequality in Theorem 3.4, we get

    DΩ(vα+12n(x)vα+12n(y))2|xy|N+2s|x|μ(λ)|y|μ(λ)dxdyC(Ωv(α+1)ˆσ2n|x|βdx)2ˆσ.

    Now by using Hölder's inequality, it holds that

    Ω˜fnvαn(x)dx(Ω˜fqn(x)|x|β(q1)dx)1q(Ωvαqn(x)|x|βdx)1q. (27)

    Hence

    C(Ω(vn(x))(α+1)ˆσ2|x|βdx)2ˆσ(Ω˜fqn(x)|x|β(q1)(x)dx)1q(Ωvαqn(x)|x|βdx)1q. (28)

    If β<2(μ(λ)+s) and ˆσ2>q, namely q>Nβ2(μ(λ)+s)β, in this case we can prove that vL(Ω). The proof follows using the classical Stampacchia argument as in [44]. Let us give some details. Using Gk(vn) as a test function (26), it follows that

    12DΩ(vn(x)vn(y))(Gk(vn(x))Gk(vn(y)))|xy|N+2s|x|μ(λ)|y|μ(λ)dxdy=Ω˜fn(x)Gk(vn(x))dx.

    Since ˆσ2>q, then 1ˆσ+1q<112q. Thus Using the Hölder inequality, we get

    CDΩ(Gk(vn(x))Gk(vn(y)))2|xy|N+2s|x|μ(λ)|y|μ(λ)dxdy(Ω˜fq(x)|x|β(q1)dx)1q(Ω(Gk(vn(x)))ˆσ|x|βdx)1ˆσ|{xΩ:Gk(vn(x))>0}|11ˆσ1q|x|βdx.

    Now, by the Caffarelli-Kohn-Nirenberg inequality in (25), we deduce that

    (Ω(Gk(vn(x)))ˆσ|x|βdx)1ˆσ(Ω˜fqn(x)|x|β(q1)dx)1q|{xΩ:Gk(vn(x))>0}|11ˆσ1q|x|βdx.

    Hence

    |{xΩ:vn(x)>k}|1ˆσ|x|βdxC|{xΩ:vn(x)>k}|11ˆσ1q|x|βdx.

    Thus using the standard Stampacchia argument, see [44], we get the existence of k0>0, independents of n such that

    |{xΩ:vn(x)>k0}|=0 for all n.

    Hence |{xΩ:v(x)>k0}| and then vL(Ω).

    If β<2(μ(λ)+s) and ˆσ2=q, since (28) holds for all α1, then using Hölder's inequality, we reach that for all n1, vrn|x|βL1(Ω), for all r< and

    (Ωvrn |x|βdx)1rC˜fnLq(Ω,|x|β(q1)dx),for all 1r<+.

    Now using Fatou's Lemma we deduce that

    (Ωvr |x|βdx)1rC˜fLq(Ω,|x|β(q1)dx),for all 1r<+

    as requested.

    Now, if β<2(μ(λ)+s) and ˆσ2<q, that is q<Nβ2(μ+s)β, and choosing α=ˆσ2qˆσ, then (α+1)ˆσ2=qˆσ2q(q1)ˆσ=q(Nβ)Nβq(2(s+μ(λ))β):=r. Going back to (28), it follows that, for all n1, vrn|x|βL1(Ω) and

    (Ωvrn |x|βdx)1rC˜fnLq(Ω,|x|β(q1)dx).

    As above, using Fatou's lemma, we get

    (Ωvr |x|βdx)1rC˜fLq(Ω,|x|β(q1)dx).

    If β=2(μ(λ)+s), then ˆσ=2. Again from (28) and choosing α=1q1, it follows that r=q and vrn|x|βL1(Ω) for all n1 with

    (Ωvqn |x|βdx)1qC˜fnLq(Ω,|x|β(q1)dx).

    Thus

    (Ωvq |x|βdx)1qC˜fLq(Ω,|x|β(q1)dx).

    As a consequence, we get the next corollary that improves the regularity results obtained in [8].

    Corollary 3.5. Let s(0,1), 0<λ<λN,s and u be theunique weak solution to problem (18) with f|x|μ(λ)L1(Ω). Suppose in addition that f|x|βμ(λ)Lq(Ω,|x|βdx) where q>1 and 2Nμ(λ)N2sβ2(μ(λ)+s). Then

    1) If β<2(μ(λ)+s) and q>(Nβ)2(μ(λ)+s)β, then u|x|μ(λ)L(Ω). Moreover, there exists a positive constant C:=C(N,β,λ,s,q,Ω) such that

    u|x|μ(λ)L(Ω)Cf|x|βμ(λ)Lq(Ω,|x|βdx).

    2) If β<2(μ(λ)+s) and q=(Nβ)2(μ(λ)+s)β, then u|x|μ(λ)Lr(Ω,|x|βdx), for all 1r<+. Moreover, there exists a positive constant C:=C(N,β,λ,s,q,r,Ω) such that

    (Ωur|x|rμ(λ)βdx)1rCf|x|βμ(λ)Lq(Ω,|x|βdx).

    3) If either β=2(μ(λ)+s) or β<2(μ(λ)+s) and 1<q<Nβ2(μ(λ)+s)β, then

    u|x|μ(λ)Lr(Ω,|x|βdx) with r=(Nβ)qNβq(2(μ(λ)+s)β). Moreover, thereexists a positive constant C:=C(N,β,λ,s,q,Ω) such that

    (Ωur |x|rμ(λ)βdx)1rCf|x|βμ(λ)Lq(Ω,|x|βdx).

    As a consequence we get the next fractional regularity.

    Theorem 3.6. Suppose that f satisfies the same condition as in Corollary 3.5. Let uL1(Ω) be the unique weak solution to (18) with λ<ΛN,s. Then

    1) If β<2(μ(λ)+s) and q>(Nβ)2(μ(λ)+s)β, then |(Δ)s2u|Lp(IRN) forall 1p<Nμ(λ)+s. In particular, thereexists a positive constant C:=C(N,β,λ,s,q,p,Ω) such that

    ||(Δ)s2u||Lp(IRN)Cf|x|βμ(λ)Lq(Ω,|x|βdx).

    2) If \beta < 2(\mu(\lambda)+s) and q = \frac{(N-\beta)}{2(\mu(\lambda)+s)-\beta} , then \big|(-\Delta)^{\frac{s}{2}} u\big| \in L^{p}({I\!\!R}^N) forall {1\le p} < \frac{N}{\mu(\lambda) +s} . In particular, thereexists a positive constant C: = C(N, \beta, \lambda, s, q, p, \Omega) such that

    ||(-\Delta)^{\frac{s}{2}} u||_{L^p( {I\!\!R}^N)} \leqslant C\|f|x|^{\beta-\mu(\lambda )}\|_{L^q(\Omega,|x|^{-\beta}dx)}.

    3) If either \beta = 2(\mu(\lambda)+s) or \beta < 2(\mu(\lambda)+s) and \frac{\beta}{\beta -\mu(\lambda)}\leq q < \frac{N-\beta}{2(\mu(\lambda)+s)-\beta} , then \big|(-\Delta)^{\frac{s}{2}} u\big| \in L^p({I\!\!R}^N) forall {1\le p} < \frac{q N}{N-\beta-q(\mu+s-\beta)} . Inparticular, there exists a positive constant C: = C(N, \beta, \lambda, s, q, p, \Omega) such that

    ||(-\Delta)^{\frac{s}{2}} u||_{L^p( {I\!\!R}^N)}\le C\|f|x|^{\beta-\mu(\lambda )}\|_{L^q(\Omega,|x|^{-\beta}dx)}.

    4) If either \beta = 2(\mu(\lambda)+s) or \beta < 2(\mu(\lambda)+s) and 1 < q\le\frac{\beta}{\beta -\mu (\lambda)} , then \big|(-\Delta)^{\frac{s}{2}} u\big| \in L^p({I\!\!R}^N) for all {1\le p} < \frac{q N}{N-qs} . In particular, there exists apositive constant C: = C(N, \beta, \lambda, s, q, p, \Omega) such that

    ||(-\Delta)^{\frac{s}{2}} u||_{L^p( {I\!\!R}^N)}\le C\|f|x|^{\beta-\mu(\lambda )}\|_{L^q(\Omega,|x|^{-\beta}dx)}.

    Proof. We start with the first case. Since \beta < 2(\mu(\lambda)+s) and q > \frac{(N-\beta)}{2(\mu(\lambda)+s)-\beta} , then by Corollary 3.5, we obtain that u(x)\le C|x|^{-\mu(\lambda)} .

    Hence \frac{u}{|x|^{2s}}\le C|x|^{-\mu(\lambda)-2s}\in L^\sigma (\Omega) for all {1 \leqslant\sigma} < \frac{N}{\mu(\lambda)+2s} . Since q > \frac{(N-\beta)}{2(\mu(\lambda)+s)-\beta} , then using Hölder inequality we can show the existence of a > \frac{N}{\mu(\lambda)+2s} such that f\in L^a(\Omega) . Thus g: = \dfrac{u}{|x|^{2s}}+f\in L^\sigma (\Omega) for all {1 \leqslant\sigma} < \frac{N}{\mu(\lambda)+2s} . Using now the regularity result in Theorem 2.10, it holds that \big|(-\Delta)^{\frac{s}{2}} u\big| \in L^p({I\!\!R}^N) for all {1 \le p} < \frac{N}{\mu(\lambda)+s} and

    ||(-\Delta)^{\frac{s}{2}} u||_{L^p( {I\!\!R}^N)}\le C\|f|x|^{\beta-\mu(\lambda )}\|_{L^q(\Omega,|x|^{-\beta}dx)}.

    The second case follows as the first case using the fact that u|x|^{\mu(\lambda)}\in L^r(\Omega, |x|^{-\beta}dx) , for all 1\leq r < +\infty .

    We consider the third case which is more involved. Assume that \beta < 2(\mu(\lambda)+s) and 1\leq q < \frac{N-\beta}{2(\mu(\lambda)+s)-\beta} , then by Corollary 3.5, we reach that u|x|^{\mu(\lambda)}\in L^{r^*}(\Omega, |x|^{-\beta}dx) with r^* = \frac{(N-\beta)q}{N-\beta-q(2(\mu(\lambda)+s)-\beta)} . We claim that \dfrac{u}{|x|^{2s}}\in L^\theta(\Omega) for all {1\le\theta} < \frac{qN}{N+q(\beta-\mu(\lambda))-\beta} . To see this we will use Hölder's inequality. More precisely, for {1\le\theta} < r^* , we have

    \begin{eqnarray*} \int\limits_ \Omega\bigg(\frac{u}{|x|^{2s}}\bigg)^\theta dx & = & \int\limits_ \Omega(u|x|^{\mu(\lambda )})^\theta (|x|^{\beta-\theta(\mu(\lambda )+2s)})|x|^{-\beta}dx\\ & \leqslant& \bigg( \int\limits_ \Omega(u|x|^{\mu(\lambda )})^{r^*}|x|^{-\beta}dx\bigg)^{\frac{\theta}{r^*}} \bigg( \int\limits_ \Omega(|x|^{\beta-\theta(\mu(\lambda )+2s)})^{\frac{r^*}{r^*-\theta}}|x|^{-\beta}dx\bigg)^{\frac{r^*-\theta}{r^*}}\\ & \leqslant& C(\Omega)\|f|x|^{\beta-\mu(\lambda )}\|^\theta_{L^q(\Omega,|x|^{-\beta}dx)} \bigg( \int\limits_\Omega (|x|^{\beta-\theta(\mu(\lambda )+2s)})^{\frac{r^*}{r^*-\theta}}|x|^{-\beta}dx\bigg)^{\frac{r^*-\theta}{r^*}}. \end{eqnarray*}

    The last integral is finite if and only if (\beta-\theta(\mu(\lambda)+2s))\frac{r^*}{r^*-\theta}-\beta > -N . This is equivalent to the fact that \theta < \frac{qN}{N+q(\beta-\mu(\lambda))-\beta} . Notice that in this case we have \frac{qN}{N+q(\beta-\mu(\lambda))-\beta} < r^* . Then the claim follows.

    In the same way and taking into consideration that \frac{qN}{N+q(\beta-\mu(\lambda))-\beta} < q , we can prove that f\in L^\theta(\Omega) for all {1\le\theta} < \frac{qN}{N+q(\beta-\mu(\lambda))-\beta} . As in the previous cases, setting g: = \dfrac{u}{|x|^{2s}}+f , then g\in L^\theta(\Omega) for all {1\le\theta} < \frac{qN}{N+q(\beta-\mu(\lambda))-\beta} . Thus by the regularity result in Theorem 2.10, we obtain that \big|(-\Delta)^{\frac{s}{2}} u\big| \in L^p({I\!\!R}^N) for all {1\le p} < \frac{\theta N}{N-\theta s} . Hence \big|(-\Delta)^{\frac{s}{2}} u\big| \in L^p({I\!\!R}^N) for all {1\le p} < \frac{q N}{N-\beta-q(\mu+s-\beta)} and

    ||(-\Delta)^{\frac{s}{2}} u||_{L^p( {I\!\!R}^N)}\le C\|f|x|^{\beta-\mu(\lambda )}\|_{L^q(\Omega,|x|^{-\beta}dx)}.

    Finally, the fourth case follows easily, using the approach of the previous case.

    To end this section we give the next weighted estimate for the fractional gradient if additional assumptions on f are satisfied. This will be used in order to show the existence of a solution to problem (1).

    Suppose that f\in L^1(|x|^{-\mu(\lambda)-a_0}dx, \Omega) for some a_0 > 0 . Hence there exists \lambda_1\in (\lambda, \Lambda_{N, s}) such that \mu(\lambda_1) = \mu(\lambda)+a_0 . Define \psi to be the unique solution to problem

    \begin{equation} \left\{ \begin{array}{rcll} (-\Delta)^s \psi & = & \lambda_1 \dfrac{\psi}{|x|^{2s}}+1 & \text{ in }\Omega , \\ \psi & = & 0 &\hbox{ in }\mathbb{R}^N\setminus\Omega, \end{array} \right. \end{equation} (29)

    then \psi\simeq |x|^{-\mu(\lambda)-a_0} near the origin. It is clear also that \psi\in L^\infty(\Omega\backslash B_r(0)) .

    Using \psi as a test function in problem (18), it holds that

    (\lambda_1-\lambda ) \int\limits_ \Omega\frac{u \psi }{|x|^{2s}}dx \leqslant \int\limits_\Omega f\psi dx.

    Hence

    \begin{equation} \int\limits_ \Omega\frac{u}{|x|^{2s+\mu(\lambda )}}dx \le C(\Omega \,\lambda ,a_0)||f||_{L^1(|x|^{-\mu(\lambda )-a_0}dx,\Omega)}. \end{equation} (30)

    The next proposition will be the crucial key in order to show a priori estimates when dealing with problem (1) with general datum f .

    Proposition 3.7. Assume that f\in L^1(|x|^{-\mu(\lambda)-a_0}dx, \Omega) for some a_0 > 0 . Let v be the unique weak solution to problem (18), then

    \begin{equation} \bigg\|(-\Delta )^{\frac{s}{2}}u\bigg\|_{L^\alpha (|x|^{-\mu(\lambda )}dx,\Omega)}\le C(\Omega,\lambda ,a_0)||f||_{L^1(|x|^{-\mu(\lambda )-a_0}dx,\Omega)}\mathit{\text{ for all }}{1 \leqslant\alpha } < \frac{N}{N-s}. \end{equation} (31)

    To prove Proposition 3.7, we need the following lemma proved in [27].

    Lemma 3.8. Let N \geqslant1 , R > 0 and \alpha, \beta \in (-\infty, N) . There exists C: = C(N, R, \alpha, \beta) > 0 such that:

    \bullet If (N-\alpha - \beta)\neq 0 , then

    \int_{B_R (0)} \frac{dz}{|x-z|^{\alpha} |y-z|^{\beta} } \leqslant C \Big(1+ |x-y|^{N-\alpha - \beta} \Big), \quad \text{ for all } x,y \in B_R(0) \text{ with } x \neq y.

    \bullet If (N - \alpha - \beta) = 0 , then

    \int_{B_R (0)} \frac{dz}{|x-z|^{\alpha} |y-z|^{\beta} } \leqslant C \Big(1+ \big|\ln |x-y|\big| \Big), \quad \text{ for all } x,y \in B_R(0) \text{ with } x \neq y.

    Proof of Proposition 3.7. Since \int\limits_\Omega f |x|^{-\mu(\lambda)}dx < \infty , then by Theorem 2.10, we know that

    \bigg\|(-\Delta )^{\frac{s}{2}}u\bigg\|_{L^\alpha (\Omega)}\le C(\Omega,\lambda ,a_0)||f||_{L^1(|x|^{-\mu(\lambda )-a_0}dx,\Omega)}\text{ for all }{1 \leqslant\alpha } < \frac{N}{N-s}.

    Thus, to prove the claim we just need to show that

    \int_{B_r(0)}|(-\Delta )^{\frac{s}{2}}u|^\alpha|x|^{-\mu(\lambda )} dx\le C(\Omega,\lambda ,a_0)||f||^{\alpha}_{L^1(|x|^{-\mu(\lambda )-a_0}dx,\Omega)} \text{ for all }{1\le\alpha } < \frac{N}{N-s},

    where B_r(0) \subset \subset \Omega .

    We set g(x): = \lambda \dfrac{u}{|x|^{2s}}+\mu f , then u(x) = \int\limits_\Omega \mathcal{G}_s(x, y)g(y)dy . Hence, for a.e. x\in B_r(0) ,

    \begin{equation} |(-\Delta )^{\frac{s}{2}}u(x)| \leqslant\int\limits_ \Omega|(-\Delta )^{\frac{s}{2}}\mathcal{G}_s(x,y)|g(y)dy. \end{equation} (32)

    Notice that from [6], we know that

    \begin{equation} \big|(-\Delta)_x^{\frac{s}{2}} G_s(x,y)\big| \leqslant\frac{C}{|x-y|^{N-s}} \left(\bigg|\ln \frac{1}{|x-y|} \bigg| + \ln \Big(\frac{1}{\delta(x)}\Big) \right), \quad \text{ for a.e. } x, y \in \Omega. \end{equation} (33)

    Since B_r(0)\subset\subset \Omega , one has

    \begin{equation} \big|(-\Delta)_x^{\frac{s}{2}} G_s(x,y)\big| \leqslant\frac{C}{|x-y|^{N-s}}\ln \bigg(\frac{C}{|x-y|}\bigg) \quad , \text{ for a.e. } (x,y)\in B_r(0)\times \Omega. \end{equation} (34)

    For the remaining part of this proof, we will use systematically this estimate for a.e. (x, y)\in B_r(0)\times \Omega .

    Thus we conclude that

    \begin{equation} \big|(-\Delta)_x^{\frac{s}{2}} G_s(x,y)\big| \leqslant G_s(x,y) h(x,y), \quad { \text{ for a.e. } (x,y)\in B_r(0)\times \Omega}, \end{equation} (35)

    with

    h(x,y) = \left\{ \begin{array}{lll} \frac{1}{|x-y|^s}\ln \left( \frac{C}{|x-y|}\right) \text{ if }|x-y| < \delta(y)\\ \\ \frac{1}{\delta^s(y)}\ln \left( \frac{C}{|x-y|}\right) \text{ if }|x-y|\ge\delta(y). \end{array} \right.

    Fix 1 < \alpha < p_* = \frac{N}{N-s} . Going back to (32), we deduce that, for a.e. x\in B_r(0) , we have

    \begin{eqnarray*} |(-\Delta )^{\frac{s}{2}}u(x)|& \leqslant& \int\limits_ \Omega|(-\Delta )^{\frac{s}{2}}\mathcal{G}_s(x,y)|g(y)dy\le \lambda\int\limits_ \Omega|(-\Delta )^{\frac{s}{2}}\mathcal{G}_s(x,y)|\frac{u(y)}{|y|^{2s}}dy + \int\limits_ \Omega|(-\Delta )^{\frac{s}{2}}\mathcal{G}_s(x,y)|f(y)dy. \end{eqnarray*}

    Hence

    |(-\Delta )^{\frac{s}{2}}u(x)||x|^{-\frac{\mu(\lambda )}{\alpha }} \leqslant\lambda|x|^{-\frac{\mu(\lambda )}{\alpha }} \int\limits_ \Omega|(-\Delta )^{\frac{s}{2}}\mathcal{G}_s(x,y)|\frac{u(y)}{|y|^{2s}}dy + |x|^{-\frac{\mu(\lambda )}{\alpha }} \int\limits_ \Omega|(-\Delta )^{\frac{s}{2}}\mathcal{G}_s(x,y)|f(y)dy.

    We set

    K_1(x) = \int\limits_ \Omega|(-\Delta )^{\frac{s}{2}}\mathcal{G}_s(x,y)|\frac{u(y)}{|y|^{2s}}dy

    and

    K_2(x) = \int\limits_ \Omega|(-\Delta )^{\frac{s}{2}}\mathcal{G}_s(x,y)|f(y)dy.

    We begin estimating K_1 . We have

    \begin{eqnarray*} K^\alpha_1(x)& \leqslant& \left( \int\limits_ \Omega|(-\Delta )^{\frac{s}{2}}\mathcal{G}_s(x,y)|\frac{u(y)}{|y|^{2s}}dy\right) ^{\alpha }{ \le} \left( \int\limits_\Omega h(x,y) \mathcal{G}_s(x,y)\frac{u(y)}{|y|^{2s}} dy\right) ^{\alpha }\\& \leqslant& \left( \int\limits_\Omega (h(x,y))^{\alpha } \mathcal{G}_s(x,y)\frac{u(y)}{|y|^{2s}}dy\right) \left( \int\limits_ \Omega\mathcal{G}_s(x,y)g(y)dy\right) ^{\alpha-1}\\ & \leqslant& \int\limits_ \Omega\left( h^{\alpha }(x,y) \mathcal{G}_s(x,y)\frac{u(y)}{|y|^{2s}}dy\right) u^{\alpha-1}(x). \end{eqnarray*}

    Thus, using Fubini's theorem, it holds that

    \begin{eqnarray*} \int_{B_r(0)} K^\alpha_1(x)|x|^{-\mu(\lambda )} dx & \leqslant& \lambda \int\limits_ \Omega\frac{u(y)}{|y|^{2s}}\left( \int_{B_r(0)} h^{\alpha }(x,y)\mathcal{G}_s(x,y)u^{\alpha-1}(x)|x|^{-\mu(\lambda )} dx\right) dy. \end{eqnarray*}

    Recall that, by (30), we have

    \int\limits_ \Omega\frac{u(y)}{|y|^{2s+\mu(\lambda )}}dy \le C \int\limits_ \Omega\frac{f(y)}{|y|^{\mu(\lambda )+a_0}}dy.

    Therefore we obtain that

    \begin{eqnarray*} \int_{B_r(0)} K^\alpha_1(x)|x|^{-\mu(\lambda )} dx & \leqslant& C_2 \int\limits_ \Omega\frac{u(y)}{|y|^{2s}} \left( \int_{B_r(0)\cap \{|x-y| < \delta(y)\}} \dfrac{u^{\alpha-1}(x)\mathcal{G}_s(x,y)}{|x|^{\mu(\lambda )}} \frac{1}{|x-y|^{s\alpha }}\ln \left( \frac{C}{|x-y|}\right) ^{\alpha }dx\right) dy \\ \\ &+ & C_2 \int\limits_ \Omega\frac{u(y)}{|y|^{2s} \delta^{s\alpha }(y)} \left( \int_{B_r(0)\cap \{|x-y|\ge\delta(y)\}} \dfrac{u^{\alpha-1}(x)\mathcal{G}_s(x,y)}{|x|^{\mu(\lambda )}} \ln \left( \frac{C}{|x-y|}\right) ^{\alpha }dx\right) dy\\ & = & J_{1}+J_{2}. \end{eqnarray*}

    Respect to J_{1} , using the fact that for all \eta > 0 ,

    \frac{1}{|x-y|^{s\alpha }}\ln \left(\frac{C}{|x-y|}\right)^{\alpha } \leqslant{\frac {\alpha}{\eta}}\frac{C^{{\eta}}}{|x-y|^{s\alpha+\eta}},

    and by Proposition 2.8, we reach that

    \begin{eqnarray*} J_{1}&\le& C_2 \int\limits_ \Omega\frac{u(y)}{|y|^{2s}} \left( \int_{B_r(0)} \dfrac{u^{\alpha-1}(x)}{|x|^{\mu(\lambda )}|x-y|^{N-(2s-s\alpha-\eta)}}dx\right) dy\\ & \leqslant& C \int\limits_ \Omega\frac{u(y)}{|y|^{2s}}\left( \int_{B_r(0)\cap \{|x| \geqslant\frac 12 |y|\}} \dfrac{u^{\alpha-1}(x)}{|x|^{\mu(\lambda )}|x-y|^{N-(2s-s\alpha-\eta)}}dx\right) dy\\ &+ & C \int\limits_ \Omega\frac{u(y)}{|y|^{2s}}\left( \int_{B_r(0)\cap \{|x| < \frac 12 |y|\}} \dfrac{u^{\alpha-1}(x)}{|x|^{\mu(\lambda )}|x-y|^{N-(2s-s\alpha-\eta)}}dx\right) dy\\ & \leqslant& I_{1}+I_{2}. \end{eqnarray*}

    To estimate I_1 , we observe that

    \begin{eqnarray*} I_1& \leqslant& C \int\limits_ \Omega\frac{u(y)}{|y|^{2s+\mu(\lambda )}}\left( \int_{B_r(0)}\dfrac{u^{\alpha-1}(x)}{|x-y|^{N-(2s-s\alpha-\eta)}}dx\right) dy. \end{eqnarray*}

    Recall that u\in L^\sigma (\Omega) for all \sigma < \frac{N}{N-2s} . Since \alpha < \frac{N}{N-s} , fixing \sigma_0 < \frac{N}{N-2s} and using Hölder inequality, we get

    \int_{B_r(0)}\dfrac{u^{\alpha-1}(x)}{|x-y|^{N-(2s-s\alpha-\eta)}}dx \leqslant\bigg(\int_{B_r(0)}u^{\sigma_0}dx \bigg)^{\frac{\alpha-1}{\sigma_0}} \bigg(\int_{B_r(0)}\frac{1}{|x-y|^{\frac{(N-(2s-s\alpha-\eta))\sigma_0}{\sigma_0-(\alpha-1)}}}dx\bigg)^{\frac{\sigma_0-(\alpha-1)}{\sigma_0}}.

    Since \alpha < \frac{N}{N-s} , then we can chose \sigma_0 close to \frac{N}{N-2s} and \eta small enough such that \frac{(N-(2s-s\alpha-\eta))\sigma_0}{\sigma_0-(\alpha-1)} < N . Thus

    \int_{B_r(0)}\frac{1}{|x-y|^{\frac{(N-(2s-s\alpha-\eta))\sigma_0}{\sigma_0-(\alpha-1)}}}dx\le C(r,\Omega),

    and then

    \begin{equation} I_1\le C \bigg(\int_{B_r(0)}u^{\sigma_0}dx \bigg)^{\frac{\alpha-1}{\sigma_0}} \bigg( \int\limits_ \Omega\frac{u(y)}{|y|^{2s+\mu(\lambda )}}dy\bigg)\le C\bigg( \int\limits_ \Omega\frac{f(y)}{|y|^{\mu(\lambda )+a_0}}dy\bigg)^\alpha. \end{equation} (36)

    We deal now with I_2 . Notice that \{|x| \leqslant\frac 12 |y|\}\subset \{|x-y| \geqslant\frac 12 |y|\} . Thus

    \begin{eqnarray*} I_2 & \leqslant& C \int\limits_ \Omega\frac{u(y)}{|y|^{\mu(\lambda )+2s}}\left( \int_{B_r(0)}\dfrac{u^{\alpha-1}(x)}{|x|^{\mu(\lambda )}|x-y|^{N-(2s+\mu(\lambda )-s\alpha-\eta)}}dx\right) dy. \end{eqnarray*}

    As in the estimate of I_1 , setting \theta = \dfrac{\sigma_0}{\sigma_0-(\alpha-1)} , we get

    \begin{eqnarray*} & \int_{B_r(0)}\dfrac{u^{\alpha-1}(x)}{|x|^{\mu(\lambda )}|x-y|^{N-(2s+\mu(\lambda )-s\alpha-\eta)}}dx\\ & \leqslant\bigg(\int_{B_r(0)}u^{\sigma_0}dx \bigg)^{\frac{\alpha-1}{\sigma_0}} \bigg(\int_{B_r(0)}\frac{1}{|x|^{\mu(\lambda )\theta}|x-y|^{(N-(2s+\mu(\lambda )-s\alpha-\eta))\theta}}dx\bigg)^{\frac{1}{\theta}}. \end{eqnarray*}

    For \alpha < \frac{N}{N-s} fixed, we can chose \eta small enough and \sigma_0 close to p_2 such that

    N-(N-(2s+\mu(\lambda )-s\alpha-\eta))\theta-\mu(\lambda )\theta < N.

    Hence using Lemma 3.8, it holds that

    \begin{eqnarray*} & \int_{B_r(0)}\frac{1}{|x|^{\mu(\lambda )\theta}|x-y|^{(N-(2s+\mu(\lambda )-s\alpha-\eta))\theta}}dx \le C(r_0). \end{eqnarray*}

    Therefore we conclude that

    \begin{equation} \begin{array}{lll} I_{2} & \leqslant& C \int\limits_ \Omega\frac{u(y)}{|y|^{\mu(\lambda )+2s}} \bigg(\int_{B_r(0)}u^{\sigma_0}dx \bigg)^{\frac{\alpha-1}{\sigma_0}} \leqslant C\bigg( \int\limits_ \Omega\frac{f(y)}{|y|^{\mu(\lambda )+a_0}}dy\bigg)^\alpha. \end{array} \end{equation} (37)

    As a consequence, we have

    \begin{equation} J_{1}\le C\bigg( \int\limits_ \Omega\frac{f(y)}{|y|^{\mu(\lambda )+a_0}}dy\bigg)^\alpha. \end{equation} (38)

    We deal now with J_{2} . Let c_1 > 0 be a positive constant to be chosen later, then

    \begin{eqnarray*} J_{2} & \leqslant& \int\limits_ \Omega\frac{u(y)}{|y|^{2s} \delta^{s\alpha }(y)} \left( \int_{B_r(0)\cap \{|x-y|\ge\delta(y)\}} \dfrac{u^{\alpha-1}(x)\mathcal{G}_s(x,y)}{|x|^{\mu(\lambda )}} \ln \left( \frac{C}{|x-y|}\right) ^{\alpha }dx\right) dy\\ & \leqslant& \int_{\Omega \cap \{\delta(y) > c_1\}} \frac{u(y)}{|y|^{2s} \delta^{s\alpha }(y)} \left( \int_{B_r(0)\cap \{|x-y|\ge\delta(y)\}} \dfrac{u^{\alpha-1}(x)\mathcal{G}_s(x,y)}{|x|^{\mu(\lambda )}} \ln \left( \frac{C}{|x-y|}\right) ^{\alpha }dx\right) dy \\ &+ & \int_{\Omega \cap \{\delta(y) < c_1\}}\frac{u(y)}{|y|^{2s} \delta^{s\alpha }(y)} \left( \int_{B_r(0)\cap \{|x-y|\ge\delta(y)\}} \dfrac{u^{\alpha-1}(x)\mathcal{G}_s(x,y)}{|x|^{\mu(\lambda )}} \ln \left( \frac{C}{|x-y|}\right) ^{\alpha }dx\right) dy\\ & \leqslant& C_1J_{1}+ \int_{\Omega \cap \{\delta(y) < {c_1}\}}\frac{u(y)}{|y|^{2s} \delta^{s\alpha }(y)} \left( \int_{B_r(0)\cap \{|x-y|\ge\delta(y)\}} \dfrac{u^{\alpha-1}(x)\mathcal{G}_s(x,y)}{|x|^{\mu(\lambda )}} \ln \left( \frac{C}{|x-y|}\right) ^{\alpha }dx\right) dy. \end{eqnarray*}

    We set

    A = \int_{\Omega \cap \{\delta(y) < {c_1}\}}\frac{u(y)}{|y|^{2s} \delta^{s\alpha }(y)} \left( \int_{B_r(0)\cap \{|x-y|\ge\delta(y)\}} \dfrac{u^{\alpha-1}(x)\mathcal{G}_s(x,y)}{|x|^{\mu(\lambda )}} \ln \left( \frac{C}{|x-y|}\right) ^{\alpha }dx\right) dy.

    Choosing c_1 small, we get the existence of a positive constant c_2 such that for \delta(y) < c_1 and x\in B_{r}(0) , we have |x-y| > c_2 > 0 . Hence using again Proposition 2.8, we deduce that

    A \leqslant \int_{\Omega \cap \{\delta(y) < c_1\}}\frac{u(y)}{\delta^{s(\alpha-1)}(y)} \left( \int_{B_r(0)\cap \{|x-y|\ge\delta(y)\}} \dfrac{u^{\alpha-1}(x)}{|x|^{\mu(\lambda )}} dx\right) dy.

    As above, for \alpha_0 < \frac{N}{N-2s} , we have

    \begin{eqnarray*} \int_{B_r(0)}\dfrac{u^{\alpha-1}(x)}{|x|^{\mu(\lambda )}}dx & \leqslant& \bigg(\int_{B_r(0)}u^{\sigma_0}dx \bigg)^{\frac{\alpha-1}{\sigma_0}} \bigg(\int_{B_r(0)}\frac{1}{|x|^{\mu(\lambda )\theta}}dx\bigg)^{\frac{1}{\theta}}\\ & \leqslant& C\bigg(\int_{B_r(0)}u^{\sigma_0}dx \bigg)^{\frac{\alpha-1}{\sigma_0}}\le C\bigg( \int\limits_ \Omega\frac{f(y)}{|y|^{\mu(\lambda )+a_0}}dy\bigg)^{\alpha-1}. \end{eqnarray*}

    On the other hand, we have

    \int_{\Omega \cap \{\delta(y) < c_2\}}\frac{u(y)}{\delta^{s(\alpha-1)}(y)}dy \leqslant C\int_{\Omega}\frac{u(y)}{\delta^{s}(y)}dy \leqslant C \int\limits_ \Omega\frac{f(y)}{|y|^{\mu(\lambda )+a_0}}dy.

    Hence

    J_{2}\le C\bigg( \int\limits_ \Omega\frac{f(y)}{|y|^{\mu(\lambda )+a_0}}dy\bigg)^\alpha.

    As a consequence we deduce that

    \int_{B_r(0)} K^\alpha_1(x)|x|^{-\mu(\lambda )} dx\le C\bigg( \int\limits_ \Omega\frac{f(y)}{|y|^{\mu(\lambda )+a_0}}dy\bigg)^\alpha.

    We treat now the term K_2 . Recall that

    K_2(x) = \int\limits_ \Omega|(-\Delta )^{\frac{s}{2}}\mathcal{G}_s(x,y)|f(y)dy.

    Notice that for \eta > 0 , small enough, to be chosen later, we have

    |(-\Delta )^{\frac{s}{2}}\mathcal{G}_s(x,y)| \leqslant\frac{C}{|x-y|^{N-s}}\ln (\frac{C}{|x-y|})\le \frac{C}{|x-y|^{N-(s-\eta)}} {\text{ for a.e. }(x,y)\in B_r(0)\times \Omega}.

    Thus

    \begin{eqnarray*} K_2(x)|x|^{-\frac{\mu(\lambda )}{\alpha }} & \leqslant& \frac{C}{|x|^{\frac{\mu(\lambda )}{\alpha }}} \int\limits_\Omega \frac{f(y)}{|x-y|^{N-(s-\eta)}}dy\\ & \leqslant& \frac{C}{|x|^{\frac{\mu(\lambda )}{\alpha }}}\int_{\Omega \cap \{|y| \leqslant4|x|\}}\frac{f(y)}{|x-y|^{N-(s-\eta)}}dy + \frac{C}{|x|^{\frac{\mu(\lambda )}{\alpha }}}\int_{\Omega \cap \{|y| \geqslant4|x|\}}\frac{f(y)}{|x-y|^{N-(s-\eta)}}dy\\ & \leqslant& C\int_{\Omega \cap \{|y| \leqslant4|x|\}}\frac{f(y)}{|y|^{\mu(\lambda )}}\frac{1}{|x-y|^{N-(s-\eta)}}dy+ \frac{C}{|x|^{\frac{\mu(\lambda )}{\alpha }}}\int_{\Omega \cap \{|y| \geqslant4|x|\}}\frac{f(y)}{|x-y|^{N-(s-\eta)}}dy\\ & = & L_1(x)+L_2(x). \end{eqnarray*}

    We start with the estimate of term L_1 . Since \dfrac{f(y)}{|y|^{\mu(\lambda)}}{\in L^1(\Omega)} , then by Theorem 2.4, we deduce that L_1\in L^{\sigma}(B_r(0)) for all {1\le\sigma} < \frac{N}{N-(s-\eta)} . Thus L_1\in L^\alpha (\Omega) and

    \int_{B_r(0)}L^\alpha_1(x) dx \leqslant C\bigg( \int\limits_ \Omega\frac{f(y)}{|y|^{\mu(\lambda )+a_0}}dy\bigg)^\alpha.

    We consider now L_2 . Since |y| \geqslant4|x| , then |x-y| \geqslant\frac 34 |y| and |x-y|{ \geqslant 3 |x|} . Hence

    L_2(x) \leqslant \frac{C}{|x|^{\frac{\mu(\lambda )}{\alpha }+N-(\mu(\lambda )+s+a_0-\eta)}} \int\limits_ \Omega\frac{f(y)}{|y|^{\mu(\lambda )+a_0}}dy.

    Since (\frac{\mu(\lambda)}{\alpha }+N-(\mu(\lambda)+s+a_0-\eta))\alpha < N , then we conclude that I_2\in L^{\alpha }(B_r(0)) and

    \int_{B_r(0)}L^\alpha_2(x) dx \leqslant C\bigg( \int\limits_ \Omega\frac{f(y)}{|y|^{\mu(\lambda )+a_0}}dy\bigg)^\alpha.

    As a consequence, we have proved that

    \int_{B_r(0)} K^\alpha_2(x)|x|^{-\mu(\lambda )} dx \le C \bigg( \int\limits_ \Omega\frac{f(y)}{|y|^{\mu(\lambda )+a_0}}dy\bigg)^\alpha.

    Therefore we conclude that

    \int_{B_r(0)} |(-\Delta ^{\frac{s}{2}})u|^{\alpha } |x|^{-\mu(\lambda )} dx \le C \bigg( \int\limits_ \Omega\frac{f(y)}{|y|^{\mu(\lambda )+a_0}}dy\bigg)^\alpha.

    Hence the main estimate follows and this finishes the proof of our proposition.

    In this section we consider the question of existence and non existence of a positive solution to problem (1) with \mathfrak{F}(u)\equiv |(-\Delta)^{\frac s2}u| . Namely we will treat the problem

    \begin{equation} \left\{ \begin{array}{rcll} (-\Delta )^s u & = &\lambda \dfrac{u}{|x|^{2s}}+|(-\Delta )^{\frac s2}u|^p+ \rho f & \text{ in } \Omega,\\ u& > &0 & \text{ in }\Omega,\\ u& = &0 & \text{ in }( {I\!\!R}^N\setminus\Omega), \end{array}\right. \end{equation} (39)

    where \Omega \subset {I\!\!R}^N is a bounded regular domain containing the origin, s\in (0, 1) , \lambda\leqslant\Lambda_{N, s} , \rho > 0 , 1 < p < \infty and f is positive measurable function satisfies some hypothesis that will be precised later.

    Let us begin with the next definition.

    Definition 4.1. Assume that f\in L^1(\Omega) is a nonnegative function. We say that u is a weak solution to problem (39) if |(-\Delta)^{\frac s2}u|^p\in L^1(\Omega), \dfrac{u}{|x|^{2s}}\in L^1(\Omega) and, setting g\equiv \lambda \dfrac{u}{|x|^{2s}}+|(-\Delta)^{\frac s2}u|^p+\rho f , then u is a weak solution to problem (8) in the sense of Definition 2.6.

    The existence of a solution in the case \lambda = 0 was proved in [7] without any limitation on p under suitable hypotheses on f . However, if \lambda > 0 , taking into consideration the singularity generated by the Hardy potential, it is possible to show a non existence result for p large. In the next computation we will find the exact critical exponent for the non existence.

    Recall that we are considering the case \mathfrak{F}(u)(x) = |(-\Delta)^{\frac s2}u(x)| . We begin by analyzing the radial case in the whole space as in [8]. Consider the equation

    \begin{equation} (-\Delta)^s w-\lambda \frac{w}{|x|^{2s}} = |(-\Delta )^{\frac s2} w|^p \text{ in } {I\!\!R}^N, \end{equation} (40)

    then we search radial positive solution in the form w = A|x|^{\beta-\frac{N-2s}{2}} , with A > 0 . By a direct computation, it follows that

    A\gamma_{\beta,s}|x|^{-2s-\frac{N-2s}{2}+\beta}-\lambda A|x|^{\beta-2s-\frac{N-2s}{2}} = \dfrac{A^p |\gamma_{\beta,\frac s2}|^p}{|x|^{(\frac{N-2s}{2}- \beta +s)p}},

    with

    \begin{equation} \gamma_{\beta,t}: = \gamma_{-\beta,t}: = \frac{2^{2t}\Gamma(\frac{N+2t+2\beta}{4})\Gamma(\frac{N+2t-2\beta}{4})}{\Gamma(\frac{N-2t-2\beta}{4})\Gamma(\frac{N-2t+2\beta}{4})} \end{equation} (41)

    and t\in\{\frac s2, \, s\} .

    Hence, by homogeneity, we need to have

    p = \dfrac{\frac{N-2s}{2}-\beta+2s}{\frac{N-2s}{2}-\beta+s},

    which means that \beta = \frac{N-2s}{2}+ {\frac{ps}{p-1}}-\frac{2s}{p-1} . Hence the constant A satisfies

    \gamma_{\beta,s} -\lambda = A^{p-1} |\gamma_{\beta,\frac s2}|^p .

    Using the fact that A > 0 , it holds that \gamma_\beta-\lambda > 0 . Define the application

    \begin{array}{rcl}\Upsilon:(-\frac{N-2s}{2}, \frac{N-2s}{2})&\mapsto& (0,\Lambda_{N,s})\\ \beta&\mapsto& \gamma_{\beta} \end{array}

    Then \Upsilon is even and the restriction of \Upsilon to the set [0, \frac{N-2s}{2}) is decreasing, see [24] and [26]. So there exists a unique \alpha_\lambda\in (0, \Lambda_{N, s}] such that \gamma_{\alpha_{\lambda}} = \gamma_{-\alpha_{\lambda}} = \lambda .

    Let \beta_0 = -\beta_1 = \alpha_\lambda . Setting

    \begin{equation} p_+(\lambda, s): = \dfrac{\frac{N-2s}{2}-\beta_0+2s}{\frac{N-2s}{2}-\beta_0+s} = \frac{N+2s-2\alpha_{\lambda}}{N-2\alpha_{\lambda}}, \end{equation} (42)

    and

    \begin{equation} p_-(\lambda, s): = \dfrac{\frac{N-2s}{2}-\beta_1+2s}{\frac{N-2s}{2}-\beta_1+s} = \frac{N+2s+2\alpha_\lambda }{N+2\alpha_\lambda }, \end{equation} (43)

    it holds that p_-(\lambda, s) < p_+(\lambda, s) and \gamma_\beta-\lambda > 0 if and only if

    p_-(\lambda, s) < p < p_+(\lambda, s).

    It is easy to check that p_{+}(\lambda, s) and p_-(\lambda, s) are respectively an increasing and a decreasing function in \alpha_{\lambda} and, therefore, are respectively a decreasing and an increasing function in the variable \lambda . Thus

    \dfrac{N}{N-s} < p_{-}(\lambda,s) < \dfrac {N+2s}{N} < p_{+}(\lambda,s) < 2, \text{ for } 0 < \lambda < \Lambda_{N,s}.

    Notice that

    p_+(\lambda, s) = \dfrac{\mu(\lambda )+2s}{\mu(\lambda )+s} \hbox{ and } p_-(\lambda, s) = \dfrac{\bar{\mu}(\lambda )+2s}{\bar{\mu}(\lambda )+s},

    where \mu(\lambda) and \bar{\mu}(\lambda) are defined by (21).

    Hence, for p_-(\lambda, s) < p < p_+(\lambda, s) fixed, using the fact that \Omega \subset B_R(0) for R large, we get the existence of a positive constant C_1 > 0 such that w(x) = C_1|x|^{\beta-\frac{N-2s}{2}} is a radial supersolution for the Dirichlet problem (39) if f(x) \leqslant\frac{1}{|x|^{\frac{N-2s}{2}+2s-\beta}} with \rho small.

    To show that p_+(\lambda, s) is critical, we prove the next non existence result.

    Theorem 4.2. Let s\in (0, 1) and suppose that p > p_+(\lambda, s) . Then for all \rho \geqslant0 , problem (39) has no positive weak solution u in the sense of Definition 4.1.

    Proof. We argue by contradiction. Assume that problem (39) has a positive solution u in the sense of Definition 4.1, then (-\Delta)^{\frac s2} u \in L^p(\Omega) and \dfrac{u}{|x|^{2s}}\in L^1(\Omega) . By Lemma 3.1, it follows that

    u(x)\ge C|x|^{-\mu(\lambda )}\text{ in }B_r(0)\subset\subset \Omega.

    Since |(-\Delta)^{\frac s2} u|^p+\lambda \dfrac{u}{|x|^{2s}}+\rho f\in L^1(\Omega) , then from the regularity result in Theorem 2.10, we deduce that (-\Delta)^{\frac{s}{2}}u\in L^t({I\!\!R}^N) for all {1\le t} < \frac{N}{N-s} .

    Let \theta\in L^\infty(\Omega) be a nonnegative function such that \text{Supp}\, \theta\subset\subset B_{\frac{r}{2}}(0)\subset B_{r}(0)\subset\subset \Omega and define \phi_{\theta} \in H_0^s(\Omega) \cap L^\infty(\Omega) to be the unique solution of the problem

    \begin{equation} \left\{ \begin{aligned} (-\Delta )^{\frac{s}{2}} \phi_{\theta} & = \theta\,, & \quad \text{ in } \Omega\,, \\ \phi_{\theta} & = 0\,, & \text{ in } {I\!\!R}^N \setminus {\Omega}. \end{aligned} \right. \end{equation} (44)

    From [40], it holds that \phi_{\theta}\simeq \delta^{\frac{s}{2}} near the boundary of \Omega . Using \phi_{\theta} as test function in (39), we get

    \begin{equation} \begin{array}{lll} & & \lambda \int_{\Omega} \frac{u \phi_{\theta}\,}{|x|^{2s}}dx+ \int_{\Omega} |(-\Delta )^{\frac s2} u |^p \phi_{\theta}\, dx+\rho \int\limits_\Omega f\phi_{\theta} = \int_{\Omega} u(-\Delta )^s\, \phi_{\theta}\, dx = \int_{ {I\!\!R}^N} (-\Delta )^{\frac s2} u\, (-\Delta )^{\frac s2}\phi_{\theta} \, dx\\ & = & \int_{\Omega} (-\Delta )^{\frac s2} u\, (-\Delta )^{\frac s2}\phi_{\theta} \, dx + \int_{ {I\!\!R}^N\backslash \Omega} (-\Delta )^{\frac s2} u\, (-\Delta )^{\frac s2}\phi_{\theta} \, dx. \end{array} \end{equation} (45)

    We treat separately each term in the right hand of the above identity.

    Since \text{Supp}\, \theta\subset\subset B_r(0)\subset\subset \Omega , using the fact that \phi_{\theta} > 0 in \Omega and then Hölder inequality, we reach that

    \begin{equation} \begin{array}{lll} |\int_{\Omega} (-\Delta )^{\frac s2} u\, (-\Delta )^{\frac s2}\phi_{\theta} \, dx| & = & |\int_{\Omega} (-\Delta )^{\frac s2} u\, \theta \, dx|\\ & \leqslant& \int_{\Omega} |(-\Delta )^{\frac s2} u|\phi_{\theta} \, \frac{\theta}{\phi_{\theta}}dx. \end{array} \end{equation} (46)

    Next, applying Young's inequality, it holds that

    \begin{equation} |\int_{\Omega} (-\Delta )^{\frac s2} u\, (-\Delta )^{\frac s2}\phi_{\theta} \, dx| \leqslant \varepsilon \int_{\Omega} |(-\Delta )^{\frac s2} u|^p\phi_{\theta}\, dx+C( \varepsilon ) \int\limits_ \Omega \, \frac{\theta^{p'}}{\phi_{\theta}^{p'-1}}dx, \end{equation} (47)

    where \epsilon > 0 will be chosen later.

    Now we deal with the term \int_{ {I\!\!R}^N\backslash \Omega} (-\Delta)^{\frac s2} u\, (-\Delta)^{\frac s2}\phi_{\theta} \, dx .

    Since x\in {I\!\!R}^N\backslash \Omega , using again Hölder inequality, it follows that

    |\int_{ {I\!\!R}^N\backslash \Omega} (-\Delta )^{\frac s2} u\, (-\Delta )^{\frac s2}\phi_{\theta} \, dx| \leqslant ||(-\Delta )^{\frac s2} u||_{L^t( {I\!\!R}^N\backslash \Omega)} ||(-\Delta )^{\frac s2}\phi_{\theta}||_{L^{t'}( {I\!\!R}^N\backslash \Omega)}

    with t < \frac{N}{N-s} . By hypothesis ||(-\Delta)^{\frac s2} u||_{L^t({I\!\!R}^N\backslash \Omega)} < \infty . Now respect to ||(-\Delta)^{\frac s2}\phi_{\theta}||_{L^{t'}({I\!\!R}^N\backslash \Omega)} , we have

    \begin{eqnarray*} ||(-\Delta )^{\frac s2}\phi_{\theta}||^{t'}_{L^{t'}( {I\!\!R}^N\backslash \Omega)} & = & \int_{ {I\!\!R}^N\backslash \Omega} \bigg( \int\limits_ \Omega\frac{\phi_{\theta}(y)}{|y-x|^{N+s}}dy\bigg)^{t'} \, dx\\ & = & \int_{ {I\!\!R}^N\backslash \Omega} \bigg(\int_{B_r(0)} \frac{\phi_{\theta}(y)}{|y-x|^{N+s}}dy\bigg)^{t'} \, dx + \int_{ {I\!\!R}^N\backslash \Omega} \bigg(\int_{\Omega \backslash B_r(0)} \frac{\phi_{\theta}(y)}{|y-x|^{N+s}}dy\bigg)^{t'} \, dx. \end{eqnarray*}

    Recalling that \text{Supp}\, \theta\subset B_{\frac{r}{2}}(0) \subset B_r(0)\subset\subset \Omega , then we can prove that

    ||\phi_{\theta}||_{L^\infty(\Omega \backslash B_r(0))}\le C||\theta||_{L^1(B_r(0))},

    where C depends only on the N, s . Thus

    \int_{ {I\!\!R}^N\backslash \Omega} \bigg(\int_{\Omega \backslash B_r(0)} \frac{\phi_{\theta}(y)}{|y-x|^{N+s}}dy\bigg)^{t'} \, dx\le C \int_{ {I\!\!R}^N\backslash \Omega} \bigg(\int_{\Omega \backslash B_r(0)} \frac{1}{|y-x|^{N+s}}dy\bigg)^{t'} \, dx\le C(r,\Omega,s,N,t).

    Now, choosing r small enough, we obtain that, for x\in {I\!\!R}^N\backslash \Omega and y\in B_r(0) , |x-y|\ge c(|x|+ 1) . Hence

    \int_{ {I\!\!R}^N\backslash \Omega} \bigg(\int_{B_r(0)} \frac{\phi_{\theta}(y)}{|y-x|^{N+s}}dy\bigg)^{t'} \, dx\le \int_{ {I\!\!R}^N\backslash \Omega}\frac{C}{(|x|+1)^{t'(N+s)}} \bigg(\int_{B_r(0)} \phi_{\theta}(y) dy\bigg)^{t'} \, dx\le C||\phi_{\theta}||^{t'}_{L^1(\Omega)}.

    Now, going back to (45), choosing \varepsilon < < 1 in estimate (47), we obtain that

    \begin{equation} \lambda \int_{\Omega} \frac{u \phi_{\theta}\,}{|x|^{2s}}dx\le C( \varepsilon ) \int\limits_ \Omega \, \frac{\theta^{p'}}{\phi_{\theta}^{p'-1}}dx +C||\phi_{\theta}||^{t'}_{L^1(\Omega)}+ C||(-\Delta )^{\frac s2} u||_{L^t( {I\!\!R}^N)}+C. \end{equation} (48)

    Recall that p > p_+(\lambda, s) = \frac{2s+\mu(\lambda)}{s+\mu(\lambda)} , hence p' < \frac{2s+\mu(\lambda)}{s} . Using an approximating argument we can take \theta = \frac{1}{|x|^{\beta}}\chi_{B_{\frac{r}{4}}(0)} with N-(\mu(\lambda)+s) \leqslant\beta < N-(p'-1)s . In this case \phi_{\theta}\backsimeq \frac{1}{|x|^{\beta-s}} near the origin and \phi_{\theta}\in L^\infty(B\backslash B_r(0) . Therefore \phi\in L^1(\Omega) . From (48), it holds that

    C \int_{B_{\frac{r}{4}}} \frac{1}{|x|^{\beta+\mu+s}}dx\le C( \varepsilon ) \int\limits_ \Omega \, \frac{\theta^{p'}}{\phi_{\theta}^{p'-1}}dx +C||\phi_{\theta}||^{t'}_{L^1(\Omega)}+ C||(-\Delta )^{\frac s2} u||_{L^t( {I\!\!R}^N)}+C.

    Since (\beta+\mu+s)\ge N , then in order to conclude we have just to show that \int\limits_ \Omega \, \frac{\theta^{p'}}{\phi_{\theta}^{p'-1}}dx < \infty . Notice that

    \int\limits_ \Omega \, \frac{\theta^{p'}}{\phi_{\theta}^{p'-1}}dx \leqslant\int_{B_{\frac{r}{4}}(0)} \, \frac{1}{|x|^{p'\beta-(p'-1)(\beta-s)}}dx = \int_{B_{\frac{r}{4}}(0)} \, \frac{1}{|x|^{(p'-1)s+\beta}}dx.

    Taking into consideration that p' < \frac{2s+\mu(\lambda)}{s} , it follows that (p'-1)s+\beta < N and so we are done.

    Remarks 4.3. Following the same arguments as above, we can prove that problem(39) has no positive supersolutions u in the following sense: u = 0 \; a.e \mathit{\text{ in }} {I\!\!R}^N\setminus \Omega , \big|(-\Delta)^{\frac{s}{2}} u\big| \in L^ r({I\!\!R}^N) for some r > 1 , g: = |(-\Delta)^{\frac s2} u|^p+ \lambda\dfrac{u}{|x|^{2s}}+\rho f\in L^1(\Omega) and for all nonnegative \phi\in \mathcal{T} (defined in (9)), we have

    \int_{ {I\!\!R}^N} (-\Delta)^{\frac{s}{2}} u (-\Delta)^{\frac{s}{2}} \phi\,dx \geqslant \int_{\Omega}\,g\phi\,dx.

    For \rho large, we are also able to prove another non existence result.

    Theorem 4.4. Assume that f\gneqq 0 and p > \frac{2s+2}{s+2} , then there exists \rho^* > 0 such that problem (39) has non positive solution for \rho > \rho^* .

    Proof. Without loss of generality we assume that f\in L^\infty(\Omega) .

    Assume that u is a positive solution to problem (39). For \theta\in \mathcal{C}^\infty_0(\Omega) with \theta\gneqq 0 , we define \phi_\theta to be the unique solution to the problem

    \left\{ \begin{aligned} (-\Delta)^{s} \phi_\theta & = \theta\,, & \quad \text{ in } \Omega,\\ \phi_\theta & = 0\,, & \quad \text{ in } {I\!\!R}^N \setminus \Omega. \end{aligned} \right.

    Notice that \phi_\theta\simeq \delta^s (x) where \delta(x)\equiv dist(x, \partial\Omega) , see for instance [41].

    Using \phi_\theta as a test function in (39), it holds that

    \int\limits_ \Omega(-\Delta)^{s} \phi_\theta\, u dx \geqslant\int\limits_\Omega |(-\Delta)^{\frac{s}{2}}u|^p\phi_\theta dx +\rho \int\limits_\Omega f\phi_\theta dx.

    Hence

    \begin{equation} \int\limits_\Omega \theta\, u \,dx \geqslant\int\limits_ \Omega|(-\Delta)^{\frac{s}{2}}u|^p\phi_\theta dx +\rho \int\limits_\Omega f\phi_\theta \,dx. \end{equation} (49)

    Let \psi_\theta to be the unique solution to the problem

    \left\{ \begin{aligned} (-\Delta)^{\frac s2} \psi_\theta & = \theta\,, & \quad \text{ in } \Omega,\\ \psi_\theta & = 0\,, & \quad \text{ in } {I\!\!R}^N \setminus \Omega. \end{aligned} \right.

    Thus

    \int\limits_ \Omega(-\Delta)^{\frac s2} \psi_\theta\, u \,dx \geqslant\int\limits_\Omega |(-\Delta)^{\frac{s}{2}}u|^p\phi_\theta \,dx +\rho \int\limits_\Omega f\phi_\theta \,dx.

    Then

    \int\limits_ \Omega(-\Delta)^{\frac s2} u \, \psi_\theta \,dx \geqslant\int\limits_\Omega |(-\Delta)^{\frac{s}{2}}u|^p\phi_\theta \,dx +\rho \int\limits_\Omega f\phi_\theta \,dx.

    Notice that

    \int\limits_ \Omega(-\Delta)^{\frac s2} u \, \psi_\theta \,dx \leqslant \int\limits_ \Omega|(-\Delta)^{\frac s2} u|\phi_\theta\frac{\psi_\theta}{\phi_\theta} \,dx.

    Hence, using Young's inequality, for any \varepsilon > 0 , we get the existence of a positive constant C(\varepsilon) such that

    \int\limits_ \Omega(-\Delta)^{\frac s2} u \, \psi_\theta \,dx \leqslant \varepsilon \int\limits_ \Omega|(-\Delta)^{\frac{s}{2}}u|^p\phi_\theta \,dx +C( \varepsilon ) \int\limits_ \Omega\phi_\theta \bigg(\frac{\psi_\theta}{\phi_\theta}\bigg)^{p'} \,dx.

    Since \theta is bounded, according with [40], then \psi_\theta\backsimeq \delta^{\frac s2} and \phi_\theta\backsimeq\delta^s , it follows that \int\limits_ \Omega\phi_\theta \bigg(\frac{\psi_\theta}{\phi_\theta}\bigg)^{p'} \, dx < \infty if, p > \frac{2s+2}{s+2} . Therefore, in this case, we deduce that

    \rho \int\limits_\Omega f\,\phi\,dx\le C( \varepsilon ) \int\limits_ \Omega\phi_\theta \bigg(\frac{\psi_\theta}{\phi_\theta}\bigg)^{p'} \,dx,

    which implies that

    \rho\le\dfrac{C( \varepsilon ) \int\limits_ \Omega\phi_\theta \bigg(\frac{\psi_\theta}{\phi_\theta}\bigg)^{p'} \,dx}{ \int\limits_\Omega f\,\phi_\theta\,dx} = :\rho^*.

    Hence the result follows in this case.

    Remarks 4.5. The condition p > \frac{2s+2}{s+2} in Theorem 4.4 seems to be technical. We conjecture that the non existence result in Theorem 4.4 holds for all p > 1 . However the above arguments does not hold if p \leqslant\frac{2s+2}{s+2} .

    To show the optimality of the exponent p_+(\lambda, s) , we show the existence of a supersolution to problem (39). Notice that, in some cases, under suitable conditions on the datum f and the exponent p , we are able to prove the existence of a weak solution to problem (39).

    Fix p_-(\lambda, s) < p < p_{+}(\lambda, s) < 2 and let w_1(x) = \dfrac{A}{|x|^{\theta_0}} , with \theta_0 = \frac{N-2s}{2}-\beta , be the solution to the Eq (40) obtained in the previous section. Recall that

    \begin{eqnarray*} (-\Delta )^s w_1(x)- \lambda\frac{w_1}{|x|^{2s}} & = & \frac{A(\gamma_{\beta,s}-\lambda )}{|x|^{\theta_0+2s}}\\ & = & \dfrac {A^{p} |\gamma_{\beta,\frac s2}|^p}{|x|^{(\theta_0+s)p}} = |(-\Delta )^{\frac s2} w_1|^p. \end{eqnarray*}

    Taking into consideration the definition of \gamma_{\beta, t} given in (41) (with t\in \{\frac s2, s\}) , it holds that (\gamma_{\beta, t}-\lambda) > 0 if and only if \theta_0\in (\mu(\lambda), \bar\mu(\lambda)) . Now, if f \leqslant\dfrac{1}{{|x|^{2s+\theta_0}}} , using the fact that \Omega is bounded, we can choose C_1 > 0 such that \widehat{w}_1 = C_1w is a supersolution to problem (39) for \rho < \rho^* . In this way we have obtained the following result.

    Theorem 4.6. Let \Omega be a bounded domain containing the origin. Suppose that p_-(\lambda, s) < p < p_+(\lambda, s) . If f \leqslant\dfrac{1}{|x|^{2s+\theta}} , with \theta given as above, then problem (39) has asupersolution w such that w, \dfrac{w}{|x|^{ 2s}}, |(-\Delta)^{\frac s2} w_1|^p \in L^1(\Omega) .

    Notice that in order to show the existence of a solution under the presence of a supersolution, we need a comparison principle in the spirit of the work of [14] for the fractional gradient. This is missing at the present time but will be investigated in a forthcoming paper. However, using the compactness approach developed in [7] we are able to show the existence of a solution in some particular cases. More precisely, we have:

    Theorem 4.7. Let s\in (0, 1) , 0 < \lambda < \Lambda_{N, s} and f\in L^1(\Omega) be a nonnegative function such that \int\limits_\Omega f|x|^{-\mu(\lambda)-a_0}\, dx < \infty , for some a_0 > 0 . Assume that 1 < p < p_* = \frac{N}{N-s} . Then, there exists \rho^*{: = \rho^*(N, p, s, f, \lambda, \Omega)} > 0 such that if \rho < \rho^* , problem (39) has asolution u\in L^{s, \sigma}_0(\Omega) , for all {1 < \sigma} < \frac{N}{N-s} .Moreover \int_{{I\!\!R}^N} |(-\Delta)^{\frac{s}{2}}u|^p|x|^{-\mu(\lambda)} dx < \infty .

    Proof. We follow again the arguments used in [11]. Fix 1 < p < p_* and let f\in L^1(\Omega) be a nonnegative function with \int\limits_\Omega f|x|^{-\mu(\lambda)-a_0}\, dx < \infty .

    Fix r > 1 be fixed such that 1 < p < r < p_* . Then, we get the existence of \rho^* > 0 such that for some l > 0 , we have

    C_0(l+\rho^*||f||_{L^1(|x|^{-\mu(\lambda )-a_0}dx,\Omega)}) = l^{\frac{1}{p}},

    where C_0 is a positive constant depending only on \Omega, \lambda and the regularity constant in Theorems (2.10).

    Let \rho < \rho^* be fixed and define the set

    \begin{equation} E = \{v\in L^{s,1}_0(\Omega): v\in L^{s,r}_0(|x|^{-\mu(\lambda )}dx,\Omega)\text{ and }||(-\Delta )^{\frac{s}{2}}v||_{{L^r(|x|^{-\mu(\lambda )}dx,\Omega)}}\le l^{\frac{1}{p}}\}. \end{equation} (50)

    It is clear that E is a closed convex set of L^{s, 1}_0(\Omega) . Consider the operator

    \begin{array}{rcl} T:E &\rightarrow & L^{s,1}_0(\Omega)\\ v&\rightarrow &T(v) = u, \end{array}

    where u is the unique solution to problem

    \begin{equation} \left\{ \begin{array}{rcll} (-\Delta)^s u & = & \lambda \dfrac{u}{|x|^{2s}} +|(-\Delta )^{\frac{s}{2}}v|^{p}+\rho f & \text{ in }\Omega , \\ u & = & 0 &\hbox{ in } \mathbb{R}^N\setminus\Omega,\\ u& > &0 &\hbox{ in }\Omega. \end{array} \right. \end{equation} (51)

    Setting

    g(x) = |(-\Delta )^{\frac{s}{2}}v|^{p}+\gamma f,

    then taking into consideration the definition of E , it holds that g\in L^1(|x|^{-\mu(\lambda)}dx, \Omega) . Hence the existence and the uniqueness of u follows using the result of [8] with u\in L^{s, \sigma}_0(\Omega) for all {1 < \sigma} < \frac{N}{N-s} . Thus T is well defined.

    We claim that T(E)\subset E . Since r > p , using Hölder inequality we get the existence of \hat{a}_0 > 0 such that

    \int\limits_ \Omega|(-\Delta )^{\frac{s}{2}}v|^{p}|x|^{-\mu(\lambda )-\hat{a}_0}dx\le C(\Omega)\bigg( \int\limits_ \Omega|(-\Delta )^{\frac{s}{2}}v|^{r}|x|^{-\mu(\lambda )}dx\bigg)^{\frac{p}{r}} < \infty.

    Setting \bar{a}_0 = \min\{a_0, \hat{a}_0\} , it holds that g\in L^1(|x|^{-\mu(\lambda)-\bar{a}}dx, \Omega) . Thus by Proposition 3.7, we reach that, for all 1\le\sigma < \frac{N}{N-s} ,

    \bigg( \int\limits_ \Omega|(-\Delta )^{\frac{s}{2}}u|^{\sigma}|x|^{-\mu(\lambda )}dx\bigg)^{\frac{1}{\sigma}}\le C(N,p,\bar{a})\bigg\||(-\Delta )^{\frac{s}{2}}v|^p +\rho f\bigg\|_{L^1(|x|^{-\mu(\lambda )-\bar{a}}dx, \Omega)}.

    Since v\in E , we conclude that

    \begin{eqnarray*} \bigg( \int\limits_ \Omega|(-\Delta )^{\frac{s}{2}} u|^{\sigma}|x|^{-\mu(\lambda )}dx\bigg)^{\frac{1}{\sigma}} & \leqslant& C(N,p,\bar{a})\big( \bigg( \int\limits_ \Omega|(-\Delta )^{\frac{s}{2}}v|^{r}|x|^{-\mu(\lambda )}dx\bigg)^{\frac{p}{r}} +\rho ||f||_{L^1(|x|^{-\mu(\lambda )-a_0}dx, \Omega)}\bigg)\\ & \leqslant& C(l+\rho^* ||f||_{L^1(|x|^{-\mu(\lambda )-a_0}dx, \Omega)})\le l. \end{eqnarray*}

    Choosing \sigma = r , it holds that u\in E .

    The continuity and the compactness of T follow using closely the same arguments as in [11].

    As a conclusion and using the Schauder Fixed Point Theorem as in [11], there exists u\in E such that T(u) = u , u\in L^{s, p}_0(\Omega) and

    ||(-\Delta )^{\frac{s}{2}}u||_{{L^r(|x|^{-\mu(\lambda )}dx,\Omega)}}\le C.

    Therefore, u solves (39).

    Let us consider now the case where \mathfrak{F}(u)\equiv (\mathbb{D}_s (u)) . Then problem (1) takes the form

    \begin{equation} \left\{ \begin{array}{rcll} (-\Delta )^s u & = &\lambda \dfrac{u}{|x|^{2s}}+ (\mathbb{D}_s (u))^p+ \rho f & \text{ in } \Omega,\\ u& > &0 & \text{ in }\Omega,\\ u& = &0 & \text{ in }(\mathbb{R}^N\setminus\Omega). \end{array}\right. \end{equation} (52)

    Recall that \mathbb{D}_s (u)(x) = \bigg(\frac{a_{N, s}}{2} \int_{ {I\!\!R}^N} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}} dy \, \bigg)^{\frac 12}. If we consider the equation

    \begin{equation} (-\Delta)^s w-\lambda \frac{w}{|x|^{2s}} = (\mathbb{D}_s (w))^p \text{ in } {I\!\!R}^N, \end{equation} (53)

    then, using the same radial computation as in the previous section, searching for a radial solution in the form w = A|x|^{\beta-\frac{N-2s}{2}} , one sees that we need p = \dfrac{\frac{N-2s}{2}-\beta+2s}{\frac{N-2s}{2}-\beta+s} , which means that \beta = \frac{N-2s}{2}+ \frac{ps}{p-1}-\frac{2s}{p-1} . Hence, as in the previous case, we obtain that w is a solution to (53) if p_-(\lambda, s) < p < p_+(\lambda, s) where p_-(\lambda, s), p_+(\lambda, s) are defined by (42) and (43) respectively.

    Notice that if f \leqslant\dfrac{1}{|x|^{2s+\theta}} with \theta = \frac{N-2s}{2}-\beta , then we can chose C_1 > 0 such that C_1w is a supersolution to problem (52) for \rho small enough.

    Let us show that p_+(\lambda, s) is the critical exponent for the existence of a weak solution. More precisely we have the next non existence result.

    Theorem 5.1. Assume that s\in (0, 1) and p > p_+(\lambda, s) . For \lambda > 0 , problem (52) has no positive solution u in the sense of Definition 4.1.

    Proof. We follow closely the arguments in [5]. Without loss of generality we assume that f\in L^\infty(\Omega) .

    According to the value of p , we will divide the proof in two parts.

    The case p_+(\lambda, s) < p < 2^*_s . In this case p' > \frac{2N}{N+2s} . Assume by contradiction that problem (52) has a weak positive u . Let \phi \in \mathcal{C}_0^{\infty}(\Omega) be a nonnegative function such that \text{Supp}\subset B_{\frac{r}{2}}(0)\subset B_{r}(0)\subset \subset \Omega to be chosen later. Using \phi^{p'} as test function in (52), it holds that

    \begin{equation} \int_{\Omega} (-\Delta )^s u \, \phi^{p'}(x) dx \geqslant\int_{\Omega}(\mathbb{D}_s(u)(x))^p\phi^{p'}(x)dx + \lambda \int\limits_ \Omega\frac{u \phi^{p'}}{|x|^{2s}} dx +\rho \int_{\Omega} f(x) \phi^{q'}(x)dx. \end{equation} (54)

    Using the algebraic inequality, for a, b\ge0, m > 1 ,

    (a^m-b^m)\simeq (a-b)(a^{m-1}+b^{m-1}),

    it holds that

    \begin{eqnarray*} & & \int_{\Omega} (-\Delta )^s u\, \phi^{p'}(x)dx = \iint_{D_{\Omega}} \frac{(u(x)-u(y))(\phi^{p'}(x) - \phi^{p'}(y))}{|x-y|^{N+2s}} dy dx\\ & \leqslant& C\iint_{D_{\Omega}} \frac{|u(x)-u(y)||\phi(x)-\phi(y)|(\phi^{p'-1}(x)+\phi^{p'-1}(y))}{|x-y|^{N+2s}} dydx \\ & \leqslant& C\iint_{D_{\Omega}} \frac{|u(x)-u(y)| |\phi(x)-\phi(y)|}{|x-y|^{N+2s}}\phi^{p'-1}(x)dydx\\ & + & C\iint_{D_{\Omega}} \frac{|u(x)-u(y)| |\phi(x)-\phi(y)|}{|x-y|^{ N+2s}}\phi^{p'-1}(y)dydx\\ & \leqslant& 2C \iint_{D_{\Omega}} \frac{|u(x)-u(y)| |\phi(x)-\phi(y)|}{|x-y|^{N+2s}}\phi^{p'-1}(x)dydx\\ & \leqslant& C \int_{{I\!\!R}^N} \bigg( \int_{{I\!\!R}^N}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dy\bigg)^{\frac{1}{2}} \bigg( \int_{{I\!\!R}^N}\frac{|\phi(x)-\phi(y)|^2}{|x-y|^{N+2s}}dy\bigg)^{\frac 12 }\phi^{p'-1}(x)dx\\ & \leqslant& C \int\limits_ \Omega\mathbb{D}_s(u) \mathbb{D}_s(\phi)\phi^{p'-1}(x)dx. \end{eqnarray*}

    Therefore, using Young's inequality, we deduce that for any \varepsilon > 0 , we get the existence of a positive constant C(\varepsilon) such that

    \begin{eqnarray*} \int_{\Omega} (-\Delta )^s u\, \phi^{p'}(x)dx & \leqslant& \varepsilon \int\limits_\Omega (\mathbb{D}_s(u))^p\phi^{p'}(x)dx +C( \varepsilon ) \int\limits_\Omega (\mathbb{D}_s(\phi))^{p'}(x)dx. \end{eqnarray*}

    Choosing \varepsilon small enough and going back to (54), we get

    \begin{equation} (1- \varepsilon )\int_{\Omega}(\mathbb{D}_s(u)(x))^p\phi^{p'}(x)dx + \lambda \int\limits_ \Omega\frac{u \phi^{p'}}{|x|^{2s}} dx +\rho \int_{\Omega} f(x) \phi^{q'}(x)dx\le C( \varepsilon ) \int\limits_ \Omega(\mathbb{D}_s(\phi)(x))^{p'}dx. \end{equation} (55)

    Recall that u(x)\ge C|x|^{-\mu(\lambda)} in B_r(0)\subset \subset \Omega . Hence fixed \phi\in \mathcal{C}_0^{\infty}(B_{\frac{r}{2}}(0)) , we have that

    \lambda C\int_{B_{\frac{r}{2}}(0)}\frac{\phi^{p'}}{|x|^{2s+\mu(\lambda )}} dx\le C( \varepsilon ) \int\limits_ \Omega(\mathbb{D}_s(\phi)(x))^{p'}dx.

    Replacing \phi by |\phi| in the above estimate, we deduce that

    \begin{equation} \lambda C\int_{B_{\frac{r}{2}}(0)}\frac{|\phi|^{p'}}{|x|^{2s+\mu(\lambda )}} dx\le C( \varepsilon ) \int\limits_ \Omega(\mathbb{D}_s(\phi)(x))^{p'}dx \leqslant C( \varepsilon ) \int_{{I\!\!R}^N} (\mathbb{D}_s(\phi)(x))^{p'}dx. \end{equation} (56)

    Since p > p_+(\lambda, s) , then sp' < 2s+\mu(\lambda) . Recall that p' > \frac{2N}{N+2s} . Hence (56) is in contradiction with the Hardy inequality in Proposition 2.3. Thus we conclude.

    The case p > 2^*_s > p_+(\lambda, s) . Notice that for all \lambda < \Lambda_{N, s} , we have p_+(\lambda, s) < 2 < 2^*_s . By a continuity argument we get the existence of \lambda_1 < \lambda and p_1 < 2^*_s such that p_1 > p_+(\lambda_1, s) . Assume that u is a weak solution to problem (52), then

    (-\Delta )^s u \geqslant\lambda \dfrac{u}{|x|^{2s}}+ (\mathbb{D}_s (u))^{p_1}-C(p_1) \text{ in } B_r(0).

    Notice that u(x)\ge C_1|x|^{-\mu(\lambda)} \text{ in } B_r(0) . Hence

    (-\Delta )^s u \geqslant\lambda_1 \dfrac{u}{|x|^{2s}}+ (\mathbb{D}_s (u))^{p_1}+\frac{C}{|x|^{2s+\mu(\lambda )}}-C(p_1) \text{ in } B_r(0).

    Choosing r small, it holds that

    (-\Delta )^s u \geqslant\lambda_1 \dfrac{u}{|x|^{2s}}+ (\mathbb{D}_s (u))^{p_1} \text{ in } B_r(0).

    Since p_+(\lambda_1, s) < p_1 < \frac{2N}{N+2s} , repeating the same argument as in the first case, we reach the same contradiction. Hence we conclude.

    Taking advantage of the previous estimate, we can show that the problem (52) has no solution for large value of \rho .

    Theorem 5.2. Assume that f\gneqq 0 and p > 1 , then there exists \rho^* > 0 such that problem (52) does not have a positive solutionfor \rho > \rho^* .

    Proof. Suppose that u is a nonnegative weak solution to problem (52). Let \phi \in \mathcal{C}_0^{\infty}(\Omega) be a nonnegative function such that

    \int_{\Omega} f(x) \phi^{q'}(x) dx > 0.

    From estimate (55), fixing \varepsilon \in(0, 1) , we obtain that

    \rho \int_{\Omega} f(x) \phi^{q'}(x)dx\le C( \varepsilon ) \int\limits_ \Omega(\mathbb{D}_s(\phi)(x))^{p'}dx.

    In particular

    \rho \leqslant \inf\limits_{\{\phi\in \mathcal{C}^\infty(\Omega), \phi\gneqq 0\}}\frac{ C( \varepsilon ) \int\limits_\Omega(\mathbb{D}_s(\phi))^{p'}(x)dx}{ \int_{\Omega} f(x) \phi^{q'}(x) dx}: = \rho^*,

    and this is in contradiction with our initial assumption.

    Remarks 5.3. 1) As in Theorem 4.6, if 1 < p < p_+(\lambda, s) and f\le\dfrac{1}{|x|^{2s+\theta}} , with \theta given as above, thenproblem (39) has a supersolution w such that w, \dfrac{w}{|x|^{ 2s}}, \mathbb{D}_s (u)\in L^p(\Omega) .

    2) Using the same compactness approach, we can also treat thecase (\mathfrak{F}(u)(x)) = |\nabla^s u(x)| , where \nabla^s u(x) isdefined in (4).

    1) In the local case s = 1 or in the nonlocal case under the existence of a local gradient term, an interesting maximum principle is obtained in the sense that if w\in W^{s, 1}_0(\Omega) is a subsolution to the problem

    \begin{equation} \left\{ \begin{array}{rcll} (-\Delta )^s w & = &a(x)|\nabla w| & \text{ in } \Omega,\\ w& = &0 & \text{ in }(\mathbb{R}^N\setminus\Omega), \end{array}\right. \end{equation} (57)

    with a\in L^{\sigma}(\Omega), \sigma > \frac{N}{s} , then w \leqslant0 in \Omega (see for instance [14] and [11]). It would be very interesting to get a similar result replacing the gradient term |\nabla w| by the nonlocal fractional gradient |(-\Delta)^{\frac{s}{2}}w| , namely for the problem

    \begin{equation} \left\{ \begin{array}{rcll} (-\Delta )^s w & = &a(x)|(-\Delta )^{\frac{s}{2}}w| & \text{ in } \Omega,\\ w& = &0 & \text{ in }(\mathbb{R}^N\setminus\Omega). \end{array}\right. \end{equation} (58)

    2) Since non comparison principle is known for problem (58), then to get general existence result to problem (1), under natural integrability conditions for f , it is necessary to prove a new class of weighted CKN inequalities as in [3], using the norm \| |(-\Delta)^{\frac{s}{2}}u| |x|^{\beta} \|_{L^p(\mathbb{R}^N)} . This will be considered in a forthcoming work.

    The work was partially supported by AEI Research Grant PID2019-110712GB-I00 and grant 1001150189 by PRICIT, Spain. Authors 1 and 2 were also supported by a research project from DGRSDT, Algeria.

    The authors would like to thank the anonymous reviewer for his/her careful reading of the paper and his/her many insightful comments and suggestions.

    The authors declare no conflict of interest.

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