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Research article

A semilnear singular problem for the fractional laplacian

  • Received: 16 October 2018 Accepted: 22 October 2018 Published: 24 October 2018
  • MSC : Primary 35A15; Secondary 35S15, 47G20, 46E35

  • We study the problem (Δ)su=auγ+λh in Ω, u=0 in RnΩ, u>0 in Ω, where 0<s<1, Ω is a bounded domain in Rn with C1,1 boundary, a and h are nonnegative bounded functions, h0, and λ>0. We prove that if γ(0,s) then, for λ positive and large enough, there exists a weak solution such that c1dsΩuc2dsΩ in Ω for some positive constants c1 and c2. A somewhat more general result is also given.

    Citation: Tomas Godoy. A semilnear singular problem for the fractional laplacian[J]. AIMS Mathematics, 2018, 3(4): 464-484. doi: 10.3934/Math.2018.4.464

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  • We study the problem (Δ)su=auγ+λh in Ω, u=0 in RnΩ, u>0 in Ω, where 0<s<1, Ω is a bounded domain in Rn with C1,1 boundary, a and h are nonnegative bounded functions, h0, and λ>0. We prove that if γ(0,s) then, for λ positive and large enough, there exists a weak solution such that c1dsΩuc2dsΩ in Ω for some positive constants c1 and c2. A somewhat more general result is also given.


    1. Introduction and statement of the main results

    Elliptic problems with singular nonlinearities appear in many nonlinear phenomena, for instance, in the study of chemical catalysts process, non-Newtonian fluids, and in the study of the temperature of electrical conductors whose resistance depends on the temperature (see e.g., [3,6,10,15] and the references therein). The seminal work [7] is the start point of a large literature concerning singular elliptic problems, see for instance, [1,3,5,6,8,9,10,13,15,17,18,21,22,23,24], and [30]. For additional references and a systematic study of singular elliptic problems see also [26].

    In [10], Diaz, Morel and Oswald considered problems of the form

    {Δu=uγ+λh(x) in Ω,u=0 on Ω,u>0 in Ω (1.1)

    where Ω is a bounded and regular enough domain, 0<γ<1, λ>0 and hL(Ω) is a nonnegative and nonidentically zero function. They proved (see [10], Theorem 1, Corollary 1, Lemma 2 and Theorem 3) that there exists λ0>0 such that. for λ>λ0, problem (1.1) has a unique maximal solution uH10(Ω) and has no solution when λ<λ0.

    Concerning nonlocal singular problems, Barrios, De Bonis, Medina, and Peral proved in [2] that if Ω is a bounded and regular enough domain in Rn, 0<s<1, n>2s, f is a nonnegative function in a suitable Lebesgue space, λ>0, M>0 and 1<p<n+2sn2s, then the problem

    {(Δ)su=λf(x)uγ+Mup in Ω,u=0 on RnΩ,u>0 in Ω, (1.2)

    has a solution, in a suitable weak sense whenever λ>0 and M>0, and that, if M=1 and f=1, then there exists Λ>0 such that (1.2) has at least two solutions when λ<Λ and has no solution when λ>Λ.

    A natural question is to ask if an analogous of the quoted result of [10] hold in the nonlocal case, i.e., when Δ is replaced by the fractional laplacian (Δ)s, s(0,1), and with the boundary condition u=0 on Ω replaced by u=0 on RnΩ. Our aim in this paper is to obtain such a result. Note that the approach of [10] need to be modified in order to be used in the fractional case. Indeed, a step in [10] was to observe that, if φ1 denotes a positive principal eigenfunction for Δ on Ω, with Dirichlet boundary condition, then

    Δφ21+γ1=21+γλ1φ21+γ12(1γ)(γ+1)2|φ1|2φ2γ1+γ1 in Ω, (1.3)

    where λ1 is the corresponding principal eigenvalue. From this fact, and using the properties of a principal eigenfunction, Diaz, Morel and Oswald proved that, for ε positive and small enough, εφ21+γ1 is a subsolution of problem (1.1). Since formula (1.3), is not avalaible for the principal eigenfunction of (Δ)s, the arguments of [10] need to be modified in order to deal with the fractional case.

    Let us state the functional setting for our problem. For s(0,1) and nN, let

    Hs(Rn):={uL2(Rn):Rn×Rn|u(x)u(y)|2|xy|n+2sdxdy<},

    and for uHs(Rn), let uHs(Rn):=(Rnu2+Rn×Rn|u(x)u(y)|2|xy|n+2sdxdy)12. Let Ω be a bounded domain in Rn with C1,1 boundary and let

    Xs0(Ω):={uHs(Rn):u=0 a.e. in RnΩ},

    and for uXs0(Ω), let uXs0(Ω):=(Rn×Rn|u(x)u(y)|2|xy|n+2sdxdy)12.

    With these norms, Hs(Rn) and Xs0(Ω) are Hilbert spaces (see e.g., [29], Lemma 7), Cc(Ω) is dense in Xs0(Ω) (see [16], Theorem 6). Also, Xs0(Ω) is a closed subspace of Hs(Rn), and from the fractional Poincaré inequality (as stated e.g., in [11], Theorem 6.5; see Remark 2.1 below), if n>2s then .Xs0(Ω) and .Hs(Rn) are equivalent norms on Xs0(Ω). For fL1loc(Ω) we say that f(Xs0(Ω)) if there exists a positive constant c such that |Ωfφ|cuXs0(Ω) for any φXs0(Ω). For f(Xs0(Ω)) we will write ((Δ)s)1f for the unique weak solution u (given by the Riesz theorem) of the problem

    {(Δ)su=f in Ω,u=0 in RnΩ. (1.4)

    Here and below, the notion of weak solution that we use is the given in the following definition:

    Definition 1.1. Let s(0,1), let f:ΩR be a Lebesgue measurable function such that fφL1(Ω) for any φXs0(Ω). We say that u:ΩR is a weak solution to the problem

    {(Δ)su=f in Ω,u=0 in RnΩ

    if uXs0(Ω), u=0 in RnΩ and, for any φ Xs0(Ω),

    Rn×Rn(u(x)u(y))(φ(x)φ(y))|xy|n+2sdxdy=Ωfφ.

    For uXs0(Ω) and fL1loc(Ω), we will write (Δ)suf in Ω (respectively (Δ)suf in Ω) to mean that, for any nonnegative φHs0(Ω), it hold that fφL1(Ω) and

    Rn×Rn(u(x)u(y))(φ(x)φ(y))|xy|n+2sdxdyΩfφ (resp. Ωfφ).

    For u,vXs0(Ω), we will write (Δ)su(Δ)sv in Ω (respectively (Δ)su(Δ)sv in Ω), to mean that (Δ)s(uv)0 in Ω (resp. (Δ)s(uv)0 in Ω).

    Let

    E:={uXs0(Ω):cdsΩucdsΩ a.e. in Ω, for some positive constants c and c}

    where, for xΩ, dΩ(x):=dist(x,Ω). With these notations, our main results read as follows:

    Theorem 1.2. Let Ω be a bounded domain in Rn with C1,1 boundary, let s(0,1), and assume n>2s. Let hL(Ω) be such that 0h0 in Ω (i.e., |{xΩ:h(x)>0}|>0) and let g:Ω×(0,)[0,) be a function satisfying the following conditions g1)-g5)

    g1) g:Ω×(0,)[0,) is a Carathéodory function, g(.,s)L(Ω) for any s>0 and limσ(g(.,σ))=0.

    g2) σg(x,σ) is non increasing on (0,) a.e. xΩ.

    g3) g(.,σdsΩ)(Xs0(Ω)) and dsΩ((Δ)s)1(dsΩg(.,σdsΩ))L(Ω) for all σ>0.

    g4) It hold that:

    limσ(σdsΩ)1((Δ)s)1(dsΩg(.,σdsΩ))=0, and

    limσdsΩ((Δ)s)1(g(.,σ))L(Ω)=0.

    g5) dsΩg(.,σdsΩ)L2(Ω) for any σ>0.

    Consider the problem

    {(Δ)su=g(.,u)+λhinΩ,u=0inRnΩ,u>0inΩ (1.5)

    Then there exists λ0 such that:

    i) If λ>λ then (1.5) has a weak solution u(λ)E, which is maximal in the following sense: If vE satisfies (Δ)svg(.,v)+λh in Ω, then u(λ)v a.e. in Ω .

    ii) If λ<λ, no weak solution exists in E.

    iii) If, in addition, there exists bL(Ω) such that 0b0 in Ω and g(.,s)bsβ a.e. in Ω for any s(0,), then λ>0.

    Theorem 1.2 allows g(x,s) to be singular at s=0. In fact, in Lemma 3.2, using some estimates from [4] for the Green function of (Δ)s in Ω (with homogeneous Dirichlet boundary condition on RnΩ), we show that if g(x,s)=asβ with a a nonnegative function in L(Ω) and β[0,s), then g satisfies the assumptions of Theorem 1.2. Thus, as a consequence of Theorem 1.2, we obtain the following:

    Theorem 1.3. Let Ω, s, and h be as in the statement of Theorem 1.2, and let g:Ω×(0,)[0,). Then the assertions i)iii) of Theorem 1.2 remain true if we assume, instead of the conditions g1)-g5), that the following conditions g6) and g7) hold:

    g6) g:Ω×(0,)[0,) is a Carathéodory function and sg(x,s) is nonincreasing for a.e. xΩ.

    g7) There exist positive constants a and β[0,s) such that g(.,s)asβ a.e. in Ω for any s(0,).

    Let us sketch our approach: In Section 2 we consider, for ε>0, the following approximated problem

    {(Δ)su=g(.,u+ε)+λh in Ω,u=0 in RnΩ,u>0 in Ω. (1.6)

    Let us mention that, in order to deal with problems involving the (p;q)-Laplacian and a convection term, this type of approximation was considered in [14] (see problem Pε therein).

    Lemma 2.5 gives a positive number λ0, independent of ε and such that, for λ=λ0, problem (1.6) has a weak solution wε. From this result, and from some properties of the function wε, in Lemma 2.11 we show that, for λλ0 and for any ε>0, there exists a weak solution uε of problem (1.6), with the following properties:

    a) cdsΩuεcdsΩ for some positive constants c and c independent of ε,

    b) uε¯u, where ¯u is the solution of the problem (Δ)s¯u=λh in Ω, ¯u=0 in RnΩ,

    c) uεψ for any ψXs0(Ω) such that (Δ)sψ=g(.,ψ+ε)+λh in Ω.

    In section 3 we prove Theorems 1.2 and 1.3. To prove Theorem 1.2, we consider a decreasing sequence {εj}jN such that limjεj=0, and we show that, for λλ0, the sequence of functions {uεj}jN given by Lemma 2.11 converges, in Xs0(Ω), to a weak solution u of problem (1.5) which has the properties required by the theorem. An adaptation of some of the arguments of [10] gives that, if problem (1.5) has a weak solution in E, then it has a maximal (in the sense stated in the theorem) weak solution in E and that if for some λ=λ (1.5) has a weak solution in E, then it has a weak solution in E for any λλ. Finally, the assertion iii) of Theorem 1.2 is proved with the same argument given in [10].


    2. Preliminaries and auxiliary results

    We fix, from now on, hL(Ω) such that 0h0 in Ω. We assume also from now on (except in Lemma 3.2) that g:Ω×(0,)[0,) satisfies the assumptions g1)-g5 of Theorem 1.2.

    In the next remark we collect some general facts concerning the operator (Δ)s.

    Remark 2.1. ⅰ) (see e.g., [27], Proposition 4.1 and Corollary 4.2) The following comparison principle holds: If u,vXs0(Ω) and (Δ)su(Δ)sv in Ω then uv in Ω. In particular, the following maximum principle holds: If vXs0(Ω), (Δ)sv0 in Ω and v0 in RnΩ, then v0 in Ω.

    ⅱ) (see e.g., [27], Lemma 7.3) If f:ΩR is a nonnegative and not identically zero measurable function in f(Xs0(Ω)), then the weak solution u of problem (1.4) satisfies, for some positive constant c,

    ucdsΩ in Ω. (2.1)

    ⅲ) (see e.g., [28], Proposition 1.1) If fL(Ω) then the weak solution u of problem (1.4) belongs to Cs(Rn). In particular, there exists a positive constant c such that

    |u|cdsΩ in Ω. (2.2)

    ⅳ) (Poincaré inequality, see [11], Theorem 6.5) Let s(0,1) and let 2s:=2nn2s. Then there exists a positive constant C=C(n,s) such that, for any measurable and compactly supported function f:RnR,

    fL2s(Rn)CRn×Rn(f(x)f(y))2|xy|n+spdxdy.

    ⅴ) If vL(2s)(Ω) then v(Xs0(Ω)), and v(Xs0(Ω))Cv(2s), with C as in i). Indeed, for φXs0(Ω), from the Hölder inequality and iii), Ω|vφ|v(2s)φ2sCv(2s)φXs0(Ω).

    ⅵ) (Hardy inequality, see [25], Theorem 2.1) There exists a positive constant c such that, for any φXs0(Ω),

    dsΩφ2cφXs0(Ω). (2.3)

    Remark 2.2. Let zHs(Rn) be the solution of the problem

    {(Δ)sz=τ1h in Ωz=0 in RnΩ, (2.4)

    with τ1 chosen such that zL(Rn)=1. Since hL(Rn), Remark 2.1 ⅲ) gives zC(Rn) (see also [12], Theorem 1.2). Thus, since supp(z)¯Ω and zC(¯Ω), we have zL(Rn). Moreover, by Remark 2.1 ⅱ), there exists a positive constant c such that

    zcdsΩ in Ω. (2.5)

    Remark 2.3. There exist positive numbers M0 and M1 such that

    12cM1dsΩ((Δ)s)1(g(.,12cM1dsΩ)),M1<M0,12cM1dsΩ((Δ)s)1(g(.,M0))L(Ω). (2.6)

    Indeed, by g4), limσ(σdsΩ)1((Δ)s)1(dsΩg(.,σdsΩ))=0 and so the first one of the above inequalities hold for M1 large enough. Fix such a M1. Since, from g4), limσdsΩ((Δ)s)1(g(.,σ))L(Ω)=0, the remaining inequalities of (2.6) hold for M0 large enough.

    Lemma 2.4. Let ε>0 and let z, τ1 and c be as in Remark 2.2. Let M0 and M1 be as in Remark 2.3. Let z:=M1z and let w0,ε:RnR be the constant function w0,ε=M0. Then there exist sequences {wj,ε}jN and {ζj,ε}jN in Xs0(Ω)L(Ω) such that, for all jN:

    i) wj1,εwj,ε0 in Rn,

    ii) wj,ε12cM1dsΩ in Ω,

    iii) wj,ε is a weak solution of the problem

    {(Δ)swj,ε=g(.,wj1,ε+ε)+τ1M1hinΩ,wj,ε=0inRnΩ. (2.7)

    iv) wj,ε=zζj,ε in Rn and ζj,ε is a weak solution of the problem

    {(Δ)sζj,ε=g(.,wj1,ε+ε)inΩ,ζj,ε=0inRnΩ. (2.8)

    v) wj,εXs0(Ω)c for some positive constant c independent of j and ε.

    Proof. The sequences {wj,ε}jN and {ζj,ε}jN with the properties i)v) will be constructed inductively. Let ζ1,εX10(Ω) be the solution of the problem

    {(Δ)sζ1,ε=g(.,w0,ε+ε) in Ωζ1,ε=0 in RnΩ

    (thus iv) holds for j=1). From g1) and g2) we have 0g(.,w0,ε+ε)g(.,ε)L(Ω). Thus g(.,w0,ε+ε)L(Ω). Then, by Remark 2.1 iii), ζ1,εC(Rn). Therefore, since supp(ζ1,ε)¯Ω, we have ζ1,εL(Ω). By g1), g(.,M0)L(Ω) and so g(.,M0)(X10(Ω)). Let u0:=((Δ)s)1(g(.,M0)). Then, by g1) and g3), dsΩu0L(Ω). We have, in weak sense,

    {(Δ)s(ζ1,εu0)=g(.,w0,ε+ε)g(.,M0)0 in Ωζ1,εu0=0 in RnΩ.

    Then, by the maximum principle of Remark 2.1 i),

    0ζ1,εu0dsΩu0L(Ω)dsΩ in Ω. (2.9)

    Let z:=M1z. By Remark 2.2, zHs(Rn)C(¯Ω) and

    zcM1dsΩ in Ω. (2.10)

    Also, zM1 in Ω, and z is a weak solution of the problem

    {(Δ)sz=τ1M1h in Ω,z=0 in RnΩ.

    Let w1,ε:=zζ1,ε. Then w1,εHs(Rn) and w1,ε=0 in RnΩ. Thus w1,εXs0(Ω). Also w1,εL(Ω). Since ζ1,ε0 in Ω, we have

    w0,εw1,ε=M0z+ζ1,εM0zM0M1>0 in Ω.

    Then w1,εw0,ε in Ω. Thus i) holds for j=1. Now, in weak sense,

    {(Δ)sw1,ε=(Δ)s(zζ1,ε)=τ1M1h(Δ)s(ζ1,ε)=g(.,w0,ε+ε)+τ1M1h in Ω,w1,ε=0 in RnΩ,

    and so iii) holds for j=1. Also, from (2.9), (2.10), and taking into account that (2.6),

    w1,ε=zζ1,εcM1dsΩdsΩ((Δ)s)1(g(.,M0))L(Ω)dsΩ12cM1dsΩ in Ω.

    and then w1,ε12cM1dsΩ in Ω. Thus ii) holds for j=1.

    Suppose constructed, for k1, functions w1,ε,..., wk,ε and ζ1,ε,...,ζk,ε, belonging to Xs0(Ω)L(Ω), and with the properties i)-iv). Let ζk+1,εXs0(Ω) be the solution of the problem

    {(Δ)sζk+1,ε=g(.,wk,ε+ε) in Ω,ζk+1,ε=0 on RnΩ. (2.11)

    (and so iv) holds for j=k+1) and let wk+1,ε:=zζk+1,ε. Then wk+1,εHs(Rn) and wk+1,ε=0 in RnΩ. Thus wk+1,εXs0(Ω). Also,

    wk,εwk+1,ε=ζk+1,εζk,ε in Rn (2.12)

    and

    {(Δ)s(ζk+1,εζk,ε)=g(.,wk,ε+ε)g(.,wk1,ε+ε)0 in Ω,ζk+1,εζk,ε=0 in RnΩ,

    the last inequality because, by g1), sg(.,s) is nonincreasing and (by our inductive hypothesis) wk,εwk1,ε in Ω. Then, by the maximum principle, ζk+1,εζk,ε0 in Rn. Therefore, by (2.12), wk,εwk+1,ε in Rn,and then i) holds for j=k+1. Also,

    {(Δ)swk+1,ε=(Δ)sz(Δ)sζk+1,ε=g(.,wk,ε+ε)+τ1M1h in Ω,wk+1,ε=0 in RnΩ.

    Then iii) holds for j=k+1. By g4), g(.,12cM1dsΩ)(Xs0(Ω)). Let u1:=((Δ)s)1(g(.,12cM1dsΩ))Xs0(Ω). By the inductive hypothesis we have wk,ε12cM1dsΩ in Ω. Now,

    {(Δ)s(ζk+1,εu1)=g(.,wk,ε+ε)g(.,12cM1dsΩ)0 in Ω,ζk+1,εu1=0 on RnΩ,

    then the comparison principle gives ζk+1,εu1. Thus, in Ω,

    wk+1,ε=zζk+1,εcM1dsΩu1=cM1dsΩ((Δ)s)1(g(.,12cM1dsΩ))cM1dsΩdsΩ((Δ)s)1(g(.,12cM1dsΩ))dsΩ12cM1dsΩ,

    the last inequality by (2.6). Thus ii) holds for j=k+1. This complete the inductive construction of the sequences {wj,ε}jN and {ζj,ε}jN with the properties i)iv).

    To see v), we take ζj,ε as a test function in (2.8). Using ii), the Hölder inequality, the Poincaré inequality of Remark 2.1 iv), we get, for any jN,

    ζj,ε2Xs0(Ω)=Ωg(.,wj1,ε+ε)ζj,εΩg(.,12cM1dsΩ)ζj,ε=ΩdsΩg(.,12cM1dsΩ)dsΩζj,εdsΩg(.,12cM1dsΩ)2dsΩζj,ε2cζj,εXs0(Ω).

    where c is a positive constant c independent of j and ε, and where, in the last inequality, we have used g5). Then ζj,εXs0(Ω) has an upper bound independent of j and ε. Since wj,ε=zζj,ε, the same assertion holds for wj,ε.

    Lemma 2.5. Let ε>0 and let τ1 and c be as in Remark 2.2. Let M0 and M1 be as in Remark 2.3 and let {wj,ε}jN and {ζj,ε}jN be as in Lemma 2.4. Let wε:=limjwj,ε and let ζε:=limjζj,ε. Then

    i) wε and ζε belong to Hs(Rn)C(¯Ω),

    ii) 12cM1dsΩwεM0 in Ω, and there exists a positive constant c independent of ε such that wεcdsΩ in Ω.

    iii) wε satisfies, in weak sense,

    {Δwε=g(.,wε+ε)+τ1M1hinΩ,wε=0inRnΩ. (2.13)

    iv) ζε satisfies, in weak sense,

    {(Δ)sζε=g(.,wε+ε)inΩ,ζε=0inRnΩ. (2.14)

    Proof. Let z be as in Remark 2.2, and let z:=M1z. Let M0 and M1 be as in Remark 2.3. By Lemma 2.4, {wj,ε}jN is a nonincreasing sequence of nonnegative functions in Rn, and so there exists wε=limjwj,ε. Since ζj,ε=zwj1,ε, there exists also ζε=limjζj,ε. Again by Lemma 2.4 we have, for any jN, 0wj,ε=zζj,εzL(Ω). Thus, by the Lebesgue dominated convergence theorem,

    {wj,ε}jN converges to wε in Lp(Ω) for any p[1,), (2.15)

    and so {g(.,wj,ε+ε)}jN converges to g(.,wε+ε) in Lp(Ω) for any p[1,). We claim that

    ζεXs0(Ω) and {ζj,ε}jN converges in Xs0(Ω) to ζε. (2.16)

    Indeed, for j, kN, from (2.8),

    {(Δ)s(ζj,εζk,ε)=g(.,wj1,ε+ε)g(.,wk1,ε+ε) in Ω,ζj,εζk,ε=0 in RnΩ. (2.17)

    We take ζj,εζk,ε as a test function in (2.17) to obtain

    ζj,εζk,ε2Xs0(Ω)=Ω(ζj,εζk,ε)(g(.,wj1,ε+ε)g(.,wk1,ε+ε))ζj,εζk,ε2sg(.,wj1,ε+ε)g(.,wk1,ε+ε)(2s)

    where 2s:=2nn2s. Then

    ζj,εζk,εXs0(Ω)cg(.,wj1,ε+ε)g(.,wk1,ε+ε)(2s).

    where c is a constant independent of j and k. Since {g(.,wj1,ε+ε)}jN converges to g(.,wε+ε) in L(2s)(Ω), we get

    limj,kζj,εζk,εXs0(Ω)=0,

    and thus {ζj,ε}jN converges in Xs0(Ω). Since {ζj,ε}jN converges to ζε in pointwise sense, (2.16) follows. Also, wj,ε=zζj,ε, and then {wj,ε}jN converges to wε,ρ in Xs0(Ω). Thus

    wεXs0(Ω) and {wj,ε}jN converges in Xs0(Ω) to wε. (2.18)

    To prove (2.14) observe that, from (2.8), we have, for any φXs0(Ω) and jN,

    Rn×Rn(ζε,j(x)ζε,j(y))(φ(x)φ(y))|xy|n+2sdxdy=Ωg(.,wε,j1+ε)φ. (2.19)

    Taking limj in (2.19) and using (2.16) and (2.15), we obtain (2.14). From (2.14) and since, by g1) and g2), g(.,wε+ε)L(Ω), Remark 2.1 iii) gives that, in addition, ζεC(¯Ω) (and so, since wε=zζε, then also wεC(¯Ω)).

    Let us see that (2.13) holds. Let φXs0(Ω). From (2.7), we have, for any jN,

    Rn×Rn(wj,ε(x)wj,ε(y))(φ(x)φ(y))|xy|n+2sdxdy=Ω¯Bρ(y)(g(.,wj1,ε+ε)+τ1M1h)φ. (2.20)

    Since φXs0(Ω) and {wj,ε}jN converges to wε in Xs0(Ω) we have

    limjRn×Rn(wj,ε(x)wj,ε(y))(φ(x)φ(y))|xy|n+2sdxdy=Rn×Rn(wε(x)wε(y))(φ(x)φ(y))|xy|n+2sdxdy. (2.21)

    Also, wε(x)=limjwj,ε(x) for any xΩ, and

    |g(.,wj1,ε+ε)φ|g(.,ε)|φ|L1(Ω),

    and clearly |τ1M1hφ|L1(Ω). Then, by the Lebesgue dominated convergence theorem,

    limjΩ(g(.,wj1,ε+ε)+τ1M1h)φ=Ω(g(.,wε+ε)+τ1M1h)φ. (2.22)

    Now (2.13) follows from (2.20), (2.21) and (2.22). Finally, by Lemma 2.4 we have, for all jN, 12cM1dsΩwj,ε in Ω and so the same inequality hold with wj,ε replaced by wε. Also, since wj,εz0 in Ω we have wj,εcdsΩ with c independent of j and ε.

    Remark 2.6. Let G:Ω×ΩR{} be the Green function for (Δ)s in Ω, with homogeneous Dirichlet boundary condition on RnΩ. Then, for fC(¯Ω) the solution u of problem (1.4) is given by u(x)=ΩG(x,y)f(y)dy for xΩ and by u(x)=0 for xRnΩ. Let us recall the following estimates from [4]:

    ⅰ) (see [4], Theorems 1.1 and 1.2) There exist positive constants c and c, depending only on Ω and s, such that for x;yΩ,

    G(x,y)cdΩ(x)s|xy|ns, (2.23)
    G(x,y)cdΩ(x)sdΩ(y)s1|xy|n2s, (2.24)
    G(x,y)cdΩ(x)sdΩ(y)s|xy|n (2.25)
    G(x,y)c1|xy|n2s if |xy|max{dΩ(x)2,dΩ(y)2} (2.26)
    G(x,y)cdΩ(x)sdΩ(y)s|xy|n if |xy|>max{dΩ(x)2,dΩ(y)2} (2.27)

    ⅱ) From ⅰ) it follows that there exists a positive constant c, depending only on Ω and s, such that for x;yΩ,

    G(x;y)cdΩ(x)sdΩ(y)s. (2.28)

    Indeed:

    If |xy|>max{dΩ(x)2,dΩ(y)2} then, from (2.27), G(x;y)cdΩ(x)sdΩ(y)s|xy|n and so G(x;y)cdΩ(x)sdΩ(y)s(diam(Ω))n.

    If |xy|max{dΩ(x)2,dΩ(y)2} then either |xy|dΩ(x)2 or |xy|dΩ(y)2. If |xy|dΩ(x)2 consider zΩ such that dΩ(y)=|zy|. then dΩ(y)=|zy||xz||xy|dΩ(x)|xy|12dΩ(x). Then dΩ(y)12dΩ(x)|xy|. Thus, since also |xy|dΩ(x)2, we have |xy|12(dΩ(x)dΩ(y))12, and so, from (2.26), G(x,y)c1|xy|n2sc1(12(dΩ(x)dΩ(y))12)n2s=2n2scdsΩ(x)dsΩ(y)(dΩ(x)dΩ(y))n22n2sc(diam(Ω))ndsΩ(x)dsΩ(y). If |xy|dΩ(y)2, by interchanging the roles of x and y in the above argument, the same conclusion is reached.

    ⅲ) If 0<β<s, then

    G(x,y)cdΩ(x)sdΩ(y)β|xy|ns+β. (2.29)

    Indeed, If dΩ(y)|xy| then, from (2.23),

    G(x,y)cdΩ(x)s|xy|ns=cdΩ(x)sdΩ(y)β|xy|nsdΩ(y)βcdΩ(x)sdΩ(y)β|xy|ns+β,

    and if dΩ(y)|xy| then, from (2.27),

    G(x,y)cdΩ(x)sdΩ(y)s|xy|n=cdΩ(x)sdΩ(y)βdΩ(y)sβ|xy|ns+β|xy|sβcdΩ(x)sdΩ(y)β|xy|ns+β,

    ⅳ) If f\in C\left(\overline{\Omega}\right) then the unique solution u\in X_{0}^{s}\left(\Omega\right) of problem (1.4) is given by u\left(x\right) : = \int_{\Omega}G\left(x, y\right) f\left(y\right) dy for x\in\Omega, and u\left(x\right) : = 0 for x\in\mathbb{R}^{n}\setminus\Omega.

    Lemma 2.7. Let a\in L^{\infty}\left(\Omega\right) and let \beta\in\left[0, s\right). Then ad_{\Omega}^{-\beta}\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime} and the weak solution u\in X_{0}^{s}\left(\Omega\right) of the problem

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u = ad_{\Omega}^{-\beta}\; in \;\Omega, \\ u = 0\; in \;\mathbb{R}^{n}\setminus\Omega \end{array} \right. (2.30)

    satisfies d_{\Omega}^{-s}u\in L^{\infty}\left(\Omega\right).

    Proof. Let \varphi\in X_{0}^{s}\left(\Omega\right). By the Hölder and Hardy inequalities we have \int_{\Omega}\left\vert ad_{\Omega}^{-\beta} \varphi\right\vert = \int_{\Omega}\left\vert ad_{\Omega}^{s-\beta}d_{\Omega }^{-s}\varphi\right\vert \leq\left\Vert a\right\Vert _{\infty}\left\Vert d_{\Omega}^{s-\beta}\right\Vert _{2}\left\Vert d_{\Omega}^{-s}\varphi \right\Vert _{2}\leq c\left\Vert \varphi\right\Vert _{X_{0}^{s}\left(\Omega\right) } with c a positive constant independent of \varphi. Thus ad_{\Omega}^{-\beta}\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime}. Let u\in X_{0}^{s}\left(\Omega\right) be the unique weak solution (given by the Riesz Theorem) of problem (2.30) and consider a decreasing sequence \left\{ \varepsilon_{j}\right\} _{j\in N} in \left(0, 1\right) such that \lim_{j\rightarrow\infty}\varepsilon_{j} = 0. For j\in\mathbb{N}, let u_{\varepsilon_{j}}\in X_{0}^{s}\left(\Omega\right) be the weak solution of the problem

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{\varepsilon_{j}} = a\left( d_{\Omega }+\varepsilon_{j}\right) ^{-\beta}\text{ in }\Omega, \\ u_{\varepsilon_{j}} = 0\text{ in }\mathbb{R}^{n}\setminus\Omega. \end{array} \right. (2.31)

    Thus u_{\varepsilon_{j}} = \int_{\Omega}G\left(., y\right) a\left(y\right) \left(d_{\Omega}\left(y\right) +\varepsilon_{j}\right) ^{-\beta}dy in \Omega, where G is the Green function for \left(-\Delta\right) ^{s} in \Omega, with homogeneous Dirichlet boundary condition on \mathbb{R} ^{n}\setminus\Omega. Since \beta < s we have \int_{\Omega}\frac {1}{\left\vert x-y\right\vert ^{n-s+\beta}}dy < \infty. Thus, recalling (2.29), there exists a positive constant c such that, for any j\in\mathbb{N} and \left(x, y\right) \in\Omega\times\Omega,

    \begin{align*} 0 & \leq G\left( x, y\right) a\left( y\right) \left( d_{\Omega}\left( y\right) +\varepsilon_{j}\right) ^{-\beta}\leq c\frac{d_{\Omega}^{s}\left( x\right) d_{\Omega}^{\beta}\left( y\right) }{\left\vert x-y\right\vert ^{n-s+\beta}}\left( d_{\Omega}\left( y\right) +\varepsilon_{j}\right) ^{-\beta}\\ & \leq cd_{\Omega}^{s}\left( x\right) \frac{1}{\left\vert x-y\right\vert ^{n-s+\beta}}\in L^{1}\left( \Omega, dy\right) . \end{align*}

    Since also \lim_{j\rightarrow\infty}G\left(x, y\right) a\left(y\right) \left(d_{\Omega}\left(y\right) +\varepsilon_{j}\right) ^{-\beta } = G\left(x, y\right) a\left(y\right) d_{\Omega}^{-\beta}\left(y\right) for a.e. y\in\Omega, by the Lebesgue dominated convergence theorem, \left\{ u_{\varepsilon_{j}}\left(x\right) \right\} _{j\in\mathbb{N}} converges to \int_{\Omega}G\left(x, y\right) a\left(y\right) d_{\Omega }^{-\beta}\left(y\right) dy for any x\in\Omega. Let u\left(x\right) : = \lim_{j\rightarrow\infty}u_{\varepsilon_{j}}\left(x\right). Thus u\left(x\right) = \int_{\Omega}G\left(x, y\right) a\left(y\right) d_{\Omega}^{-\beta}\left(y\right) dy. Again from (2.29), u\leq cd_{\Omega}^{s} a.e. in \Omega, with c constant c independent of x. Now we take u_{\varepsilon_{j}} as a test function in (2.31) to obtain that

    \begin{align*} \int_{\Omega\times\Omega}\frac{\left( u_{\varepsilon_{j}}\left( x\right) -u_{\varepsilon_{j}}\left( y\right) \right) ^{2}}{\left\vert x-y\right\vert ^{n+2s}} & = \int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{\left( u_{\varepsilon_{j}}\left( x\right) -u_{\varepsilon_{j}}\left( y\right) \right) ^{2}}{\left\vert x-y\right\vert ^{n+2s}}\\ & = \int_{\Omega}u_{\varepsilon_{j}}\left( y\right) \left( d_{\Omega }\left( y\right) +\varepsilon_{j}\right) ^{-\beta}dy\\ & \leq c\int_{\Omega}d_{\Omega}^{s}\left( y\right) \left( d_{\Omega }\left( y\right) +\varepsilon\right) j^{-\beta}dy\leq c^{\prime} \int_{\Omega}d_{\Omega}^{s-\beta}\left( y\right) dy = c^{\prime\prime}, \end{align*}

    with c and c^{\prime} constants independent of j. For j\in\mathbb{N}, let U_{\varepsilon_{j}} and U be the functions, defined on \mathbb{R} ^{n}\times\mathbb{R}^{n}, by

    U_{\varepsilon_{j}}\left( x, y\right) : = u_{\varepsilon_{j}}\left( x\right) -u_{\varepsilon_{j}}\left( y\right) , \text{ }U\left( x, y\right) : = u\left( x\right) -u\left( y\right) .

    Then \left\{ U_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} is bounded in \mathcal{H} = L^{2}\left(\mathbb{R}^{n}\times\mathbb{R}^{n}, \frac {1}{\left\vert x-y\right\vert ^{n+2s}}dxdy\right). Thus, after pass to a subsequence if necessary, we can assume that \left\{ U_{\varepsilon_{j} }\right\} _{j\in\mathbb{N}} is weakly convergent in \mathcal{H} to some V\in\mathcal{H}. Since \left\{ U_{\varepsilon_{j}}\right\} _{j\in \mathbb{N}} converges pointwise to U on \mathbb{R}^{n}\times\mathbb{R} ^{n}, we conclude that U\in\mathcal{H} and that \left\{ U_{\varepsilon _{j}}\right\} _{j\in\mathbb{N}} converges weakly to U in \mathcal{H}. Thus u\in X_{0}^{s}\left(\Omega\right) and, for any \varphi\in X_{0}^{s}\left(\Omega\right),

    \int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{\left( u\left( x\right) -u\left( y\right) \right) \left( \varphi\left( x\right) -\varphi\left( y\right) \right) }{\left\vert x-y\right\vert ^{n+2s}}dxdy\\ = \lim\limits_{j\rightarrow\infty}\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}} \frac{\left( u_{\varepsilon_{j}}\left( x\right) -u_{\varepsilon_{j}}\left( y\right) \right) \left( \varphi\left( x\right) -\varphi\left( y\right) \right) }{\left\vert x-y\right\vert ^{n+2s}}dxdy\\ = \lim\limits_{j\rightarrow\infty}\int_{\Omega}a\left( d_{\Omega}+\varepsilon _{j}\right) ^{-\beta}\varphi = \int_{\Omega}ad_{\Omega}^{-\beta}\varphi,

    Then u is the weak solution of (2.30). Finally, since for all j, u_{\varepsilon_{j}}\leq c^{\prime}d_{\Omega}^{s} a.e. in \Omega, we have u\leq c^{\prime}d_{\Omega}^{s} a.e. in \Omega.

    Lemma 2.8. Let \lambda>0 and let \varepsilon \geq0. Suppose that \left\{ u_{j}\right\} _{j\in\mathbb{N}}\subset X_{0}^{s}\left(\Omega\right) is a nonincreasing sequence with the following properties i) and ii):

    i) There exist positive constants c and c^{\prime} such that cd_{\Omega}^{s}\leq u_{j}\leq c^{\prime}d_{\Omega }^{s} a.e. in \Omega for any j\in\mathbb{N}.

    ii) for any j\in\mathbb{N}, u_{j} is a weak solution of the problem

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{j} = -g\left( ., u_{j}+\varepsilon\right) +\lambda h\; in\;\Omega, \\ u_{j} = 0\; in \;\mathbb{R}^{n}\setminus\Omega, \\ u_{j}>.0\; in \;\Omega \end{array} \right. (2.32)

    Then \left\{ u_{j}\right\} _{j\in\mathbb{N}} converges in X_{0} ^{s}\left(\Omega\right) to a weak solution u of the problem

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u = -g\left( ., u+\varepsilon\right) +\lambda h\; in \;\Omega, \\ u_{j} = 0\; in \;\mathbb{R}^{n}\setminus\Omega, \\ u>0\; in \;\Omega, \end{array} \right. (2.33)

    which satisfies cd_{\Omega}^{s}\leq u\leq c^{\prime}d_{\Omega}^{s} a.e. in \Omega. Moreover, the same conclusions holds if, instead of ii), we assume the following ii'):

    ii') for any j\geq2, u_{j} is a weak solution of the problem

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{j} = -g\left( ., u_{j-1}+\varepsilon\right) +\lambda h\; in\;\Omega, \\ u_{j} = 0\; in \;\mathbb{R}^{n}\setminus\Omega, \\ u_{j}>0\; in \;\Omega. \end{array} \right.

    Proof. Assume i) and ii). For x\in\mathbb{R}^{n}, let u\left(x\right) : = \lim_{j\rightarrow\infty}u_{j}\left(x\right). For j, k\in\mathbb{N} we have, in weak sense,

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}\left( u_{j}-u_{k}\right) = g\left( ., u_{k}+\varepsilon\right) -g\left( u_{j}+\varepsilon\right) \text{ in }\Omega, \\ u_{j}-u_{k} = 0\text{ in }\mathbb{R}^{n}\setminus\Omega. \end{array} \right. (2.34)

    We take u_{j}-u_{k} as a test function in (2.34) to get

    \begin{align} \left\Vert u_{j}-u_{k}\right\Vert _{X_{0}^{s}\left( \Omega\right) }^{2} & = \int_{\Omega}\left( g\left( ., u_{k}+\varepsilon\right) -g\left( ., u_{j}+\varepsilon\right) \right) \left( u_{j}-u_{k}\right) \label{inecuacion domingo dos new}\\ & = \int_{\Omega}d_{\Omega}^{s}\left( g\left( ., u_{k}+\varepsilon\right) -g\left( ., u_{j}+\varepsilon\right) \right) d_{\Omega}^{-s}\left( u_{j}-u_{k}\right) \nonumber\\ & \leq\left\Vert d_{\Omega}^{-s}\left( \overline{u}_{j}-\overline{u} _{k}\right) \right\Vert _{2}\left\Vert d_{\Omega}^{s}\left( g\left( ., \overline{u}_{k}+\varepsilon\right) -g\left( ., \overline{u}_{j} +\varepsilon\right) \right) \right\Vert _{2}.\nonumber \end{align} (2.35)

    By the Hardy inequality, \left\Vert d_{\Omega}^{-s}\left(u_{j} -u_{k}\right) \right\Vert _{2}\leq c^{\prime\prime}\left\Vert u_{j} -u_{k}\right\Vert _{X_{0}^{s}\left(\Omega\right) } where c^{\prime\prime } is a constant independent of j and k. Thus, from (2.35),

    \left\Vert u_{j}-u_{k}\right\Vert _{X_{0}^{s}\left( \Omega\right) }\leq c^{\prime\prime}\left\Vert d_{\Omega}^{s}\left( g\left( ., u_{k} +\varepsilon\right) -g\left( ., u_{j}+\varepsilon\right) \right) \right\Vert _{2}. (2.36)

    Now, \lim_{j, k\rightarrow\infty}\left\vert d_{\Omega}^{s}\left(g\left(., u_{k}+\varepsilon\right) -g\left(., u_{j}+\varepsilon\right) \right) \right\vert ^{2} = 0 a.e. in \Omega. Also, since u_{l}\geq cd_{\Omega} ^{s} a.e. in \Omega for any l\in\mathbb{N}, and taking into account g5) and g2),

    \left\vert d_{\Omega}^{s}\left( g\left( ., u_{k}+\varepsilon\right) -g\left( ., u_{j}+\varepsilon\right) \right) \right\vert ^{2}\leq c^{\prime }\left( d_{\Omega}^{s}g\left( ., cd_{\Omega}^{s}\right) \right) ^{2}\in L^{1}\left( \Omega\right) ,

    where c^{\prime} is a constant independent of j and k. Then, by the Lebesgue dominated convergence theorem \lim_{j, k\rightarrow\infty}\left\Vert d_{\Omega}^{s}\left(g\left(., u_{k}+\varepsilon\right) -g\left(., u_{j}+\varepsilon\right) \right) \right\Vert _{2} = 0. Therefore, from (2.36), \lim_{j, k\rightarrow\infty}\left\Vert u_{j}-u_{k}\right\Vert _{X_{0}^{s}\left(\Omega\right) } = 0 and so \left\{ u_{j}\right\} _{j\in\mathbb{N}} converges in X_{0}^{s}\left(\Omega\right) to some u^{\ast}\in X_{0}^{s}\left(\Omega\right). Then, by the Poincaré inequality of Remark 2.1 iv), \left\{ u_{j}\right\} _{j\in\mathbb{N}} converges to u^{\ast} in L^{2_{s}^{\ast}}\left(\Omega\right), and thus there exists a subsequence \left\{ u_{j_{k}}\right\} _{k\in\mathbb{N}} that converges to u^{\ast} a.e. in \Omega. Since \left\{ u_{j_{k}}\right\} _{k\in\mathbb{N}} converges pointwise to u_{\varepsilon}, we conclude that u^{\ast} = u. Then \left\{ u_{j}\right\} _{j\in\mathbb{N}} converges to u_{\varepsilon} in X_{0}^{s}\left(\Omega\right). Now, for \varphi\in X_{0}^{s}\left(\Omega\right) and j\in\mathbb{N},

    \int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{\left( u_{j}\left( x\right) -u_{j}\left( y\right) \right) \left( \varphi\left( x\right) -\varphi\left( y\right) \right) }{\left\vert x-y\right\vert ^{n+2s} }dxdy = \int_{\Omega}\left( -g\left( ., u_{j}+\varepsilon\right) +\lambda h\right) \varphi. (2.37)

    Since \left\{ u_{j}\right\} _{j\in\mathbb{N}} converges to u in X_{0}^{s}\left(\Omega\right), we have

    \lim\limits_{j\rightarrow\infty}\int_{\mathbb{R}^{n}\times\mathbb{R}^{n}} \frac{\left( u_{j}\left( x\right) -u_{j}\left( y\right) \right) \left( \varphi\left( x\right) -\varphi\left( y\right) \right) }{\left\vert x-y\right\vert ^{n+2s}}dxdy\label{ecuacion domingo cinco}\\ = \int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{\left( u\left( x\right) -u\left( y\right) \right) \left( \varphi\left( x\right) -\varphi\left( y\right) \right) }{\left\vert x-y\right\vert ^{n+2s}}dxdy.\nonumber (2.38)

    On the other hand, \left\vert \left(-g\left(., u_{j}+\varepsilon\right) +\lambda h\right) \varphi\right\vert \leq\left(g\left(., cd_{\Omega }\right) +\lambda\left\Vert h\right\Vert _{\infty}\right) \left\vert \varphi\right\vert \in L^{1}\left(\Omega\right) (with c as in i)). Also, \left\{ \left(-g\left(u_{j}+\varepsilon\right) +\lambda h\right) \varphi\right\} _{j\in\mathbb{N}} converges to \left(-g\left(u+\varepsilon\right) +\lambda h\right) \varphi a.e. in \Omega. Then, by the Lebesgue dominated convergence theorem,

    \lim\limits_{j\rightarrow\infty}\int_{\Omega}\left( -g\left( ., u_{j}+\varepsilon \right) +\lambda h\right) \varphi = \int_{\Omega}\left( -g\left( ., u+\varepsilon\right) +\lambda h\right) \varphi. (2.39)

    From (2.37), (2.38) and (2.39) we get, for any \varphi\in X_{0}^{s}\left(\Omega\right),

    \int_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{\left( u\left( x\right) -u\left( y\right) \right) \left( \varphi\left( x\right) -\varphi\left( y\right) \right) }{\left\vert x-y\right\vert ^{n+2s}}dxdy = \int_{\Omega }\left( -g\left( ., u+\varepsilon\right) +\lambda h\right) \varphi.

    and so u is a weak solution of problem (2.33) which clearly satisfies cd_{\Omega}^{s}\leq u\leq c^{\prime}d_{\Omega}^{s} a.e. in \Omega. If instead of ii) we assume ii'), the proof is the same. Only replace, for j\geq2, k\geq2 and in each appearance, g\left(., u_{j}\right) and g\left(., u_{k}\right) by g\left(., u_{j-1}\right) and g\left(., u_{k-1}\right) respectively.

    Lemma 2.9. Let \lambda>0, and let \overline{u} be the weak solution of

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}\overline{u} = \lambda h\; in \;\Omega, \\ \overline{u} = 0\; in \;\mathbb{R}^{n}\setminus\Omega. \end{array} \right. (2.40)

    Assume that, for each \varepsilon>0, we have a function \widetilde {v}_{\varepsilon}\in X_{0}^{s}\left(\Omega\right) satisfying, in weak sense,

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}\widetilde{v}_{\varepsilon}\leq-g\left( ., \widetilde{v}_{\varepsilon}+\varepsilon\right) +\lambda h\;in \; \Omega, \\ \widetilde{v}_{\varepsilon} = 0\; in \;\mathbb{R}^{n}\setminus\Omega. \end{array} \right. (2.41)

    and such that \widetilde{v}_{\varepsilon}\geq cd_{\Omega}^{s} a.e. in \Omega, where c is a positive constant independent of \varepsilon. Then for any \varepsilon>0 there exists a sequence \left\{ u_{j}\right\} _{j\in\mathbb{N}}\subset X_{0}^{s}\left(\Omega\right) such that:

    i) u_{1} = \overline{u} and u_{j}\leq u_{j-1} for any j\geq2.

    ii) \widetilde{v}_{\varepsilon}\leq u_{j}\leq\overline{u} for all j\in N.

    iii) For any j\geq2, u_{j} satisfies, in weak sense,

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{j} = -g\left( ., u_{j-1}+\varepsilon\right) +\lambda h\; in \;\Omega, \\ u_{j} = 0\; in \;\mathbb{R}^{n}\setminus\Omega. \end{array} \right.

    iv) There exist positive constants c and c^{\prime}independent of \varepsilon and j such that, for all j, cd_{\Omega}^{s}\leq u_{j}\leq c^{\prime}d_{\Omega}^{s} a.e. in \Omega.

    Proof. By Remark 2.1 iii), there exists a positive constant c^{\prime} such that \overline{u}\leq c^{\prime}d_{\Omega}^{s} in \Omega. We construct inductively a sequence \left\{ u_{j}\right\} _{j\in\mathbb{N}} satisfying the assertions i)-iii) of the lemma: Let u_{1}: = \overline{u}. Thus, in weak sense, \left(-\Delta\right) ^{s}u_{1} = \lambda h\geq-g\left(., \widetilde{v}_{\varepsilon}+\varepsilon \right) +\lambda h in \Omega. Thus \left(-\Delta\right) ^{s}\left(u_{1}-\widetilde{v}_{\varepsilon}\right) \geq0 in \Omega. Then, by the maximum principle in Remark 2.1 i), u_{1} \geq\widetilde{v}_{\varepsilon} in \Omega, and so u_{1}\geq cd_{\Omega }^{s} in \Omega. Then, for some positive constant c^{\prime\prime}, \left\vert -g\left(., u_{1}+\varepsilon\right) +\lambda h\right\vert \leq c^{\prime\prime}\left(1+g\left(., cd_{\Omega}^{s}\right) \right) in \Omega and, by g1), g\left(., cd_{\Omega}^{s}\right) \in L^{\infty}\left(\Omega\right). Thus there exists a weak solution u_{2}\in X_{0}^{s}\left(\Omega\right) to the problem

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{2} = -g\left( ., u_{1}+\varepsilon\right) +\lambda h\text{ in }\Omega, \\ u_{2} = 0\text{ in }\mathbb{R}^{n}\setminus\Omega. \end{array} \right.

    Since, in weak sense, \left(-\Delta\right) ^{s}u_{2}\leq\lambda h = \left(-\Delta\right) ^{s}u_{1} in \Omega, the maximum principle in Remark 2.1 i) gives u_{2}\leq u_{1} in \Omega. Since u_{1}\geq\widetilde{v}_{\varepsilon} in \Omega we have, in weak sense, \left(-\Delta\right) ^{s}u_{2} = -g\left(., u_{1}+\varepsilon \right) +\lambda h\geq-g\left(., \widetilde{v}_{\varepsilon}+\varepsilon \right) +\lambda h in \Omega. Also, \left(-\Delta\right) ^{s}\widetilde{v}_{\varepsilon}\leq-g\left(., \widetilde{v}_{\varepsilon }+\varepsilon\right) +\lambda h in \Omega and so, by the maximum principle, u_{2}\geq\widetilde{v}_{\varepsilon} in \Omega. Then i)-iii) hold for j = 1.

    Supposed constructed u_{1}, ..., u_{k} such that i)-iii) hold for 1\leq j\leq k. Then, for some positive constant c^{\prime\prime\prime}, \left\vert -g\left(., u_{k}+\varepsilon\right) +\lambda h\right\vert \leq c^{\prime\prime}\left(1+g\left(., cd_{\Omega}^{s}\right) \right) in \Omega and so, as before, there exists a weak solution u_{k+1}\in X_{0}^{s}\left(\Omega\right) to the problem

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{k+1} = -g\left( ., u_{k}+\varepsilon\right) +\lambda h\text{ in }\Omega, \\ u_{k+1} = 0\text{ in }\mathbb{R}^{n}\setminus\Omega. \end{array} \right.

    By our inductive hypothesis, u_{k}\geq\widetilde{v}_{\varepsilon} in \Omega. Then, in weak sense, \left(-\Delta\right) ^{s}u_{k+1} = -g\left(., u_{k}+\varepsilon\right) +\lambda h\geq-g\left(., \widetilde {v}_{\varepsilon}+\varepsilon\right) +\lambda h\geq\left(-\Delta\right) ^{s}\widetilde{v}_{\varepsilon} in \Omega and thus, by the maximum principle, u_{k+1}\geq\widetilde{v}_{\varepsilon} in \Omega. If k = 2 we have u_{k}\leq u_{k-1} in \Omega. If k>2, by the inductive hypothesis we have, in weak sense, \left(-\Delta\right) ^{s}u_{k} = -g\left(., u_{k-1}+\varepsilon\right) +\lambda h\leq-g\left(., u_{k-2}+\varepsilon \right) +\lambda h in \Omega. Also, \left(-\Delta\right) ^{s} u_{k} = -g\left(., u_{k-1}+\varepsilon\right) +\lambda h in \Omega. Thus, by the maximum principle, u_{k+1}\leq u_{k} in \Omega. Again by the inductive hypothesis u_{k}\leq\overline{u} in \Omega and then, since u_{k+1}\leq u_{k} in \Omega, we get u_{k+1}\leq\overline{u} in \Omega.

    Since for all j, v_{\varepsilon}\leq u_{j}\leq\overline{u} in \Omega, iv) follows from the facts that \overline{u}\leq c^{\prime}d_{\Omega}^{s} in \Omega, and that \widetilde{v}_{\varepsilon }\geq cd_{\Omega}^{s} in \Omega, with c and c^{\prime} positive constants independent of \varepsilon and j.

    Lemma 2.10. Let \lambda>0. Assume that we have, for each \varepsilon>0, a function \widetilde{v}_{\varepsilon}\in X_{0}^{s}\left(\Omega\right) satisfying, in weak sense, (2.41), and such that \widetilde{v}_{\varepsilon}\geq cd_{\Omega}^{s} a.e. in \Omega, with c a positive constant independent of \varepsilon. Then for any \varepsilon>0 there exists a weak solution u_{\varepsilon} of the problem

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{\varepsilon} = -g\left( ., u_{\varepsilon }+\varepsilon\right) +\lambda h\;in \;\Omega, \\ u_{\varepsilon} = 0\; in \;\mathbb{R}^{n}\setminus\Omega, \\ u_{\varepsilon}>0\;in \;\Omega. \end{array} \right. (2.42)

    such that:

    i) u_{\varepsilon}\geq\widetilde{v}_{\varepsilon} and there exist positive constants c and c^{\prime} independent of \varepsilon such that cd_{\Omega}^{s}\leq u_{\varepsilon}\leq c^{\prime }d_{\Omega}^{s} in \Omega,

    ii) If \underline{u}_{\varepsilon}\in X_{0}^{s}\left(\Omega\right) and \left(-\Delta\right) ^{s} \underline{u}_{\varepsilon}\leq-g\left(., \underline{u}_{\varepsilon }+\varepsilon\right) +\lambda h in \Omega, then \underline{u} _{\varepsilon}\leq u_{\varepsilon} in \Omega,

    iii) If 0 < \varepsilon < \eta then u_{\varepsilon}\leq u_{\eta}.

    Proof. Let \left\{ u_{j}\right\} _{j\in\mathbb{N}} be as given by Lemma 2.9. For x\in\Omega, let u_{\varepsilon}\left(x\right) : = \lim_{j\rightarrow\infty}u_{j}\left(x\right). By Lemma 2.8, \left\{ u_{j}\right\} _{j\in\mathbb{N}} converges to u_{\varepsilon} in X_{0} ^{s}\left(\Omega\right) and u_{\varepsilon} is a weak solution to (2.42). From Lemma 2.9 iv) we have u_{\varepsilon}\geq\widetilde{v}_{\varepsilon} in \Omega and cd_{\Omega }^{s}\leq u_{\varepsilon}\leq c^{\prime}d_{\Omega}^{s} in \Omega, for some positive constants c and c^{\prime} independent of \varepsilon. Then i) holds. If \underline{u}_{\varepsilon}\in X_{0}^{s}\left(\Omega\right) and \left(-\Delta\right) ^{s}\underline{u}_{\varepsilon }\leq-g\left(., \underline{u}_{\varepsilon}+\varepsilon\right) +\lambda h in \Omega, then \left(-\Delta\right) ^{s}\underline{u}_{\varepsilon} \leq\lambda h = \left(-\Delta\right) ^{s}u_{1} in \Omega, and so \underline{u}_{\varepsilon}\leq u_{1}. Thus \left(-\Delta\right) ^{s}\underline{u}_{\varepsilon}\leq-g\left(., \underline{u}_{\varepsilon }+\varepsilon\right) +\lambda h\leq-g\left(., u_{1}+\varepsilon\right) +\lambda h = \left(-\Delta\right) ^{s}u_{2}, then \underline{u} _{\varepsilon}\leq u_{2} and, iterating this procedure, we obtain that \underline{u}_{\varepsilon}\leq u_{j} for all j. Then \underline {u}_{\varepsilon}\leq u_{\varepsilon}. Thus ii) holds. Finally, iii) is immediate from ii).

    Lemma 2.11. Let \varepsilon>0 and let \tau_{1} and M_{1} be as in Remarks 2.2, and 2.3 respectively. Let w_{\varepsilon} be as in Lemma 2.5. Then, for \lambda\geq\tau_{1} M_{1}, there exists a weak solution u_{\varepsilon}\in X_{0}^{s}\left(\Omega\right) of problem (2.42) such that

    i) u_{\varepsilon}\geq w_{\varepsilon} and there exist positive constants c and c^{\prime}, both independent of \varepsilon, such that cd_{\Omega }^{s}\leq u_{\varepsilon}\leq c^{\prime}d_{\Omega}^{s} in \Omega,

    ii) If \underline{u}_{\varepsilon}\in X_{0}^{s}\left(\Omega\right) and \left(-\Delta\right) ^{s}\underline{u}_{\varepsilon}\leq-g\left(., \underline{u}_{\varepsilon}+\varepsilon\right) +\lambda h in \Omega, then \underline{u}_{\varepsilon}\leq u_{\varepsilon} in \Omega,

    iii) If 0 < \varepsilon_{1} < \varepsilon_{2} then u_{\varepsilon_{1}}\leq u_{\varepsilon_{2}}.

    Proof. Let \lambda\geq\tau_{1}M_{1} and let w_{\varepsilon} be as in Lemma 2.5. We have, in weak sense,

    \left\{ \begin{array} [c]{c} -\Delta w_{\varepsilon} = -g\left( ., w_{\varepsilon}+\varepsilon\right) +\tau_{1}M_{1}h\text{ in }\Omega, \\ w_{\varepsilon} = 0\text{ on }\mathbb{R}^{n}\setminus\Omega, \end{array} \right.

    Also, -g\left(., w_{\varepsilon}+\varepsilon\right) +\tau_{1}M_{1} h\leq-g\left(., w_{\varepsilon}+\varepsilon\right) +\lambda h in \Omega, and cd_{\Omega}^{s}\leq w_{\varepsilon}\leq c^{\prime}d_{\Omega}^{s} in \Omega, with c and c^{\prime} positive constants independent of \varepsilon. Then the lemma follows from Lemma 2.10.


    3. Proof of the main results

    Lemma 3.1. Let \lambda>0. If problem (1.5) has a weak solution in \mathcal{E}, then it has a weak solution u\in\mathcal{E} satisfying u\geq\psi a.e. in \Omega for any \psi \in\mathcal{E} such that -\Delta\psi\leq-g\left(., \psi\right) +\lambda h in \Omega.

    Proof. Let u^{\ast}\in\mathcal{E} be a weak solution of (1.5), and let \overline{u} be as in (2.40). By the comparison principle u^{\ast}\leq \overline{u} in \Omega. We construct inductively a sequence \left\{ u_{j}\right\} _{j\in\mathbb{N}}\subset\mathcal{E} with the following properties: u_{1} = \overline{u} and

    i) u^{\ast}\leq u_{j}\leq\overline{u} for all j\in\mathbb{N}

    ii) g\left(., u_{j}\right) \in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime} for all j\in\mathbb{N}.

    iii) u_{j}\leq u_{j-1} for all j\geq2.

    iv) For all j\geq2

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{j} = -g\left( ., u_{j-1}\right) +\lambda h\text{ in }\Omega, \\ u_{j} = 0\text{ on }\mathbb{R}^{n}\setminus\Omega. \end{array} \right.

    To do it, define u_{1} = :\overline{u}. Thus u_{1}\in\mathcal{E}. By the comparison principle, u^{\ast}\leq\overline{u}, i.e., u^{\ast}\leq u_{1}. By Remark 2.1there exist positive constants c and c^{\prime} such that cd_{\Omega}^{s}\leq\overline{u}\leq c^{\prime }d_{\Omega}^{s} in \Omega. Thus \left\vert -g\left(., \overline {u}\right) +\lambda h\right\vert \leq g\left(., cd_{\Omega}^{s}\right) +\lambda\left\Vert h\right\Vert _{\infty} and so -g\left(., u_{1}\right) +\lambda h\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime}. Thus i) and ii) hold for j = 1. Define u_{2} as the weak solution of

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{2} = -g\left( ., u_{1}\right) +\lambda h\text{ in }\Omega, \\ u_{2} = 0\text{ on }\mathbb{R}^{n}\setminus\Omega. \end{array} \right.

    Then, in weak sense,

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{2}\leq\left( -\Delta\right) ^{s}u_{1}\text{ in }\Omega, \\ u_{2} = 0 = u_{1}\text{ on }\mathbb{R}^{n}\setminus\Omega. \end{array} \right.

    and so u_{2}\leq u_{1} = \overline{u} a.e. in \Omega. Since u_{1}\geq u^{\ast}, we have, in weak sense,

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{2} = -g\left( ., u_{1}\right) +\lambda h\geq-g\left( ., u^{\ast}\right) +\lambda h = \left( -\Delta\right) ^{s}u^{\ast}\text{ in }\Omega, \\ u_{2} = 0 = u^{\ast}\text{ on }\mathbb{R}^{n}\setminus\Omega. \end{array} \right.

    and then u_{2}\geq u^{\ast} a.e. in \Omega. Thus u^{\ast}\leq u_{2} \leq\overline{u}. In particular this gives u_{2}\in\mathcal{E}. Let c^{\prime\prime}>0 such that u^{\ast}\geq c^{\prime\prime}d_{\Omega} in \Omega. Now, \left\vert -g\left(., u_{2}\right) +\lambda h\right\vert \leq g\left(., u_{2}\right) +\lambda h\leq g\left(., u^{\ast}\right) +\lambda h\leq g\left(., c^{\prime\prime}d_{\Omega}^{s}\right) +\lambda\left\Vert h\right\Vert _{\infty} and so -g\left(., u_{2}\right) +\lambda h\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime}. Thus i)-iv) hold for j = 2. Suppose constructed, for 2\leq j\leq k, functions u_{j}\in\mathcal{E} with the properties i)-iv). Define u_{k+1} by

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{k+1} = -g\left( ., u_{k}\right) +\lambda h\text{ in }\Omega, \\ u_{k+1} = 0\text{ on }\mathbb{R}^{n}\setminus\Omega. \end{array} \right.

    Thus, by the comparison principle, u_{k+1}\leq\overline{u}. Also, by the inductive hypothesis, u_{k}\geq u^{\ast}, then

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{k+1} = -g\left( ., u_{k}\right) +\lambda h\geq-g\left( ., u^{\ast}\right) +\lambda h\text{ in }\Omega, \\ u_{k+1} = 0 = u^{\ast}\text{ on }\mathbb{R}^{n}\setminus\Omega. \end{array} \right.

    and so u_{k+1}\geq u^{\ast}. Then u^{\ast}\leq u_{k+1}\leq\overline{u}. In particular u_{k+1}\in\mathcal{E}. Again by the inductive hypothesis, u_{k}\leq u_{k-1}. Then

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u_{k+1} = -g\left( ., u_{k}\right) +\lambda h\leq-g\left( ., u_{k-1}\right) +\lambda h = \left( -\Delta\right) ^{s} u_{k}\text{ in }\Omega, \\ u_{k+1} = 0 = u_{k}\text{ on }\mathbb{R}^{n}\setminus\Omega. \end{array} \right.

    and so u_{k+1}\leq u_{k}. Also, \left\vert -g\left(., u_{k+1}\right) +\lambda h\right\vert \leq g\left(., u_{k+1}\right) +\lambda h\leq g\left(., u^{\ast}\right) +\lambda h\leq g\left(., c^{\prime\prime}d_{\Omega} ^{s}\right) +\lambda\left\Vert h\right\Vert _{\infty} and so -g\left(., u_{2}\right) +\lambda h\in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime}. Thus i)-iv) hold for j = k+1, which completes the inductive construction of the sequence \left\{ u_{j}\right\} _{j\in \mathbb{N}}. For x\in\mathbb{R}^{n} let u\left(x\right) : = \lim _{j\rightarrow\infty}u_{j}\left(x\right). By i) we have c^{\prime\prime}d_{\Omega}^{s}\leq u_{j}\leq c^{\prime}d_{\Omega}^{s} in \Omega for all j\in\mathbb{N}, and so c^{\prime\prime}d_{\Omega}^{s}\leq u\leq c^{\prime}d_{\Omega}^{s} in \Omega. By Lemma 2.8 \left\{ u_{j}\right\} _{j\in \mathbb{N}} converges in X_{0}^{s}\left(\Omega\right) to some weak solution u^{\ast\ast}\in X_{0}^{s}\left(\Omega\right) of problem (1.5). Thus, by the Poincaré inequality, \left\{ u_{j}\right\} _{j\in\mathbb{N}} converges to u^{\ast\ast} in L^{2_{s}^{\ast}}\left(\Omega\right), which implies u = u^{\ast\ast}. Thus u\in X_{0}^{s}\left(\Omega\right) and u is a weak solution of problem (1.5). Since c^{\prime\prime}d_{\Omega}^{s}\leq u_{j}\leq c^{\prime}d_{\Omega}^{s} in \Omega for all j, we have c^{\prime\prime}d_{\Omega}^{s}\leq u\leq c^{\prime}d_{\Omega}^{s} in \Omega. Thus u\in\mathcal{E}. Let \psi\in\mathcal{E} such that -\Delta\psi\leq-g\left(., \psi\right) +\lambda h in \Omega. By the comparison principle, \psi\leq u a.e. in \Omega. An easy induction shows that \psi\leq u_{j} for all j. Indeed, by the comparison principle, \psi\leq\overline{u} = u_{1}. Then

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}\psi\leq-g\left( ., \psi\right) +\lambda h\leq-g\left( ., u_{1}\right) +\lambda h = \left( -\Delta\right) ^{s} u_{2}\text{ in }\Omega, \\ \psi = 0 = u_{2}\text{ on }\mathbb{R}^{n}\setminus\Omega. \end{array} \right.

    Thus, again by the comparison principle, \psi\leq u_{2}. Suppose that k\geq2 and \psi\leq u_{k}. Then, in weak sense,

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}\psi\leq-g\left( ., \psi\right) +\lambda h\leq-g\left( ., u_{k}\right) +\lambda h = \left( -\Delta\right) ^{s} u_{k+1}\text{ in }\Omega, \\ \psi = 0 = u_{k+1}\text{ on }\mathbb{R}^{n}\setminus\Omega, \end{array} \right.

    which gives \psi\leq u_{k+1}. Thus \psi\leq u_{j} for all j, and so \psi\leq u.

    Proof of Theorem 1. Let \left\{ \varepsilon_{j}\right\} _{j\in\mathbb{N}}\subset\left(0, \infty\right) be a decreasing sequence such that \lim_{j\rightarrow\infty}\varepsilon_{j} = 0. For \lambda\geq \tau_{1}M_{1} and j\in\mathbb{N}, let u_{\varepsilon_{j}} be the weak solution of problem (2.42), given by Lemma 2.11, taking there \varepsilon = \varepsilon_{j}. Then \left\{ u_{\varepsilon_{j}}\right\} _{j\in \mathbb{N}} is a nonincreasing sequence in X_{0}^{s}\left(\Omega\right) and there exist positive constants c and c^{\prime} such that cd_{\Omega }^{s}\leq u_{j}\leq c^{\prime}d_{\Omega}^{s} in \Omega for all j\in\mathbb{N}. Therefore, by Lemma 2.8, \left\{ u_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} converges in X_{0}^{s}\left(\Omega\right) to some weak solution u\in X_{0} ^{s}\left(\Omega\right) of problem (1.5). Let

    \mathcal{T}: = \left\{ \lambda>0:\text{ problem (1.5) has a weak solution }u\in\mathcal{E}\right\} .

    Thus \lambda\in\mathcal{T} whenever \lambda\geq\tau_{1}M_{1}. Consider now an arbitrary \lambda\in\mathcal{T}, and let \lambda^{\prime}>\lambda. Let u\in\mathcal{E} be a weak solution of the problem

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u = -g\left( ., u\right) +\lambda h\text{ in }\Omega, \\ u = 0\text{ on }\mathbb{R}^{n}\setminus\Omega. \end{array} \right.

    Let \left\{ \varepsilon_{j}\right\} _{j\in\mathbb{N}}\subset\left(0, \infty\right) be a decreasing sequence such that \lim_{j\rightarrow \infty}\varepsilon_{j} = 0. We have, in weak sense,

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}u = -g\left( ., u\right) +\lambda h\leq-g\left( ., u+\varepsilon_{j}\right) +\lambda^{\prime}h\text{ in }\Omega, \\ u = 0\text{ on }\mathbb{R}^{n}\setminus\Omega. \end{array} \right.

    Then, by Lemma 2.9, used with \varepsilon = \varepsilon_{j}, \widetilde{v}_{\varepsilon_{j}} = u, and with \lambda replaced by \lambda^{\prime}, there exists a nonincreasing sequence \left\{ \widetilde{u}_{\varepsilon_{j}}\right\} _{j\in\mathbb{N} }\subset X_{0}^{s}\left(\Omega\right) such that

    \left\{ \begin{array} [c]{c} \left( -\Delta\right) ^{s}\widetilde{u}_{\varepsilon_{j}} = -g\left( ., \widetilde{u}_{\varepsilon_{j}}+\varepsilon_{j}\right) +\lambda^{\prime }h\text{ in }\Omega, \\ \widetilde{u}_{\varepsilon_{j}} = 0\text{ in }\mathbb{R}^{n}\setminus\Omega, \end{array} \right.

    satisfying that \widetilde{u}_{\varepsilon_{j}}\geq u for all j\in \mathbb{N}, and cd_{\Omega}^{s}\leq\widetilde{u}_{\varepsilon_{j}}\leq c^{\prime}d_{\Omega}^{s} in \Omega for some positive constants c and c^{\prime} independent of j. Let \widetilde{u}: = \lim_{j\rightarrow\infty }\widetilde{u}_{\varepsilon_{j}}. Proceeding as in the first part of the proof, we get that \widetilde{u}\in\mathcal{E} and that \widetilde{u} is a weak solution of problem (1.5). Then \lambda^{\prime} \in\mathcal{T} whenever \lambda^{\prime}>\lambda for some \lambda \in\mathcal{T}. Thus there exists \lambda^{\ast}\geq0 such that \left(\lambda^{\ast}, \infty\right) \subset\mathcal{T}\subset\left[\lambda^{\ast }, \infty\right).

    By Lemma 3.1, for any \lambda\in\mathcal{T} there exists a weak solution u\in\mathcal{E} of problem (1.5) such that u\geq\psi a.e. in \Omega for any \psi\in\mathcal{E} such that \left(-\Delta\right) ^{s}\psi \leq-g\left(., \psi\right) +\lambda h in \Omega.

    Suppose now that g\left(., s\right) \geq bs^{-\beta} a.e. in \Omega for any s\in\left(0, \infty\right) for some b\in L^{\infty}\left(\Omega\right) such that 0\leq b\not \equiv 0 in \Omega. Then there exist a constant \eta_{0}>0 and a measurable set \Omega_{0}\subset\Omega such that \left\vert \Omega_{0}\right\vert >0 and b\geq\eta_{0} in \Omega_{0}. Let \lambda_{1} be the principal eigenvalue for \left(-\Delta\right) ^{s} in \Omega with Dirichlet boundary condition \varphi_{1} = 0 on \mathbb{R}^{n}\setminus\Omega, and let \varphi_{1}\in X_{0}^{s}\left(\Omega\right) be an associated positive principal eigenfunction. Then

    \lambda_{1}\int_{\Omega}\varphi\varphi_{1} = \int_{\mathbb{R}^{n}\times \mathbb{R}^{n}}\frac{\left( \varphi\left( x\right) -\varphi\left( y\right) \right) \left( \varphi_{1}\left( x\right) -\varphi_{1}\left( y\right) \right) }{\left\vert x-y\right\vert ^{n+2s}}dxdy

    and \varphi_{1}>0 a.e. in \Omega (see e.g., [25], Theorem 3.1). Let \lambda\in\mathcal{T} and let u\in\mathcal{E} be a weak solution of (1.5). Thus

    \begin{align*} \lambda_{1}\int_{\Omega}u\varphi_{1} & = \int_{\mathbb{R}^{n}\times \mathbb{R}^{n}}\frac{\left( u\left( x\right) -u\left( y\right) \right) \left( \varphi_{1}\left( x\right) -\varphi_{1}\left( y\right) \right) }{\left\vert x-y\right\vert ^{n+2s}}dxdy\\ & = \int_{\Omega}\left( -\varphi_{1}g\left( ., u\right) +\lambda h\varphi_{1}\right) \leq\int_{\Omega}\left( -bu^{-\beta}\varphi_{1}+\lambda h\varphi_{1}\right) \end{align*}

    and so

    \lambda\int_{\Omega}h\varphi_{1}\geq\int_{\Omega_{0}}\left( \lambda _{1}u+bu^{-\beta}\right) \varphi_{1}\geq\inf\limits_{s>0}\left( \lambda_{1} s+\eta_{0}s^{-\beta}\right) \int_{\Omega_{0}}\varphi_{1}

    thus \lambda\geq\inf_{s>0}\left(\lambda_{1}s+\eta_{0}s^{-\beta}\right) \left(\int_{\Omega}h\varphi_{1}\right) ^{-1}\int_{\Omega_{0}}\varphi_{1} for any \lambda\in\mathcal{T}. Then \lambda^{\ast}>0.

    Lemma 3.2. Let g:\Omega\times\left(0, \infty\right) \rightarrow\left[0, \infty\right) be a Carathéodory function. Assume that s\rightarrow g\left(x, s\right) is nonincreasing for a.e. x\in\Omega, and that, for some a\in L^{\infty}\left(\Omega\right) and \beta\in\left[0, s\right), g\left(., s\right) \leq as^{-\beta} a.e. in \Omega for any s\in\left(0, \infty\right). Then g satisfies the conditions g1)-g5) of Theorem 1.2.

    Proof. Clearly g satisfies g1) and g2). Let \sigma>0. By Lemma 2.7, 0\leq g\left(., \sigma d_{\Omega} ^{s}\right) \leq a\sigma^{-\beta}d_{\Omega}^{-s\beta}\in\left(X_{0} ^{s}\left(\Omega\right) \right) ^{\prime} and so g\left(., \sigma d_{\Omega}^{s}\right) \in\left(X_{0}^{s}\left(\Omega\right) \right) ^{\prime}. Also, from the comparison principle, 0\leq\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{s}g\left(., \sigma ad_{\Omega}^{s}\right) \right) \leq\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(\sigma^{-\beta}ad_{\Omega}^{s-\beta}\right) in \Omega, and, since ad_{\Omega}^{s-\beta}\in L^{\infty}\left(\Omega\right), by Remark 2.1 iii), there exists a positive constant c such that \left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(\sigma ad_{\Omega}^{s-\beta}\right) \leq cd_{\Omega}^{s} in \Omega. Thus d_{\Omega}^{-s}\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{s}g\left(., \sigma d_{\Omega}^{s}\right) \right) \in L^{\infty}\left(\Omega\right). Then g satisfies g3). In particular, d_{\Omega}^{-s}\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{s}g\left(., d_{\Omega }^{s}\right) \right) \in L^{\infty}\left(\Omega\right). Since, for \sigma\geq1,

    0\leq\left( \sigma d_{\Omega}^{s}\right) ^{-1}\left( \left( -\Delta \right) ^{s}\right) ^{-1}\left( d_{\Omega}^{s}g\left( ., \sigma d_{\Omega }^{s}\right) \right) \leq\sigma^{-1}d_{\Omega}^{-s}\left( \left( -\Delta\right) ^{s}\right) ^{-1}\left( d_{\Omega}^{s}g\left( ., d_{\Omega }^{s}\right) \right) ,

    we get \lim_{\sigma\rightarrow\infty}\left\Vert \left(\sigma d_{\Omega} ^{s}\right) ^{-1}\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(d_{\Omega}^{s}g\left(., \sigma d_{\Omega}^{s}\right) \right) \right\Vert _{\infty} = 0. Also, by the comparison principle,

    0\leq d_{\Omega}^{-s}\left( \left( -\Delta\right) ^{s}\right) ^{-1}\left( g\left( ., \sigma\right) \right) \leq\sigma^{-\beta}d_{\Omega}^{-s}\left( \left( -\Delta\right) ^{s}\right) ^{-1}\left( a\right) ,

    and, by Remark 2.1 iii), d_{\Omega} ^{-s}\left(\left(-\Delta\right) ^{s}\right) ^{-1}\left(a\right) \in L^{\infty}\left(\Omega\right). Thus

    \lim\limits_{\sigma\rightarrow\infty}\left\Vert d_{\Omega}^{-s}\left( \left( -\Delta\right) ^{s}\right) ^{-1}\left( g\left( ., \sigma\right) \right) \right\Vert _{L^{\infty}\left( \Omega\right) } = 0.

    Then g4) holds. Finally, 0\leq d_{\Omega}^{s}g\left(., \sigma d_{\Omega}^{s}\right) \leq\sigma^{-\beta}d_{\Omega}^{s-\beta}a and so g5) holds.

    Proof of Theorem 1.3. The theorem follows from Lemma 3.2 and Theorem 1.2.


    Conflict of interest

    The author declare no conflicts of interest in this paper.


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