Loading [MathJax]/jax/element/mml/optable/GeneralPunctuation.js
Research article

A new procedure for unit root to long-memory process change-point monitoring

  • Received: 27 August 2021 Revised: 09 January 2022 Accepted: 17 January 2022 Published: 20 January 2022
  • MSC : 62F03, 62L10

  • In this paper, we propose a Dickey-Fuller difference statistic to sequentially detect the change-point that shift from an unit root process to a long-memory process. The limiting distribution of monitoring statistic under the unit root process null hypothesis as well as its consistency under the alternative hypothesis are proved. Simulations indicate that the new method can control the empirical size well even for the heavy-tailed unit root process when using the sieve bootstrap method computing its critical values. In particular, it performs significantly better than the available method in the literature under the alternative hypothesis. Finally, we illustrate the new monitoring procedure by a set of foreign exchange rate data.

    Citation: Zhanshou Chen, Muci Peng, Li Xi. A new procedure for unit root to long-memory process change-point monitoring[J]. AIMS Mathematics, 2022, 7(4): 6467-6477. doi: 10.3934/math.2022360

    Related Papers:

    [1] Yasir Nadeem Anjam . Singularities and regularity of stationary Stokes and Navier-Stokes equations on polygonal domains and their treatments. AIMS Mathematics, 2020, 5(1): 440-466. doi: 10.3934/math.2020030
    [2] Khaled Kefi, Nasser S. Albalawi . Three weak solutions for degenerate weighted quasilinear elliptic equations with indefinite weights and variable exponents. AIMS Mathematics, 2025, 10(2): 4492-4503. doi: 10.3934/math.2025207
    [3] Zhiying Deng, Yisheng Huang . Existence and multiplicity results for a singular fourth-order elliptic system involving critical homogeneous nonlinearities. AIMS Mathematics, 2023, 8(4): 9054-9073. doi: 10.3934/math.2023453
    [4] Tomas Godoy, Alfredo Guerin . Multiple finite-energy positive weak solutions to singular elliptic problems with a parameter. AIMS Mathematics, 2018, 3(1): 233-252. doi: 10.3934/Math.2018.1.233
    [5] Yazid Alhojilan, Islam Samir . Investigating stochastic solutions for fourth order dispersive NLSE with quantic nonlinearity. AIMS Mathematics, 2023, 8(7): 15201-15213. doi: 10.3934/math.2023776
    [6] Haohao Jia, Feiyao Ma, Weifeng Wo . Large positive solutions to an elliptic system of competitive type with nonhomogeneous terms. AIMS Mathematics, 2021, 6(8): 8191-8204. doi: 10.3934/math.2021474
    [7] Maoji Ri, Shuibo Huang, Qiaoyu Tian, Zhan-Ping Ma . Existence of W1,10(Ω) solutions to nonlinear elliptic equation with singular natural growth term. AIMS Mathematics, 2020, 5(6): 5791-5800. doi: 10.3934/math.2020371
    [8] Nafissa Toureche Trouba, Mohamed E. M. Alngar, Reham M. A. Shohib, Haitham A. Mahmoud, Yakup Yildirim, Huiying Xu, Xinzhong Zhu . Novel solitary wave solutions of the (3+1)–dimensional nonlinear Schrödinger equation with generalized Kudryashov self–phase modulation. AIMS Mathematics, 2025, 10(2): 4374-4411. doi: 10.3934/math.2025202
    [9] Ruyun Ma, Dongliang Yan, Liping Wei . Global bifurcation of sign-changing radial solutions of elliptic equations of order 2m in annular domains. AIMS Mathematics, 2020, 5(5): 4909-4916. doi: 10.3934/math.2020313
    [10] Chen Peng, Zhao Li . Optical soliton solutions for Lakshmanan-Porsezian-Daniel equation with parabolic law nonlinearity by trial function method. AIMS Mathematics, 2023, 8(2): 2648-2658. doi: 10.3934/math.2023138
  • In this paper, we propose a Dickey-Fuller difference statistic to sequentially detect the change-point that shift from an unit root process to a long-memory process. The limiting distribution of monitoring statistic under the unit root process null hypothesis as well as its consistency under the alternative hypothesis are proved. Simulations indicate that the new method can control the empirical size well even for the heavy-tailed unit root process when using the sieve bootstrap method computing its critical values. In particular, it performs significantly better than the available method in the literature under the alternative hypothesis. Finally, we illustrate the new monitoring procedure by a set of foreign exchange rate data.



    Let Ω be a bounded and regular enough domain in Rn, let α>0, and let a:ΩR be a nonnegative and nonidentically zero function. Singular elliptic problems like to

    {Δu=auα in Ω,u=0 on Ω,u>0 in Ω, (1.1)

    arise in many applications to physical phenomena, for instance, in chemical catalysts process, non-Newtonian fluids, and in models for the temperature of electrical conductors (see e.g., [3,5,13,16] and the references therein). Starting with the pioneering works [6,13,16,26], and [11], the existence of positive solutions of singular elliptic problems has been intensively studied in the literature.

    Bifurcation problems whose model is Δu=auα+f(.,λu) in Ω, u=0 on Ω, u>0 in Ω, were studied by Coclite and Palmieri [4], under the assumptions aC1(¯Ω), a>0 in ¯Ω, fC1(¯Ω×[0,)) and λ>0. Problems of the form Δu=Kuα+λsp in Ω, u=0 on Ω, u>0 in Ω, were studied by 35 [35], when p(0,1), K is a regular enough function that may change sign, and λR. Ghergu and Rădulescu [19] addressed multi-parameter singular bifurcation problems of the form Δu=g(u)+λ|u|p+μf(.,u) in Ω, u=0 on Ω, u>0 in Ω, where g is Hölder continuous, nonincreasingt and positive on (0,), and singular at the origin; f:¯Ω×[0,)[0,) is Hölder continuous, positive on ¯Ω×(0,), and such that f(x,s) is nondecreasing with respect to s, 0<p2,and λ>0. Dupaigne, Ghergu and Rădulescu [14] studied Lane–Emden–Fowler equations with convection and singular potential; and Rădulescu [32] addressed the existence, nonexistence, and uniqueness of blow-up boundary solutions of logistic equations and of singular Lane-Emden-Fowler equations with convection term. Cîrstea, Ghergu and Rădulescu [7] considered the problem of the existence of classical positive solutions for problems of the form Δu=a(x)h(u)+λf(u) in Ω, u=0 on Ω, u>0 in Ω, in the case when Ω is a regular enough domain, f and h are positive Hölder continuous functions on [0,) and (0,) respectively satisfying some monotonicity assumptions, h singular at the origin, and h(s)csα for some positive constant c and some α(0,1).

    Multiplicity results for positive solutions of singular elliptic problems were obtained by Gasiński and Papageorgiou [17] and by Papageorgiou and G. Smyrlis [30]; in both articles the singular term of the considered nonlinearity has the form a(x)sα, with 0aL(Ω), a0 in Ω, and α positive.

    Recently, problem (1.1) has been addressed by Chu, Gao and Gao [8], under the assumption that α=α(x) (i.e., with a singular nonlinearity with a variable exponent).

    Concerning the existence of nonnegative solutions of singular elliptic problems, Dávila and Montenegro [9] studied the free boundary singular bifurcation problem

    {Δu=χ{u>0}(uα+λf(.,u)) in Ω,u=0 on Ω,u0 in Ω, u0 in Ω,

    where 0<α<1, λ>0, and f:Ω×[0,)[0,) is a Carathéodory function f such that, for a.e. xΩ, f(x,s) is nondecreasing and concave in s, and satisfies limsf(x,s)/s=0 uniformly on xΩ. and where, for h:Ω×(0,)R, χ{s>0}h(x,s) stands for the function defined on Ω×[0,) by χ{s>0}h(x,s):=h(x,s) if s>0, and χ{s>0}h(x,s):=0 if s=0. Let us mention also the work [10], where a related singular parabolic problem was treated.

    For a systematic study of singular problems and additional references, we refer the reader to [18,32], see also [12].

    Our aim in this work is to prove an existence result for nonnegative weak solutions of singular elliptic problems of the form

    {Δu=χ{u>0}(auαg(.,u)) in Ω,u=0 on Ω,u0 in Ω, u0 in Ω, (1.2)

    where Ω is a bounded domain in Rn with C1,1 boundary, α(0,1], a:ΩR, and g:Ω×[0,)R, with a and g satisfying the following conditions h1)-h4):

    h1) 0aL(Ω) and a0,

    h2) {xΩ:a(x)=0}=Ω0N for some (possibly empty) open set Ω0Ω and some measurable set NΩ such that |N|=0,

    h3) g is a nonnegative Carathéodory function on Ω×[0,), i.e., g(.,s) is measurable for any s[0,), and g(x,.) is continuous on [0,) for a.e. xΩ,

    h4) sup0sMg(.,s)L(Ω) for any M>0.

    Here and below, χ{u>0}(auαg(.,u)) stands for the function h:ΩR defined by h(x):=a(x)uα(x)g(x,u(x)) if u(x)0, and h(x):=0 otherwise; u0 in Ω means |{xΩ:u(x)0}|>0 and, by a weak solution of (1.2), we mean a solution in the sense of the following:

    Definition 1.1. Let h:ΩR be a measurable function such that hφL1(Ω) for all φ in H10(Ω)L(Ω). We say that u:ΩR is a weak solution to the problem

    {Δu=h in Ω,u=0 on Ω (1.3)

    if uH10(Ω), and Ωu,φ=Ωhφ for all φ in H10(Ω)L(Ω).

    We will say that, in weak sense,

    Δuh in Ω (respectively Δuh in Ω),u=0 on Ω

    if uH10(Ω), and Ωu,φΩhφ (respectively Ωu,φΩhφ) for all nonnegative φ in H10(Ω)L(Ω).

    Our first result reads as follows:

    Theorem 1.2. Let Ω be a bounded domain in Rn with C1,1 boundary. Let α(0,1], let a:Ω[0,) and let g:Ω×(0,)R; and assume that a and g satisfy the conditions h1)-h4). Then there exists a nonnegative weak solution uH10(Ω)L(Ω), in the sense of Definition 1.1, to problem (1.2), and such that u>0 a.e. in {a>0}. In particular, χ{u>0}(auαg(.,u))0 in Ω and χ{u>0}(auαg(.,u))φL1(Ω) for any φH10(Ω)L(Ω)).

    Let us mention that in [21] it was proved the existence of weak solutions (in the sense of Definition 1.1) of problem (1.2), in the case when 0aL(Ω), a0, 0<α<1, and g(.,u)=bup, with 0<p<n+2n2, and 0bLr(Ω) for suitable values of r. In addition, existence results for weak solutions of problems of the form

    {Δu=χ{u>0}auαh(.,u) in Ω,u=0 on Ω,u0 in Ω, and u0 in Ω, (1.4)

    were obtained, in [22] (see Remark 2.1 below), and in ([25], Theorem 1.2), for more general nonlinearities h:Ω×[0,)[0,)(x,s), in the case when h is a Carathéodory function on Ω×[0,), which satisfies h(.,0)0 as well as some additional hypothesis. Then the result of Theorem 1.2 is not covered by those in [22] and [25] because, under the assumptions of Theorem 1.2, the condition g(.,0)0 is not required and χ{s>0}g(.,s) is not, in general, a Carathéodory function on Ω×[0,) (except when g(.,0)0 in Ω).

    Our next result says that if the condition h4) is replaced by the stronger condition

    h4') a>0 a.e. in Ω and sup0<sMs1g(.,s)L(Ω) for any M>0,

    then the solution u, given by Theorem 1.2, is positive a.e. in Ω and is a weak solution in the usual sense of H10(Ω).

    Theorem 1.3. Let Ω, α, and a be as in Theorem 1.2, and let g:Ω×(0,)R. Assume the conditions h1)-h3) and h4'). Then the solution u of (1.2), given by Theorem 1.2, belongs to C(¯Ω)W2,ploc(Ω) for any p[1,), there exist positive constants c, c and τ such that cdΩucdτΩ in Ω, and u is a weak solution, in the usual H10(Ω) sense, of the problem

    {Δu=auαg(.,u) in Ω,u=0 on Ω,u>0 in Ω (1.5)

    i.e., for any φH10(Ω), (auαg(.,u))φL1(Ω) and Ωu,φ=Ω(auαg(.,u))φ.

    Finally, our last result says that, if in addition to h1)-h4), α is sufficiently small, the set where a>0 is nice enough and, for any s0, g(.,s)=0 a.e. in the set where a>0, then the solution obtained in Theorem 1.2, is a weak solution in the usual sense of H10(Ω), and that it is positive on some subset of Ω:

    Theorem 1.4. Let Ω be a bounded domain in Rn with C1,1 boundary. Assume the hypothesis h1)-h4) of Theorem 1.2 and that 0<α<12+1n when n>2, and α(0,1] when n2. Let A+:={xΩ:a(x)>0} and assume, in addition, the following two conditions:

    h5) g(.,s)=0 a.e. in A+ for any s0.

    h6) A+=Ω+N+ for some open set Ω+ and a measurable set N+ such that |N+|=0, and with Ω+ such that Ω+ has a finite number of connected components {Ω+l}1lN and each Ω+l is a C1,1 domain.

    Then the solution u of problem (1.2), given by Theorem 1.2, is a weak solution, in the usual H10(Ω) sense, to the same problem, and there exist positive constants c, c and τ such that ucdΩ+ a.e. in Ω+, and ucdτΩ a.e. in Ω.

    The article is organized as follows: In Section 2 we study, for ε(0,1], the existence of weak solutions to the auxiliary problem

    {Δu=auαgε(.,u) in Ω,u=0 on Ω,u>0 in Ω. (1.6)

    where Ω is a bounded domain in Rn with C1,1 boundary, α(0,1], a:Ω[0,) is a nonnegative function in L(Ω) such that |{xΩ:a(x)>0}|>0, and {gε}ε(0,1] is a family of real valued functions defined on Ω×[0,) satisfying the following conditions h7)-h9):

    h7) gε is a nonnegative Carathéodory function on Ω×[0,) for any ε(0,1].

    h8) sup0<sMs1gε(.,s)L(Ω) for any ε(0,1] and M>0.

    h9) The map εgε(x,s) is nonincreasing on (0,1] for any (x,s)Ω×[0,).

    Lemma 2.2 observes that, as a consequence of a result of [22], the problem

    {Δu=χ{u>0}auαgε(.,u) in Ω,u=0 on Ω,u0 in Ω, u0 in Ω (1.7)

    has (at least) a weak solution u (in the sense of Definition 1.1) which satisfies u>0 a.e. in {a>0}; and this assertion is improved in Lemmas 2.6 and 2.7, which state that any weak solution u (in the sense of Definition 1.1) of problem (1.7) is positive in Ω, belongs to C(¯Ω), and is also a weak solution in the usual sense of H10(Ω). By using a sub-supersolution theorem of [28] as well as an adaptation of arguments of [27], Lemma 2.15 shows that, for any ε(0,1], problem (1.6) has a solution uεH10(Ω), which is a weak solution in the usual sense of H10(Ω), and is maximal in the sense that, if v is a solution, in the sense of Definition 1.1, of problem (1.6) then vuε. Lemma 2.16 states that εuε is nondecreasing, Lemma 2.17 says that {uε}ε(0,1] is bounded in H10(Ω), and Lemma 2.18 says that the function u:=limε0+uε belong to H10(Ω)L(Ω) and is positive in {a>0}.

    To prove Theorems 1.2–1.4 we consider, in Section 3, the family {gε}ε(0,1] defined by gε(.,s):=s(s+ε)1g(.,s) and we show that, in each case, the corresponding function u defined above is a solution of problem (1.2) with the desired properties.

    We assume, from now on, that Ω is a bounded domain in Rn with C1,1 boundary, α(0,1] and a:Ω[0,) is a nonnegative function in L(Ω) such that |{xΩ:a(x)>0}|>0, and additional conditions will be explicitely impossed on a and α when necessary, at some steps of the paper. Our aim in this section is to study, for ε(0,1], the existence of weak solutions to problem (1.6), in the case when {gε}ε(0,1] is a family of functions satisfying the conditions h7)-h9).

    In order to present, in the next remark, a need result of [22], we need to recall the notion of principal egenvalue with weight function: For bL(Ω) such that b0, we say that λR is a principal eigenvalue for Δ on Ω, with weight function b and homogeneous Dirichlet boundary condition, if the problem Δu=λbu in Ω, u=0 on Ω has a solution u wich is positive in Ω. If bL(Ω) and b+0, it is well known that there exists a unique positive principal eigenvalue for the above problem, which we wiill denote by λ1(b). For a proof of this fact and for additional properties of principal eigenvalues and their associated principal eigenfunctions see, for instance [15].

    Remark 2.1. (See [22], Theorem 1.2, or, in a more general setting, [25], Theorem 1.2) Let β(0,3), ˜a:ΩR and f:Ω×[0,)R; and assume the following conditions H1)-H6):

    H1) 0˜aL(Ω), and ˜a0,

    H2) f is a Carathéodory function on Ω×[0,),

    H3) sup0sM|f(.,s)|L1(Ω) for any M>0,

    H4) One of the two following conditions holds:

    H4') sups>0f(.,s)sb for some bL(Ω) such that b+0, and λ1(b)>m for some integer mmax{2,1+β},

    H4") fL(Ω×(0,σ)) for all σ>0, and ¯limsf(.,s)s0 uniformly on Ω, i.e., for any ε>0 there exists s0>0 such that supss0f(.,s)sε, a.e. in Ω,

    H5) f(.,0)0.

    Then the problem

    {Δu=χ{u>0}˜auβ+f(x,u) in Ω,u=0 on Ω, u0 in Ω, u0 in Ω. (2.1)

    has a weak solution (in the sense of Definition 1.1) uH10(Ω)L(Ω) such that u>0 a.e. in {˜a>0}.

    Lemma 2.2. Let aL(Ω) be such that a0 in Ω and a0, let α(0,1], and let {gε}ε(0,1] be a family of functions defined on Ω×[0,) satisfying the conditions h7)-h9) stated at the introduction. Then, for any ε(0,1], problem (1.7) has at least a weak solution uH10(Ω)L(Ω), in the sense of Definition 1.1, such that u>0 a.e. in {a>0}.

    Proof. Notice that, since gε is a Carathéodory function, we have gε(.,0)=lims0+gε(.,s)=lims0+(ss1gε(.,s))=0, the last inequality by h8). Thus gε(.,0)=0. Taking into account this fact and h7)-h9), the lemma follows immediately from Remark 2.1.

    Let us recall, in the next remark, the uniform Hopf maximum principle:

    Remark 2.3. ⅰ) (see [2], Lemma 3.2) Suppose that 0hL(Ω); and let v1p<(W2,p(Ω)W1,p0(Ω)) be the strong solution of Δv=h in Ω, v=0 on Ω. Then vcdΩΩhdΩ a.e. in Ω, where dΩ:=dist(.,Ω), and c is a positive constant depending only on Ω.

    ⅱ) (see e.g., [25], Remark 8) Let Ψ be a nonnegative function in L1loc(Ω), and let v be a function in H10(Ω) such that ΔvΨ on Ω in the sense of distributions. Then

    v(x)cdΩΩΨdΩa.e. in Ω, (2.2)

    where c is a positive constant depending only on Ω.

    Remark 2.4. (See, e.g., [23], Lemmas 2.9, 2.10 and 2.12) Let aL(Ω) be such that a0 in Ω and a0, and let let α(0,1]. Then the problem

    {Δz=azα in Ω,z=0 on Ω,z0 in Ω. (2.3)

    has a unique weak solution, in the sense of Definition 1.1, zH10(Ω)L(Ω). Moreover:

    ⅰ) zC(¯Ω).

    ⅱ) There exists positive constants c1, c2 and τ>0 such that c1dΩzc2dτΩ in Ω.

    ⅲ) z is a solution of problem (2.3) in the usual weak sense, i.e., for any φH10(Ω), azαφL1(Ω) and Ωz,φ=Ωazαφ.

    Lemma 2.5. Let a, α, and {gε}ε(0,1] be as in Lemma 2.2, let z be as given in Remark 2.4; and let ε(0,1]. If uH10(Ω)L(Ω) is a weak solution, in the sense of Definition 1.1, of problem (1.7), then uz a.e. in Ω.

    Proof. By h5), gε(.,u)0 and so, from Lemma 2.2 and Remark 2.4, we have, in the sense of Definition 1.1,

    Δ(uz)=auαgε(.,u)azαa(uαzα) in Ω,

    Thus, taking (uz)+ as a test function, we get

    Ω|(uz)+|2Ωa(uαzα)(uz)+0

    which implies uza.e. in Ω.

    Lemma 2.6. Let a, α, and {gε}ε(0,1] be as in Lemma 2.2. If ε(0,1] and uH10(Ω)L(Ω) is a weak solution, in the sense of Definition 1.1, of problem (1.7), then:

    i) There exists a positive constant c1 (which may depend on ε) and constants c2 and τ such that c1dΩuc2dτΩ a.e. in Ω (and so, in particular, u>0 in Ω).

    ii) For any φH10(Ω) we have (auαgε(.,u))φL1(Ω) and

    Ωu,φ=Ω(auαgε(.,u))φ,

    i.e., u is a weak solution, in the usual sense of H10(Ω), to the problem Δu=auαgε(.,u) in Ω, u=0 on Ω.

    Proof. We have, in the weak sense of Definition 1.1, Δu=χ{u>0}auαgε(.,u) in Ω, u=0 on Ω. Also, u0 in Ω and u0 in Ω. Let a0:ΩR be defined by a0(x)=u1(x)gε(x,u(x)) if u(x)0 and by a0(x)=0 otherwise. Since uL(Ω) and taking into account h7) and h8), we have 0a0L(Ω), and from the definition of a0 we have gε(.,u)=a0u a.e. in Ω. Therefore u satisfies, in the sense of Definition 1.1, Δu+a0u=χ{u>0}auα in Ω, u=0 on Ω. Thus, since u is nonidentically zero, it follows that χ{u>0}auα is nonidentically zero on Ω. Then there exist η>0, and a measurable set EΩ, such that |E|>0 and χ{u>0}auαηχE in Ω. Let ψ1q<W2,,q(Ω)W1,,q0(Ω) be the solution of the problem Δψ+a0ψ=ηχE in Ω, ψ=0 on Ω. By the Hopf maximum principle (as stated, e.g., in [34], Theorem 1.1) there exists a positive constant c1 such that ψc1dΩ in Ω; and, from (1.7) we have Δu+a0uηχE in D(Ω). Then, by the weak maximum principle (as stated, e.g., in [20], Theorem 8.1), uψ in Ω. Hence uc1dΩ in Ω. Also, by Lemma 2.5, uz a.e. in Ω, and so Remark 2.4 gives positive constants c2 and τ (both independent of ε) such that uc2dτΩ in Ω. Thus i) holds.

    To see ii), consider an arbitrary function φH10(Ω), and for kN, let φ+k:=max{k,φ+}. Thus φ+kH10(Ω)L(Ω), {φ+k}kN converges to φ+ in H10(Ω) and, after pass to some subsequence if necessary, we can assume also that {φ+k}kN converges to φ+ a.e. in Ω. Since u is a weak solution, in the sense of Definition 1.1, of (1.7) and u>0 a.e. in Ω, we have, for all kN, (auαgε(.,u))φ+kL1(Ω), and, by h6), gε(.,u)L(Ω). Thus gε(.,u)φ+kL1(Ω). Then auαφ+kL1(Ω).

    From (1.7),

    Ωu,φ+k+Ωgε(.,u)φ+k=Ωauαφ+k. (2.4)

    Now, limkΩu,φ+k=Ωu,φ+. Also, for any k,

    0gε(.,u)φ+ksups[0,

    then, by the Lebesgue dominated convergence theorem, \lim_{k\rightarrow \infty}\int_{\Omega}g_{\varepsilon}\left(., u\right) \varphi_{k} ^{+} = \int_{\Omega}g_{\varepsilon}\left(., u\right) \varphi^{+} < \infty. Hence, by (2.4), \lim_{k\rightarrow\infty}\int_{\Omega }au^{-\alpha}\varphi_{k}^{+} exists and is finite. Since \left\{ au^{-\alpha}\varphi_{k}^{+}\right\} _{k\in\mathbb{N}} is nondecreasing and converges to au^{-\alpha}\varphi^{+} a.e. in \Omega, the monotone convergence theorem gives \lim_{k\rightarrow\infty}\int_{\Omega}au^{-\alpha }\varphi_{k}^{+} = \int_{\Omega}au^{-\alpha}\varphi^{+} < \infty. Thus

    \left( au^{-\alpha}-g_{\varepsilon}\left( ., u\right) \right) \varphi ^{+}\in L^{1}\left( \Omega\right)

    and

    \begin{equation} \int_{\Omega}\left\langle \nabla u, \nabla\varphi^{+}\right\rangle +\int_{\Omega}g_{\varepsilon}\left( ., u\right) \varphi^{+} = \int_{\Omega }au^{-\alpha}\varphi^{+}. \end{equation} (2.5)

    Similarly, we have that \left(au^{-\alpha}-g_{\varepsilon}\left(., u\right) \right) \varphi^{-}\in L^{1}\left(\Omega\right), and that (2.5) holds with \varphi^{+} replaced by \varphi^{-} By writing \varphi = \varphi^{+}-\varphi^{-} the lemma follows.

    Lemma 2.7. Let a , \alpha, and \left\{ g_{\varepsilon }\right\} _{\varepsilon\in\left(0, 1\right] } be as in Lemma 2.2. For any \varepsilon\in\left(0, 1\right], if u\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right) is a weak solution, in the sense of Definition 1.1 (and so, by Lemma 2.6, also in the usual sense of H_{0}^{1}\left(\left(\Omega\right) \right) ), of problem (1.7), then u\in C\left(\overline{\Omega}\right).

    Proof. By Lemma 2.6 we have u\geq cd_{\Omega} a.e. in \Omega, with c a positive constant and, by h6), 0\leq u^{-1}g_{\varepsilon}\left(., u\right) \in L^{\infty}\left(\Omega\right). Thus au^{-\alpha }-g_{\varepsilon}\left(., u\right) \in L_{loc}^{\infty}\left(\Omega\right). Also, u\in L^{\infty}\left(\Omega\right). Then, by the inner elliptic estimates (as stated, e.g., in [20], Theorem 8.24), u\in W_{loc}^{2, p}\left(\Omega\right) for any p\in\left[1, \infty\right). Thus u\in C\left(\Omega\right), and, since 0\leq u\leq z, z\in C\left(\overline{\Omega}\right) and z = 0 on \partial\Omega, it follows that u is also continuous at \partial\Omega.

    Definition 2.8. Let C_{0}^{\infty}\left(\overline{\Omega}\right) : = \left\{ \varphi\in C^{\infty}\left(\overline{\Omega}\right) :\varphi = 0\text{ on }\partial\Omega\right\}. If u\in L^{1}\left(\Omega\right) and h\in L^{1}\left(\Omega\right), we will say that u is a solution, in the sense of \left(C_{0}^{\infty}\left(\overline {\Omega}\right) \right) ^{\prime}, of the problem -\Delta u = h in \Omega, u = 0 on \partial\Omega, if -\int_{\Omega}u\Delta\varphi = \int_{\Omega}h\varphi for any \varphi\in C_{0}^{\infty}\left(\overline{\Omega}\right).

    We will say also that -\Delta u\geq h in \left(C_{0}^{\infty}\left(\overline{\Omega}\right) \right) ^{\prime } (respectively -\Delta u\leq h in \left(C_{0}^{\infty}\left(\overline{\Omega}\right) \right) ^{\prime} ) if -\int_{\Omega} u\Delta\varphi\geq\int_{\Omega}h\varphi (resp. -\int_{\Omega}u\Delta \varphi\leq\int_{\Omega}h\varphi ) for any nonnegative \varphi\in C_{0}^{\infty}\left(\overline{\Omega}\right).

    Remark 2.9. The following statements hold:

    ⅰ) (Maximum principle, [31], Proposition 5.1) If u\in L^{1}\left(\Omega\right), 0\leq h\in L^{1}\left(\Omega\right), and -\Delta u\geq h in the sense of \left(C_{0}^{\infty}\left(\overline{\Omega }\right) \right) ^{\prime}, then u\geq0 \ a.e. in \Omega .

    ⅱ) (Kato's inequality, [31], Proposition 5.7) If h\in L^{1}\left(\Omega\right), u\in L^{1}\left(\Omega\right) and if -\Delta u\leq h in D^{\prime}\left(\Omega\right) , then -\Delta\left(u^{+}\right) \leq\chi_{\left\{ u > 0\right\} }h in D^{\prime}\left(\Omega\right).

    ⅲ) ([31], Proposition 3.5) For \varepsilon > 0, let A_{\varepsilon}: = \left\{ x\in\Omega:dist\left(x, \partial\Omega\right) < \varepsilon\right\}. If h\in L^{1}\left(\Omega\right) and if u\in L^{1}\left(\Omega\right) is a solution of -\Delta u = h , in the sense of Definition 2.8, then there exists a constant c such that, for all \varepsilon > 0, \int_{A_{\varepsilon}}\left\vert u\right\vert \leq c\varepsilon^{2}\left\Vert h\right\Vert _{1}. In particular, \lim_{\varepsilon\rightarrow0^{+}}\frac{1}{\varepsilon}\int_{A_{\varepsilon} }\left\vert u\right\vert = 0.

    ⅳ) ([31], Proposition 5.2) Let u\in L^{1}\left(\Omega\right) and h\in L^{1}\left(\Omega\right). If -\Delta u\leq h (respectively -\Delta u = h ) in D^{\prime}\left(\Omega\right) and \lim_{\varepsilon\rightarrow0^{+} }\frac{1}{\varepsilon}\int_{A_{\varepsilon}}\left\vert u\right\vert = 0 then -\Delta u\leq h (resp. -\Delta u = h ) in the sense of \left(C_{0} ^{\infty}\left(\overline{\Omega}\right) \right) ^{\prime}.

    ⅴ) ([31], Proposition 5.9) Let f_{1}, f_{2}\in L^{1}\left(\Omega\right). If u_{1}, u_{2}\in L^{1}\left(\Omega\right) are such that \Delta u_{1}\geq f_{1} and \Delta u_{2}\geq f_{2} in the sense of distributions in \Omega , then \Delta\max\left\{ u_{1}, u_{2}\right\} \geq\chi_{\left\{ u_{1} > u_{2}\right\} }f_{1} +\chi_{\left\{ u_{2} > u1\right\} }f_{2}+\chi_{\left\{ u_{1} = u_{2}\right\} }\frac{1}{2}\left(f_{1}+f_{2}\right) in the sense of distributions in \Omega.

    If h:\Omega\rightarrow\mathbb{R} is a measurable function such that h\varphi\in L^{1}\left(\Omega\right) for any \varphi\in C_{c}^{\infty}\left(\Omega\right), we say that u:\Omega\rightarrow \mathbb{R} is a subsolution (respectively a supersolution), in the sense of distributions, of the problem -\Delta u = h in \Omega, if u\in L_{loc} ^{1}\left(\Omega\right) and -\int_{\Omega}u\Delta\varphi\leq\int _{\Omega}h\varphi (resp. -\int_{\Omega}u\Delta\varphi\geq\int_{\Omega }h\varphi ) for any nonnegative \varphi\in C_{c}^{\infty}\left(\Omega\right).

    Remark 2.10. ([28], Theorem 2.4) Let f:\Omega \times\left(0, \infty\right) \rightarrow\mathbb{R} be a Caratheodory function, and let \underline{w} and \overline{w} be two functions, both in L_{loc}^{\infty}\left(\Omega\right) \cap W_{loc}^{1, 2}\left(\Omega\right), and such that f\left(., \underline{w}\right) and f\left(., \overline{w}\right) belong to L_{loc}^{1}\left(\Omega\right). Suppose that \underline{w} is a subsolution and \overline{w} is a supersolution, both in the sense of distributions, of the problem

    \begin{equation} -\Delta w = f\left( ., w\right) \text{ in }\Omega. \end{equation} (2.6)

    Suppose in addition that 0 < \underline{w}\left(x\right) \leq\overline {w}\left(x\right) a.e. x\in \Omega, and that there exists h\in L_{loc}^{\infty}\left(\Omega\right) such that \sup_{s\in\left[\underline{w}\left(x\right), \overline{w}\left(x\right) \right] }\left\vert f\left(x, s\right) \right\vert \leq h\left(x\right) a.e. x\in \Omega. Then (2.6) has a solution w, in the sense of distributions, which satisfies \underline{w}\leq w\leq\overline{w} a.e. in \Omega. Moreover, as obverved in [28], if in addition f\left(., w\right) \in L_{loc}^{\infty}\left(\Omega\right), then, by a density argument, the equality \int_{\Omega}\left\langle \nabla w, \nabla \varphi\right\rangle = \int_{\Omega}f\left(., w\right) \varphi holds also for any \varphi\in W_{loc}^{1, 2}\left(\Omega\right) with compact support.

    Remark 2.11. Let us recall the Hardy inequality (as stated, e.g., in [29], Theorem 1.10.15, see also [1], p. 313): There exists a positive constant c such that \left\Vert \frac{\varphi }{d_{\Omega}}\right\Vert _{L^{2}\left(\Omega\right) }\leq c\left\Vert \nabla\varphi\right\Vert _{L^{2}\left(\Omega\right) } for all \varphi\in H_{0}^{1}\left(\Omega\right).

    Remark 2.12. Let a and \left\{ g_{\varepsilon}\right\} _{\varepsilon\in\left(0, 1\right] } be as in Lemma 2.2 and assume that \alpha\in\left(0, 1\right]. Let \varepsilon\in\left(0, 1\right]. If u\in L^{\infty}\left(\Omega\right) and if, for some positive constant c, u\geq cd_{\Omega} a.e. in \Omega, then au^{-\alpha}-g_{\varepsilon }\left(., u\right) \in\left(H_{0}^{1}\left(\Omega\right) \right) ^{\prime}. Indeed, for \varphi\in H_{0}^{1}\left(\Omega\right) we have \left\vert au^{-\alpha}\varphi\right\vert \leq c^{-\alpha}d_{\Omega }^{1-\alpha}\left\vert \frac{\varphi}{d_{\Omega}}\right\vert. Since d_{\Omega}^{1-\alpha}\in L^{\infty}\left(\Omega\right) (because \alpha\leq1 ), the Hardy inequality gives a positive constant c^{\prime} independent of \varphi such that \left\Vert au^{-\alpha}\varphi\right\Vert _{1}\leq c^{\prime}\left\Vert \nabla\varphi\right\Vert _{2}. Also, since u\in L^{\infty}\left(\Omega\right), from h6) and the Hardy inequality, \left\Vert g_{\varepsilon}\left(., u\right) \varphi\right\Vert _{1}\leq c^{\prime\prime}\left\Vert \nabla\varphi\right\Vert _{2}, with c^{\prime\prime} a positive constant independent of \varphi.

    Lemma 2.13. Let a and \left\{ g_{\varepsilon}\right\} _{\varepsilon\in\left(0, 1\right] } be as in Lemma 2.2 and assume that \alpha\in\left(0, 1\right]. Let \varepsilon\in\left(0, 1\right]. Suppose that u\in W_{loc}^{1, 2}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right) is a solution, in the sense of distributions, of the problem

    \begin{equation} -\Delta u = au^{-\alpha}-g_{\varepsilon}\left( ., u\right) ~\mathit{\text{in}}~\Omega, \end{equation} (2.7)

    and that there exist positive constants c, c^{\prime} and \gamma such that c^{\prime}d_{\Omega}\leq u\leq cd_{\Omega}^{\gamma} a.e. in \Omega. Then u\in H_{0}^{1}\left(\Omega\right) \cap C\left(\overline{\Omega }\right), and u is a weak solution, in the usual sense of H_{0} ^{1}\left(\Omega\right), of problem (1.6).

    Proof. Since u\in L^{\infty}\left(\Omega\right) and u\geq c^{\prime}d_{\Omega }, we have au^{-\alpha}-g_{\varepsilon}\left(., u\right) \in L_{loc}^{\infty}\left(\Omega\right). Thus, from the inner elliptic estimates in ([20], Theorem 8.24), u\in C\left(\Omega\right) and, from the inequalities c^{\prime}d_{\Omega}\leq u\leq cd_{\Omega}^{\gamma} a.e. in \Omega, u is also continuous on \partial\Omega. Then u\in C\left(\overline{\Omega}\right)

    The proof of that u\in H_{0}^{1}\left(\Omega\right) and that u is a weak solution, in the usual sense of H_{0}^{1}\left(\Omega\right), of problem (1.6), is a slight variation of the proof of ([24], Lemma 2.4). For the convenience of the reader, we give the details: For j\in\mathbb{N}, let h_{j}:\mathbb{R\rightarrow R} be the function defined by h_{j}\left(s\right) : = 0 if s\leq\frac{1}{j}, h_{j}\left(s\right) : = -3j^{2}s^{3}+14js^{2}-19s+\frac{8}{j} if \frac {1}{j} < s < \frac{2}{j} and h\left(s\right) = s for \frac{2}{j}\leq s. Then h_{j}\in C^{1}\left(\mathbb{R}\right), h_{j}^{\prime}\left(s\right) = 0 for s < \frac{1}{j}, h_{j}^{\prime}\left(s\right) \geq0 for \frac{1}{j} < s < \frac{2}{j} and h_{j}^{\prime}\left(s\right) = 1 for \frac{2}{j}\leq s . Moreover, for s\in\left(\frac{1}{j}, \frac{2} {j}\right) we have s^{-1}h_{j}\left(s\right) = -3j^{2}s^{2} +14js-19+\frac{8}{js} < -3j^{2}s^{2}+14js-11 < 5 (the last inequality because -3t^{2}+14t-16 < 0 whenever t\notin\left[\frac{8}{3}, 2\right] ). Thus 0\leq h_{j}\left(s\right) \leq5s for any j\in\mathbb{N} and s\geq 0.

    Let h_{j}\left(u\right) : = h_{j}\circ u. Then, for all j, \nabla\left(h_{j}\left(u\right) \right) = h_{j}^{\prime}\left(u\right) \nabla u. Since u\in W_{loc}^{1, 2}\left(\Omega\right), we have h_{j}\left(u\right) \in W_{loc}^{1, 2}\left(\Omega\right), and since h_{j}\left(u\right) has compact support, Remark 2.10 gives, for all j\in\mathbb{N}, \int_{\Omega }\left\langle \nabla u, \nabla\left(h_{j}\left(u\right) \right) \right\rangle = \int_{\Omega}\left(au^{-\alpha}-g_{\varepsilon}\left(., u\right) \right) h_{j}\left(u\right), i.e.,

    \begin{equation} \int_{\left\{ u \gt 0\right\} }h_{j}^{\prime}\left( u\right) \left\vert \nabla u\right\vert ^{2} = \int_{\Omega}\left( au^{-\alpha}-g_{\varepsilon}\left( ., u\right) \right) h_{j}\left( u\right) . \end{equation} (2.8)

    Now, h_{j}^{\prime}\left(u\right) \left\vert \nabla u\right\vert ^{2} is a nonnegative function and \lim_{j\rightarrow\infty}h_{j}^{\prime}\left(u\right) \left\vert \nabla u\right\vert ^{2} = \left\vert \nabla u\right\vert ^{2} a.e. in \Omega, and so, by (2.8) and the Fatou's lemma,

    \int_{\Omega}\left\vert \nabla u\right\vert ^{2}\leq\underline{\lim }_{j\rightarrow\infty}\int_{\Omega}\left( au^{-\alpha}-g_{\varepsilon}\left( ., u\right) \right) h_{j}\left( u\right) .

    Also,

    \lim\limits_{j\rightarrow\infty}\left( au^{-\alpha}-g_{\varepsilon}\left( ., u\right) \right) h_{j}\left( u\right) = au^{1-\alpha}-ug_{\varepsilon }\left( ., u\right) \text{ }a.e.\text{ in }\Omega.

    Now, 0\leq au^{-\alpha}h_{j}\left(u\right) \leq5au^{1-\alpha}. Since a and u belong to L^{\infty}\left(\Omega\right) and \alpha\leq1, we have au^{1-\alpha}\in L^{1}\left(\Omega\right). Also,

    0\leq g_{\varepsilon}\left( ., u\right) h_{j}\left( u\right) \leq 5ug_{\varepsilon}\left( ., u\right) \leq5\left\Vert u\right\Vert _{\infty }^{2}\sup\limits_{0 \lt s\leq\left\Vert u\right\Vert _{\infty}}s^{-1}g_{\varepsilon }\left( ., s\right) \text{ }a.e.\text{ in }\Omega,

    and, by h6), \sup_{0 < s\leq\left\Vert u\right\Vert _{\infty}} s^{-1}g_{\varepsilon}\left(., s\right) \in L^{\infty}\left(\Omega\right). Then, by the Lebesgue dominated convergence theorem,

    \lim\limits_{j\rightarrow\infty}\int_{\Omega}\left( au^{-\alpha}-g_{\varepsilon }\left( ., u\right) \right) h_{j}\left( u\right) = \int_{\Omega}\left( au^{1-\alpha}-ug_{\varepsilon}\left( ., u\right) \right) \lt \infty.

    Thus \int_{\Omega}\left\vert \nabla u\right\vert ^{2} < \infty, and so u\in H^{1}\left(\Omega\right). Since u\in C\left(\overline{\Omega}\right) and u = 0 on \partial\Omega, we conclude that u\in H_{0}^{1}\left(\Omega\right). Also, by Remark 2.12, au^{-\alpha }-g_{\varepsilon}\left(., u\right) \in\left(H_{0}^{1}\left(\Omega\right) \right) ^{\prime}. Then, by a density argument, the equality

    \int_{\Omega}\left\langle \nabla u, \nabla\varphi\right\rangle = \int_{\Omega }\left( au^{-\alpha}-g_{\varepsilon}\left( ., u\right) \right) \varphi

    which holds for \varphi\in C_{c}^{\infty}\left(\Omega\right), holds also for any \varphi\in H_{0}^{1}\left(\Omega\right).

    Lemma 2.14. Let a , \alpha, and \left\{ g_{\varepsilon }\right\} _{\varepsilon\in\left(0, 1\right] } be as in Lemma 2.2. Let \varepsilon\in\left(0, 1\right] and let f_{\varepsilon}:\Omega\times\left[0, \infty\right) \rightarrow\mathbb{R} be defined by f_{\varepsilon}\left(., s\right) : = \chi_{\left(0, \infty\right) }\left(s\right) as^{-\alpha}-g_{\varepsilon}\left(., s\right). Let v_{1} and v_{2} be two nonnegative functions in L^{\infty}\left(\Omega\right) \cap H_{0}^{1}\left(\Omega\right) such that f_{\varepsilon}\left(., v_{i}\right) \in L_{loc}^{1}\left(\Omega\right) for i = 1, 2; and let v: = \max\left\{ v_{1}, v_{2}\right\}. Then:

    i) f_{\varepsilon}\left(., v\right) \in L_{loc}^{1}\left(\Omega\right).

    ii) If v_{1} and v_{2} are subsolutions, in the sense of distributions, to problem (1.7), then v is also a subsolution, in the sense of distributions, to the problem

    -\Delta u = \chi_{\left\{ u \gt 0\right\} }au^{-\alpha}-g_{\varepsilon}\left( ., u\right) ~\mathit{\text{in}}~\Omega.

    Proof. Since 0\leq v\in L^{\infty}\left(\Omega\right), from h7) and h8) it follows that g_{\varepsilon}\left(., v\right) \in L^{1}\left(\Omega\right). Similarly, g_{\varepsilon}\left(., v_{1}\right) and g_{\varepsilon }\left(., v_{2}\right) belong to L^{1}\left(\Omega\right) and so, since f_{\varepsilon}\left(., v_{i}\right) \in L_{loc}^{1}\left(\Omega\right) for i = 1, 2; we get that \chi_{\left\{ v_{1} > 0\right\} }av_{1}^{-\alpha} and \chi_{\left\{ v_{2} > 0\right\} }av_{2}^{-\alpha} belong to L_{loc}^{1}\left(\Omega\right). Therefore, to prove i) it suffices to see that \chi_{\left\{ v > 0\right\} }av^{-\alpha }\in L_{loc}^{1}\left(\Omega\right). Note that if x\in\Omega and v\left(x\right) > 0 then either v_{1}\left(x\right) > 0 or v_{2}\left(x\right) > 0. Now, \chi_{\left\{ v > 0\right\} }av^{-\alpha } = av^{-\alpha}\leq av_{1}^{-\alpha} = \chi_{\left\{ v_{1} > 0\right\} } av_{1}^{-\alpha} in \left\{ v_{1} > 0\right\}, and similarly, \chi_{\left\{ v > 0\right\} }av^{-\alpha}\leq\chi_{\left\{ v_{2} > 0\right\} }av_{2}^{-\alpha} in \left\{ v_{2} > 0\right\}. Also, \chi_{\left\{ v > 0\right\} }av^{-\alpha} = 0 in \left\{ v = 0\right\}. Then \chi _{\left\{ v > 0\right\} }av^{-\alpha}\leq\chi_{\left\{ v_{1} > 0\right\} }av_{1}^{-\alpha}+\chi_{\left\{ v_{2} > 0\right\} }av_{2}^{-\alpha} in \Omega and so \chi_{\left\{ v > 0\right\} }av^{-\alpha}\in L_{loc} ^{1}\left(\Omega\right). Thus i) holds.

    To see ii), suppose that -\Delta v_{i}\leq f_{\varepsilon}\left(., v_{i}\right) in D^{\prime}\left(\Omega\right) for i = 1, 2; and let \varphi be a nonnegative function in C_{c}^{\infty}\left(\Omega\right). Let \Omega^{\prime} be a C^{1, 1} subdomain of \Omega, such that supp\left(\varphi\right) \subset\Omega^{\prime} and \overline {\Omega^{\prime}}\subset\Omega. Consider the restrictions (still denoted by v_{1} and v_{2} ) of v_{1} and v_{2} to \Omega^{\prime}. For each i = 1, 2, we have v_{i}\in L^{1}\left(\Omega^{\prime}\right), f_{\varepsilon}\left(., v_{i}\right) \in L^{1}\left(\Omega^{\prime }\right) and -\Delta v_{i}\leq f_{\varepsilon}\left(., v_{i}\right) in D^{\prime}\left(\Omega^{\prime}\right). Thus, from Remark 2.9 v),

    \begin{array}{l} -\Delta v & \leq\chi_{\left\{ v_{1} \gt v_{2}\right\} }f_{\varepsilon}\left( ., v_{1}\right) +\chi_{\left\{ v_{2} \gt v_{1}\right\} }f_{\varepsilon}\left( ., v_{2}\right) +\chi_{\left\{ v_{1} = v_{2}\right\} }\frac{1}{2}\left( f_{\varepsilon}\left( ., v_{1}\right) +f_{\varepsilon}\left( ., v_{2}\right) \right) \\ & = f_{\varepsilon}\left( ., v\right) \text{ in }D^{\prime}\left( \Omega^{\prime}\right) \end{array}

    and then -\int_{\Omega}v\Delta\varphi\leq\int_{\Omega}f_{\varepsilon}\left(., v\right) \varphi.

    Lemma 2.15. Let a , \alpha, and \left\{ g_{\varepsilon }\right\} _{\varepsilon\in\left(0, 1\right] } be as in Lemma 2.2. Then for any \varepsilon\in\left(0, 1\right] there exists a weak solution u_{\varepsilon} , in the sense of Definition 1.1, of problem (1.7), which is maximal in the following sense: If v is a weak solution, in the sense of Definition 1.1, of problem (1.7), then v\leq u_{\varepsilon} a.e. in \Omega. Moreover, u_{\varepsilon} is a solution, in the usual sense of H_{0}^{1}\left(\Omega\right), of problem (1.7).

    Proof. Let z be as given in Remark 2.4, and let \mathcal{S} be the set of the nonidentically zero weak solutions, in the sense of Definition 1.1, of problem (1.7). By Lemma 2.2, \mathcal{S}\neq\varnothing and, for any u\in\mathcal{S} , by Lemma 2.5 we have u\leq z in \Omega and, by Lemma 2.6, there exists a positive constant c such that u\geq cd_{\Omega } in \Omega. Then 0 < \int_{\Omega}u\leq\int_{\Omega}z < \infty for any u\in\mathcal{S}. Let \beta: = \sup\left\{ \int_{\Omega}u:u\in\mathcal{S} \right\}. Thus 0 < \beta < \infty. Let \left\{ u_{k}\right\} _{k\in\mathbb{N}}\subset\mathcal{S} be a sequence such that \lim _{k\rightarrow\infty}\int_{\Omega}u_{k} = \beta. For k\in\mathbb{N} , let w_{k}: = \max\left\{ u_{j}:1\leq j\leq k\right\}. Thus \left\{ w_{k}\right\} _{k\in\mathbb{N}} is a nondecreasing sequence in H_{0} ^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right), and a repeated use of Lemma 2.14 gives that each w_{k} is a subsolution, in the sense of D^{\prime}\left(\Omega\right) , of the problem

    \begin{equation} -\Delta u = au^{-\alpha}-g_{\varepsilon}\left( ., u\right) \text{ in }\Omega. \end{equation} (2.9)

    Since w_{k}\in L^{\infty}\left(\Omega\right) and w_{k}\geq u_{1}\geq c_{1}d_{\Omega} a.e. in \Omega, Remark 2.12 gives that aw_{k}^{-\alpha}-g_{\varepsilon}\left(., w_{k}\right) \in\left(H_{0} ^{1}\left(\Omega\right) \right) ^{\prime}. Then, by a density argument, the inequality

    \begin{equation} \int_{\Omega}\left\langle \nabla w_{k}, \nabla\varphi\right\rangle \leq \int_{\Omega}\left( aw_{k}^{-\alpha}-g_{\varepsilon}\left( ., w_{k}\right) \right) \varphi, \end{equation} (2.10)

    which holds for \varphi\in C_{c}^{\infty}\left(\Omega\right), holds also for any \varphi\in H_{0}^{1}\left(\Omega\right), i.e., w_{k} is a subsolution, in the usual sense of H_{0}^{1}\left(\Omega\right), of problem (2.9)

    Note that \left\{ \int_{\left\{ a > 0\right\} }aw_{k}^{1-\alpha}\right\} _{k\in\mathbb{N}} is bounded. Indeed, since u_{k}\leq z a.e. in \Omega for any k\in \mathbb{N} , we have w_{k}\leq z a.e. in \Omega for all k, and so \int_{\left\{ a > 0\right\} }aw_{k}^{1-\alpha}\leq\int_{\Omega}az^{1-\alpha } < \infty. Moreover, \left\{ w_{k}\right\} _{k\in\mathbb{N}} is bounded in H_{0}^{1}\left(\Omega\right). In fact, taking w_{k} as a test function in (2.10) we get, for any k\in\mathbb{N},

    \begin{equation} \int_{\Omega}\left\vert \nabla w_{k}\right\vert ^{2}+\int_{\Omega }g_{\varepsilon}\left( ., w_{k}\right) w_{k}\leq\int_{\left\{ a \gt 0\right\} }aw_{k}^{1-\alpha} \end{equation} (2.11)

    Then, after pass to a subsequence if necessary, we can assume that there exists w\in H_{0}^{1}\left(\Omega\right) such that \left\{ w_{k}\right\} _{k\in\mathbb{N}} converges in L^{2}\left(\Omega\right) and a.e. in \Omega to w; and \left\{ \nabla w_{k}\right\} _{k\in\mathbb{N}} converges weakly in L^{2}\left(\Omega, \mathbb{R} ^{n}\right) to \nabla w. Let us show that w is a subsolution, in the sense of distributions of problem (2.9). Let \varphi be a nonnegative function in C_{c}^{\infty}\left(\Omega\right) and let k\in\mathbb{N}. Since w_{k} is a subsolution, in the sense of distributions, of (2.9), we have

    \begin{equation} \int_{\Omega}\left\langle \nabla w_{k}, \nabla\varphi\right\rangle +\int_{\Omega}g_{\varepsilon}\left( ., w_{k}\right) \varphi\leq\int_{\Omega }aw_{k}^{-\alpha}\varphi. \end{equation} (2.12)

    Since \left\{ \nabla w_{k}\right\} _{k\in\mathbb{N}} converges weakly in L^{2}\left(\Omega, \mathbb{R}^{n}\right) to \nabla w, we have

    \lim\limits_{k\rightarrow\infty}\int_{\Omega}\left\langle \nabla w_{k}, \nabla \varphi\right\rangle = \int_{\Omega}\left\langle \nabla w, \nabla\varphi \right\rangle .

    Also, since \left\{ g_{\varepsilon}\left(., w_{k}\right) \varphi\right\} _{k\in\mathbb{N}} converges to g_{\varepsilon}\left(., w\right) \varphi a.e. in \Omega, and

    \left\vert g_{\varepsilon}\left( ., w_{k}\right) \varphi\right\vert \leq \sup\limits_{s\in\left[ 0, \left\Vert z\right\Vert _{\infty}\right] }\left( s^{-1}g_{\varepsilon}\left( ., s\right) \right) w_{k}\left\vert \varphi\right\vert \in L^{1}\left( \Omega\right) ,

    the Lebesgue dominated convergence theorem gives

    \lim\limits_{k\rightarrow\infty}\int_{\Omega}g_{\varepsilon}\left( ., w_{k}\right) \varphi = \int_{\Omega}g_{\varepsilon}\left( ., w\right) \varphi.

    On the other hand, \left\{ aw_{k}^{-\alpha}\varphi\right\} _{k\in \mathbb{N}} converges to aw^{-\alpha}\varphi a.e. in \Omega; and w_{k}\geq u_{1}\geq cd_{\Omega} a.e. in \Omega , and so \left\vert aw_{k}^{-\alpha}\varphi\right\vert \leq c^{-\alpha}ad_{\Omega}^{1-\alpha }\left\vert d_{\Omega}^{-1}\varphi\right\vert a.e. in \Omega; and, since d_{\Omega}^{1-\alpha}\in L^{\infty}\left(\Omega\right), the Hardy inequality gives that ad_{\Omega}^{1-\alpha}\left\vert d_{\Omega}^{-1} \varphi\right\vert \in L^{1}\left(\Omega\right). Then, by the Lebesgue dominated convergence theorem, \lim_{k\rightarrow\infty}\int_{\Omega} aw_{k}^{-\alpha}\varphi = \int_{\Omega}aw^{-\alpha}\varphi < \infty. Hence, from (2.12),

    \int_{\Omega}\left\langle \nabla w, \nabla\varphi\right\rangle +\int_{\Omega }g_{\varepsilon}\left( ., w\right) \varphi\leq\int_{\Omega}aw^{-\alpha }\varphi,

    and so w is a subsolution, in the sense of distributions to problem (2.9). Note that z is a supersolution, in the sense of distributions, of problem (2.9) and that w\leq z a.e. in \Omega (because u_{k}\leq z for all k\in\mathbb{N} ). Also, for some positive constant c and for any k, w\geq w_{k}\geq u_{1}\geq cd_{\Omega} a.e. in \Omega. Then there exists a positive constant c^{\prime} such that

    \sup\limits_{s\in\left[ w\left( x\right) , z\left( x\right) \right] }\left( \chi_{\left\{ s \gt 0\right\} }a\left( x\right) s^{-\alpha}-g_{\varepsilon }\left( x, s\right) \right) \leq c^{\prime}d_{\Omega}^{-\alpha}\text{ for }a.e\text{ }x\in\Omega

    and so, by Remark 2.10, there exists a solution u_{\varepsilon }\in W_{loc}^{1, 2}\left(\Omega\right) , in the sense of distributions, of (2.9) such that w\leq u_{\varepsilon}\leq z a.e. a.e. in \Omega. Therefore, by Remark 2.4, cd_{\Omega }\leq u_{\varepsilon}\leq c^{\prime}d_{\Omega}^{\tau} a.e. in \Omega, with c, c^{\prime} and \tau positive constants. Then, by Lemma 2.13, u_{\varepsilon}\in H_{0}^{1}\left(\Omega\right) \cap C\left(\overline{\Omega}\right) and u_{\varepsilon} is a weak solution, in the sense of Definition 1.1, of problem (1.7). Also, u_{\varepsilon}\geq w\geq w_{k}\geq u_{k} a.e. in \Omega for any k\in\mathbb{N}, and so \int_{\Omega }u_{\varepsilon}\geq\beta which, by the definition of \beta, implies \int_{\Omega}u_{\varepsilon} = \beta.

    Let us show that u_{\varepsilon } is the maximal solution of problem (1.7), in the sense required by the lemma. Suppose that w^{\ast} is a nonidentically zero weak solution, in the sense of Definition 1.1, of (1.7). By Lemmas 2.5, 2.7 and 2.6, w^{\ast}\leq z in \Omega, w^{\ast}\in C\left(\overline{\Omega}\right) and w^{\ast}\geq cd_{\Omega} a.e. in \Omega with c a positive constant c. Let w^{\ast\ast}: = \max\left\{ u_{\varepsilon}, w^{\ast}\right\}. Thus w^{\ast\ast} is a subsolution, in the sense of distributions, of problem (2.9), Remark 2.10 applies to obtain a solution \widetilde{w} , in the sense of distributions, of problem (1.7), such that w^{\ast\ast}\leq\widetilde{w}\leq z, and Lemma 2.13 applies to obtain that \widetilde{w}\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right) and that \widetilde{w} is a weak solution, in the sense of Definition 1.1, to problem (1.7). Then \int_{\Omega}\widetilde{w}\leq\beta. Since u_{\varepsilon}\leq w^{\ast\ast}\leq\widetilde{w} we get \beta = \int_{\Omega}u_{\varepsilon}\leq\int_{\Omega}w^{\ast\ast}\leq\int_{\Omega }\widetilde{w}\leq\beta, and so u_{\varepsilon} = w^{\ast\ast}. Thus u_{\varepsilon}\geq w^{\ast}.

    For \varepsilon\in\left(0, 1\right], let u_{\varepsilon} be the maximal weak solution to problem (1.7) given by Lemma 2.15.

    Lemma 2.16. Let a , \alpha, and \left\{ g_{\varepsilon }\right\} _{\varepsilon\in\left(0, 1\right] } be as in Lemma 2.2. Then the map \varepsilon\rightarrow u_{\varepsilon} is nondecreasing on \left(0, 1\right].

    Proof. For 0 < \varepsilon < \eta we have, in the sense of definition 1.1,

    -\Delta u_{\varepsilon} = au_{\varepsilon}^{-\alpha}-g_{\varepsilon}\left( ., u_{\varepsilon}\right) \leq au_{\varepsilon}^{-\alpha}-g_{\eta}\left( ., u_{\varepsilon}\right) \text{ in }\Omega,

    and so u_{\varepsilon}\in H_{0}^{1}\left(\Omega\right) \cap C\left(\overline{\Omega}\right) is a subsolution, in the sense of distributions, to the problem

    \begin{equation} -\Delta u = au^{-\alpha}-g_{\eta}\left( ., u\right) \text{ in }\Omega. \end{equation} (2.13)

    Let z be as in Remark 2.4. Thus z is a supersolution, in the sense of distributions, of problem (2.9), and z\leq cd_{\Omega}^{\tau} a.e. in \Omega, with c and \tau positive constants c. Taking into account that, for some positive constant c, u_{\varepsilon}\geq cd_{\Omega} a.e. in \Omega, Remark 2.10 applies, as before, to obtain a weak solution, in the sense of distributions, \widetilde{u}_{\eta}\in W_{loc}^{1, 2}\left(\Omega\right) of (2.13) such that u_{\varepsilon}\leq\widetilde{u}_{\eta}\leq z. Now, Lemma 2.13 gives that \widetilde{u}_{\eta}\in H_{0}^{1}\left(\Omega\right) \cap C\left(\overline{\Omega}\right) and that \widetilde{u}_{\eta} is a weak solution, in the sense of Definition 1.1, of problem (2.13), which implies \widetilde{u}_{\eta}\leq u_{\eta}. Thus u_{\varepsilon}\leq u_{\eta}.

    Lemma 2.17. Let a , \alpha, and \left\{ g_{\varepsilon }\right\} _{\varepsilon\in\left(0, 1\right] } be as in Lemma 2.2. Then \left\{ u_{\varepsilon}\right\} _{\varepsilon \in\left(0, 1\right] } is bounded in H_{0}^{1}\left(\Omega\right).

    Proof. Let z be as in Remark 2.4. by Lemma 2.5 u_{\varepsilon}\leq z in \Omega and so, since 0 < \alpha\leq1, we have \int_{\left\{ a > 0\right\} }au_{\varepsilon}^{1-\alpha}\leq\int_{\Omega }az^{1-\alpha} < \infty. By taking u_{\varepsilon} as a test function in (1.7) we get, for any \varepsilon\in\left(0, 1\right],

    \int_{\Omega}\left\vert \nabla u_{\varepsilon}\right\vert ^{2}+\int_{\Omega }u_{\varepsilon}g_{\varepsilon}\left( ., u_{\varepsilon}\right) = \int_{\left\{ a \gt 0\right\} }au_{\varepsilon}^{1-\alpha}.

    Then \int_{\Omega}\left\vert \nabla u_{\varepsilon}\right\vert ^{2}\leq \int_{\Omega}az^{1-\alpha} < \infty.

    Lemma 2.18. Let a , \alpha, and \left\{ g_{\varepsilon }\right\} _{\varepsilon\in\left(0, 1\right] } be as in Lemma 2.2. Let \boldsymbol{u}: = \lim_{\varepsilon\rightarrow0^{+} }u_{\varepsilon}. Then:

    i) \boldsymbol{u}\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right).

    ii) \boldsymbol{u} > 0 a.e. in \left\{ a > 0\right\}.

    iii) \chi_{\left\{ \boldsymbol{u} > 0\right\} }a\boldsymbol{u}^{-\alpha}\varphi\in L^{1}\left(\Omega\right) for any \varphi\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right).

    iv) If \left\{ \varepsilon_{j}\right\} _{j\in\mathbb{N}} is a decreasing sequence in \left(0, 1\right] such that \lim_{j\rightarrow\infty}\varepsilon _{j} = 0 then \lim_{j\rightarrow\infty}\int_{\left\{ a > 0\right\} }au_{\varepsilon_{j}}^{-\alpha}\varphi = \int_{\left\{ a > 0\right\} }a\boldsymbol{u}^{-\alpha}\varphi for any \varphi\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right).

    Proof. To see i), consider a nonincreasing sequence \left\{ \varepsilon _{j}\right\} _{j\in\mathbb{N}}\subset\left(0, 1\right] such that \lim_{j\rightarrow\infty}\varepsilon_{j} = 0. By Lemma 2.17, \left\{ u_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} is bounded in H_{0}^{1}\left(\Omega\right) and so, after pass to a subsequence if necessary, \left\{ u_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} converges, strongly in L^{2}\left(\Omega\right), and a.e. in \Omega, to some \widetilde{u}\in H_{0}^{1}\left(\Omega\right), and \left\{ \nabla u_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} converges weakly in L^{2}\left(\Omega, \mathbb{R}^{n}\right) to \nabla\widetilde{u}. Since u_{\varepsilon_{j}} converges to \boldsymbol{u} a.e. in \Omega we have \boldsymbol{u} = \widetilde{u} a.e. in \Omega, and so \boldsymbol{u}\in H_{0}^{1}\left(\Omega\right). Also, 0\leq\boldsymbol{u}\leq u_{\varepsilon_{1}}\in L^{\infty}\left(\Omega\right) and then \boldsymbol{u}\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right). Thus i) holds.

    To see ii) and iii), consider an arbitrary nonnegative function \varphi\in H_{0} ^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right). From (1.7) we have, for each j,

    \begin{equation} \int_{\Omega}\left\langle \nabla u_{\varepsilon_{j}}, \nabla\varphi \right\rangle +\int_{\Omega}g_{\varepsilon_{j}}\left( ., u_{\varepsilon_{j} }\right) \varphi = \int_{\Omega}au_{\varepsilon_{j}}^{-\alpha}\varphi. \end{equation} (2.14)

    \left\{ \nabla u_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} converges weakly in L^{2}\left(\Omega, \mathbb{R}^{n}\right) to \nabla \boldsymbol{u}, and thus

    \lim\limits_{j\rightarrow\infty}\int_{\Omega}\left\langle \nabla u_{\varepsilon_{j} }, \nabla\varphi\right\rangle = \int_{\Omega}\left\langle \nabla\boldsymbol{u} , \nabla\varphi\right\rangle .

    By Lemma 2.16, \left\{ au_{\varepsilon_{j}}^{-\alpha} \varphi\right\} _{j\in\mathbb{N}} is nondecreasing, then, by the monotone convergence theorem, \lim_{j\rightarrow\infty}\int_{\Omega}au_{\varepsilon _{j}}^{-\alpha}\varphi = \lim_{j\rightarrow\infty}\int_{\left\{ a > 0\right\} }au_{\varepsilon_{j}}^{-\alpha}\varphi = \int_{\left\{ a > 0\right\} }a\boldsymbol{u}^{-\alpha}\varphi.

    Let z be as in Lemma 2.5. Then u_{\varepsilon_{j}}\leq z in \Omega and so, taking into account h4), \int_{\Omega}g_{\varepsilon_{j}}\left(., u_{\varepsilon_{j}}\right) \varphi\leq\int_{\Omega}\sup_{0\leq s\leq\left\Vert z\right\Vert _{\infty}}g\left(., s\right) \varphi < \infty. Thus

    \begin{array}{l} \int_{\left\{ a \gt 0\right\} }a\boldsymbol{u}^{-\alpha}\varphi & = \lim\limits_{j\rightarrow\infty}\int_{\Omega}au_{\varepsilon_{j}}^{-\alpha} \varphi = \lim\limits_{j\rightarrow\infty}\left( \int_{\Omega}\left\langle \nabla u_{\varepsilon_{j}}, \nabla\varphi\right\rangle +\int_{\Omega}g_{\varepsilon _{j}}\left( ., u_{\varepsilon_{j}}\right) \varphi\right) \\ & \leq\overline{\lim}_{j\rightarrow\infty}\int_{\Omega}\left\langle \nabla u_{\varepsilon_{j}}, \nabla\varphi\right\rangle +\overline{\lim}_{j\rightarrow \infty}\int_{\Omega}g_{\varepsilon_{j}}\left( ., u_{\varepsilon_{j}}\right) \varphi\\ & \leq\int_{\Omega}\left\langle \nabla\boldsymbol{u}, \nabla\varphi \right\rangle +\int_{\Omega}\sup\limits_{0\leq s\leq\left\Vert z\right\Vert _{\infty }}g\left( ., s\right) \varphi \lt \infty. \end{array}

    Therefore \int_{\left\{ a > 0\right\} }a\boldsymbol{u}^{-\alpha} \varphi < \infty . Since this holds for any nonnegative \varphi\in H_{0} ^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right), we conclude that \boldsymbol{u} > 0 a.e. in \left\{ a > 0\right\}. Thus ii) holds. Now,

    \int_{\Omega}\chi_{\left\{ \boldsymbol{u} \gt 0\right\} }a\boldsymbol{u} ^{-\alpha}\varphi = \int_{\left\{ a \gt 0\right\} }\chi_{\left\{ \boldsymbol{u} \gt 0\right\} }a\boldsymbol{u}^{-\alpha}\varphi = \int_{\left\{ a \gt 0\right\} }a\boldsymbol{u}^{-\alpha}\varphi \lt \infty,

    and then iii) holds for any nonnegative \varphi\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right). Hence, by writing \varphi = \varphi^{+}-\varphi^{-}, iii) holds also for any \varphi\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right). Finally, observe that, in the case when \varphi\geq0, the monotone convergence theorem gives iv). Then, by writing \varphi = \varphi^{+}-\varphi^{-}, iv), holds also for an arbitrary \varphi\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right).

    Remark 2.19. Assume that a satisfies the conditions h1), h2) and also the condition h6) of Theorem 1.4; and let \Omega^{+} be as in h6). Taking into account h6), Remark 2.4 (applied in each connected component of \Omega^{+} ) gives that the problem

    \begin{equation} \left\{ \begin{array} [c]{c} -\Delta\zeta = a\zeta^{-\alpha}\text{ in }\Omega^{+}, \\ \zeta = 0\text{ on }\partial\Omega^{+}, \\ \zeta \gt 0\text{ in }\Omega^{+}, \end{array} \right. \end{equation} (2.15)

    has a unique weak solution, in the sense of Definition 1.1, \zeta\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right), and that it satisfies:

    ⅰ) \zeta\in C\left(\overline{\Omega^{+}}\right).

    ⅱ) There exists a positive constant c such that \zeta\geq cd_{\Omega^{+}} in \Omega^{+}.

    ⅲ) \zeta is also a solution of problem (2.15) in the usual sense of H_{0}^{1}\left(\Omega^{+}\right), i.e., a\zeta^{-\alpha }\varphi\in L^{1}\left(\Omega\right) and \int_{\Omega}\left\langle \nabla\zeta, \nabla\varphi\right\rangle = \int_{\Omega}a\zeta^{-\alpha}\varphi for any \varphi\in H_{0}^{1}\left(\Omega^{+}\right).

    Lemma 2.20. Assume that a and g satisfy the conditions h1)-h4) and also the condition h6) of Theorem 1.4. Let \Omega^{+} and A^{+} be as in the statement of Theorem 1.4 and assume, in addition, that g\left(., s\right) = 0 a.e. in A^{+} for any s\geq0. Let \zeta be as in Remark 2.19, let \varepsilon\in\left(0, 1\right], and let u\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right) be a weak solution, in the sense of Definition 1.1, of problem (1.5). Then u\geq\zeta in \Omega^{+}.

    Proof. By Remark 2.19 i), \zeta\in C\left(\overline {\Omega^{+}}\right) and, by Lemma 2.7, u\in C\left(\overline{\Omega}\right). Also, since g\left(., s\right) = 0 a.e. in \Omega^{+} for s\geq0, we have -\Delta\left(u-\zeta\right) = a\left(u^{-\alpha}-\zeta^{-\alpha}\right) \geq0 in D^{\prime}\left(\Omega ^{+}\right). We claim that u\geq\zeta in \Omega^{+}. To prove this fact we proceed by the way of contradiction: Let U: = \left\{ x\in\Omega ^{+}:u\left(x\right) < \zeta\left(x\right) \right\} and suppose that U\neq\varnothing. Then U is an open subset of \Omega^{+} and -\Delta\left(u-\zeta\right) = a\left(u^{-\alpha}-\zeta^{-\alpha}\right) \geq0 in D^{\prime}\left(U\right). Notice that u-\zeta\geq0 on \partial U. In fact, if u\left(x\right) < \zeta\left(x\right) for some x\in\partial U we would have, either x\in\Omega^{+} or x\in \partial\Omega^{+}; if x\in\Omega^{+} then, since u and \zeta are continuous on \Omega^{+}, we would have u < \zeta on some ball around x, contradicting the fact that x\in\partial U, and if x\in\partial\Omega^{+}, then u\left(x\right) \geq0 = \zeta\left(x\right) contradicting our assumption that u\left(x\right) < \zeta\left(x\right). Then U = \varnothing and so u\geq\zeta in \Omega^{+}; and then, by continuity, also u\geq\zeta on \partial\Omega^{+}. Therefore, from the weak maximum principle, u\geq\zeta in \Omega^{+}.

    Observe that if g:\Omega\times\left[0, \infty\right) \rightarrow\mathbb{R} satisfies the conditions h3) and h4) stated at the introduction, and if, for \varepsilon\in\left(0, 1\right], g_{\varepsilon}:\Omega\times\left[0, \infty\right) \rightarrow\mathbb{R} is defined by

    \begin{equation} g_{\varepsilon}\left( ., s\right) : = s\left( s+\varepsilon\right) ^{-1}g\left( ., s\right) , \end{equation} (3.1)

    then, for any s > 0, g\left(., s\right) = \lim_{\varepsilon\rightarrow0^{+} }g_{\varepsilon}\left(., s\right) a.e. in \Omega; and the family \left\{ g_{\varepsilon}\right\} _{\varepsilon\in\left(0, 1\right] } satisfies the conditions h7)-h9). Therefore all the results of the Section 2 hold for such a family \left\{ g_{\varepsilon}\right\} _{\varepsilon\in\left(0, 1\right] }.

    Lemma 3.1. Let a:\Omega \rightarrow\mathbb{R} and g:\Omega\times\left[0, \infty\right) \rightarrow\mathbb{R} satisfying the conditions h1)-h4) and, for \varepsilon \in\left(0, 1\right], let g_{\varepsilon}:\Omega\times\left[0, \infty\right) \rightarrow\mathbb{R} be defined by (3.1), let u_{\varepsilon} be as given by Lemma 2.15, and let \boldsymbol{u}: = \lim_{\varepsilon\rightarrow0^{+} }u_{\varepsilon}. Let \left\{ \varepsilon_{j}\right\} _{j\in\mathbb{N} }\subset\left(0, 1\right] be a nonincreasing sequence such that \lim_{j\rightarrow\infty}\varepsilon_{j} = 0 and, for j\in\mathbb{N} , let u_{\varepsilon_{j}} be as given by Lemma 2.15. Let \theta _{j}: = u_{\varepsilon_{j}}\left(u_{\varepsilon_{j}}+\varepsilon_{j}\right) ^{-1} . Then there exist a nonnegative function \theta^{\ast}\in L^{\infty }\left(\Omega\right) and a sequence \left\{ w_{m}\right\} _{m\in\mathbb{N}}\subset L^{2}\left(\Omega, \mathbb{R}^{n}\right) \times L^{2}\left(\Omega\right) with the following properties:

    i) for each m\in\mathbb{N}, w_{m} = \sum_{l\in\mathcal{F}_{m}}\gamma_{l, m}\left(\nabla u_{\varepsilon_{l}}, \theta_{l}g\left(., u_{\varepsilon_{l}}\right) \right), where each \mathcal{F}_{m} is a finite subset of \mathbb{N} satisfying \lim_{m\rightarrow\infty}\min\mathcal{F}_{m} = \infty; \ \gamma _{l, m}\in\left[0, 1\right] for any m\in\mathbb{N} and l\in \mathcal{F}_{m}; and \sum_{l\in\mathcal{F}_{m}}\gamma_{l, m} = 1 for any m\in\mathbb{N}.

    ii) \left\{ w_{m}\right\} _{m\in\mathbb{N}} converges strongly in L^{2}\left(\Omega, \mathbb{R}^{n}\right) \times L^{2}\left(\Omega\right) to \left(\nabla\mathbf{u}, \theta^{\ast }\right).

    iii) \lim_{m\rightarrow\infty}\sum_{l\in\mathcal{F}_{m} }\gamma_{l, m}\theta_{l}g\left(., u_{\varepsilon_{l}}\right) = \theta^{\ast} a.e. in \Omega.

    iv) \theta^{\ast} = \chi_{\left\{ \mathbf{u} > 0\right\} }g\left(., \mathbf{u}\right) a.e. in \left\{ \mathbf{u} > 0\right\}.

    Proof. By Lemma 2.17 \left\{ u_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} is bounded in H_{0}^{1}\left(\Omega\right). Then, after pass to a subsequence if necessary, we can assume that \left\{ u_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} converges to \mathbf{u} in L^{2}\left(\Omega\right) and that \left\{ \nabla u_{\varepsilon_{j} }\right\} _{j\in\mathbb{N}} converges weakly to \nabla\mathbf{u} in L^{2}\left(\Omega, \mathbb{R}^{n}\right). Moreover, by Lemma 2.5, u_{\varepsilon_{j}}\leq z a.e. in \Omega for all j, and so \mathbf{u}\leq z a.e. in \Omega. Since, for any j, 0 < \theta_{j} < 1 a.e. in \Omega, and, by h3) and h4), 0\leq g\left(., u_{\varepsilon_{j}}\right) \leq\sup_{s\in\left[0, \left\Vert z\right\Vert _{\infty}\right] }g\left(., s\right) \in L^{\infty}\left(\Omega\right), we have that \left\{ \theta_{j}g\left(., u_{\varepsilon_{j}}\right) \right\} _{j\in\mathbb{N}} is bounded in L^{2}\left(\Omega\right). Thus, after pass to a further subsequence, we can assume that \left\{ \theta_{j}g\left(., u_{\varepsilon_{j}}\right) \right\} _{j\in\mathbb{N}} is weakly convergent in L^{2}\left(\Omega\right) to a function \theta^{\ast}\in L^{2}\left(\Omega\right), and that \left\{ \nabla u_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} is weakly convergent in L^{2}\left(\Omega, \mathbb{R}^{n}\right) to \nabla\mathbf{u}. Then \left\{ \left(\nabla u_{\varepsilon_{j}}, \theta_{j}g\left(., u_{\varepsilon_{j}}\right) \right) \right\} _{j\in\mathbb{N}} is weakly convergent to \left(\nabla\mathbf{u}, \theta^{\ast}\right) in L^{2}\left(\Omega, \mathbb{R}^{n}\right) \times L^{2}\left(\Omega\right). Thus (see e.g., [33] Theorem 3.13) there exists a sequence \left\{ w_{m}\right\} _{m\in\mathbb{N}} of the form w_{m} = \sum_{l\in\mathcal{F}_{m}}\gamma_{l, m}\left(\nabla u_{\varepsilon_{l}}, \theta_{l}g\left(., u_{\varepsilon_{l}}\right) \right), where each \mathcal{F}_{m} is a finite subset of \mathbb{N} such that \lim_{m\rightarrow\infty}\min\mathcal{F}_{m} = \infty, \gamma_{l, m}\in\left[0, 1\right] for any m\in\mathbb{N} and l\in\mathcal{F}_{m}, for each m, \sum_{l\in\mathcal{F}_{m}}\gamma_{l, m} = 1 and such that \left\{ w_{m}\right\} _{m\in\mathbb{N}} converges strongly in L^{2}\left(\Omega, \mathbb{R}^{n}\right) \times L^{2}\left(\Omega\right) to \left(\nabla\mathbf{u}, \theta^{\ast}\right). Then i) and ii) hold, and \left\{ \sum_{l\in\mathcal{F}_{m}}\gamma_{l, m}\theta_{l}g\left(., u_{\varepsilon_{l}}\right) \right\} _{m\in\mathbb{N}} converges in L^{2}\left(\Omega\right) to \theta^{\ast}. Therefore, after pass to a further subsequence, we can assume that \lim_{m\rightarrow\infty}\sum _{l\in\mathcal{F}_{m}}\gamma_{l, m}\theta_{l}g\left(., \boldsymbol{u} _{\varepsilon_{l}}\right) = \theta^{\ast} a.e. in \Omega and, since \left\{ \theta_{j}g\left(., u_{\varepsilon_{j}}\right) \right\} _{j\in\mathbb{N}} is bounded in L^{\infty}\left(\Omega\right), we have that \theta^{\ast}\in L^{\infty}\left(\Omega\right). Thus iii) holds. Also \left\{ \theta_{j}\right\} _{j\in\mathbb{N}} and \left\{ g\left(., u_{\varepsilon_{j}}\right) \right\} _{j\in\mathbb{N}} converge, a.e. in \left\{ \mathbf{u} > 0\right\} , to \chi_{\left\{ \mathbf{u} > 0\right\} } and to g\left(., \mathbf{u}\right) respectively, and then iv) follows from iii).

    Proof of Theorem 1.2. Let \left\{ \varepsilon_{j}\right\} _{j\in\mathbb{N}}\subset\left(0, 1\right) be a nonincreasing sequence such that \lim_{j\rightarrow\infty}\varepsilon_{j} = 0, let \theta^{\ast} and \left\{ w_{m}\right\} _{m\in\mathbb{N}}\subset L^{2}\left(\Omega, \mathbb{R}^{n}\right) \times L^{2}\left(\Omega\right) be as given by Lemma 3.1, and let \varphi\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right). Assume temporarily that \varphi \geq0 in \Omega. Then \left\{ \sum_{l\in\mathcal{F}_{m}}\gamma _{l, m}\theta_{l}g\left(., u_{\varepsilon_{l}}\right) \varphi\right\} _{m\in\mathbb{N}} and \left\{ \sum_{l\in\mathcal{F}_{m}}\gamma _{l, m}\left\langle \nabla u_{\varepsilon_{l}}, \nabla\varphi\right\rangle \right\} _{m\in\mathbb{N}} converge in L^{1}\left(\Omega\right) to \theta^{\ast}\varphi and \left\langle \nabla\mathbf{u}, \nabla \varphi\right\rangle respectively. Thus

    \begin{align} \lim\limits_{m\rightarrow\infty}\int_{\Omega}\sum\limits_{l\in\mathcal{F}_{m}}\gamma _{l, m}\theta_{l}g\left( ., u_{\varepsilon_{l}}\right) \varphi & = \int_{\Omega}\theta^{\ast}\varphi, \end{align} (3.2)
    \begin{align} \lim\limits_{m\rightarrow\infty}\int_{\Omega}\sum\limits_{l\in\mathcal{F}_{m}}\gamma _{l, m}\left\langle \nabla u_{\varepsilon_{l}}, \nabla\varphi\right\rangle & = \int_{\Omega}\left\langle \nabla\mathbf{u}, \nabla\varphi\right\rangle \end{align} (3.3)

    and both limits are finite. Since \left\{ u_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} is nonincreasing we have, for m\in\mathbb{N} and l\in\mathcal{F}_{m},

    \begin{equation} au_{\varepsilon_{L_{m}}}^{-\alpha}\varphi\leq a\sum\limits_{l\in\mathcal{F}_{m} }\gamma_{l, m}u_{\varepsilon_{l}}^{-\alpha}\varphi\leq au_{\varepsilon _{L_{m}^{\ast}}}^{-\alpha}\varphi, \end{equation} (3.4)

    where L_{m}: = \max\mathcal{F}_{m} and L_{m}^{\ast}: = \min\mathcal{F}_{m}. Also, by the monotone convergence theorem,

    \begin{equation} \lim\limits_{j\rightarrow\infty}\int_{\Omega}au_{\varepsilon_{j}}^{-\alpha} \varphi = \lim\limits_{j\rightarrow\infty}\int_{\left\{ a \gt 0\right\} }au_{\varepsilon _{j}}^{-\alpha}\varphi = \int_{\left\{ a \gt 0\right\} }a\mathbf{u}^{-\alpha }\varphi = \int_{\Omega}\chi_{\left\{ \mathbf{u} \gt 0\right\} }a\mathbf{u} ^{-\alpha}\varphi, \end{equation} (3.5)

    the last equality because, by Lemma 2.18, \mathbf{u} > 0 a.e. in \left\{ a > 0\right\}. Then, since \lim_{m\rightarrow\infty}L_{m}^{\ast } = \infty, (3.4) and (3.5) give

    \begin{equation} \lim\limits_{m\rightarrow\infty}\int_{\left\{ a \gt 0\right\} }a\sum\limits_{l\in \mathcal{F}_{m}}\gamma_{l, m}\boldsymbol{u}_{\varepsilon_{l}}^{-\alpha} \varphi = \int_{\Omega}\chi_{\left\{ \mathbf{u} \gt 0\right\} }a\mathbf{u} ^{-\alpha}\varphi. \end{equation} (3.6)

    (notice that, by Lemma 2.18, \int_{\Omega}\chi_{\left\{ \mathbf{u} > 0\right\} }a\mathbf{u}^{-\alpha}\varphi < \infty ). Since \theta_{l}g\left(., u_{\varepsilon_{l}}\right) = g_{\varepsilon_{l}}\left(., u_{\varepsilon_{l}}\right) we have, for any m\in\mathbb{N} , and in the sense of definition 1.1,

    \begin{equation} \left\{ \begin{array} [c]{c} -\Delta\left( \sum _{l\in\mathcal{F}_{m}}\gamma_{l, m}u_{\varepsilon_{l} }\right) \\ = a\sum _{l\in\mathcal{F}_{m}}\gamma_{l, m}u_{\varepsilon_{l}}^{-\alpha} -\sum\limits_{l\in\mathcal{F}_{m}}\gamma_{l, m}\theta_{l}g\left( ., u_{\varepsilon _{l}}\right) \text{ in }\Omega, \\ \sum _{l\in\mathcal{F}_{m}}\gamma_{l, m}u_{\varepsilon_{l}} = 0\text{ on } \partial\Omega \end{array} \right. \end{equation} (3.7)

    and so

    \begin{align} & \int_{\Omega}\sum\limits_{l\in\mathcal{F}_{m}}\gamma_{l, m}\left\langle \nabla u_{\varepsilon_{l}}, \nabla\varphi\right\rangle \\ & = \int_{\Omega}a\sum\limits_{l\in\mathcal{F}_{m}}\gamma_{l, m}u_{\varepsilon_{l} }^{-\alpha}\varphi-\int_{\Omega}\sum\limits_{l\in\mathcal{F}_{m}}\gamma_{l, m} \theta_{l}g\left( ., u_{\varepsilon_{l}}\right) \varphi. \end{align} (3.8)

    Taking the limit as m\rightarrow\infty in (3.8), and using (3.2), (3.3), (3.6) and recalling that, by Lemma 3.1 iv), \theta^{\ast} = \chi_{\left\{ \mathbf{u} > 0\right\} }g\left(., \mathbf{u}\right) a.e. in \left\{ \mathbf{u} > 0\right\} , we get that

    \begin{align} \int_{\Omega}\left\langle \nabla\mathbf{u}, \nabla\varphi\right\rangle & = \int_{\Omega}\chi_{\left\{ \mathbf{u} \gt 0\right\} }a\mathbf{u}^{-\alpha }\varphi-\int_{\Omega}\theta^{\ast}\varphi \\ & = \int_{\Omega}\chi_{\left\{ \mathbf{u} \gt 0\right\} }a\mathbf{u}^{-\alpha }\varphi-\int_{\Omega}\chi_{\left\{ \mathbf{u} \gt 0\right\} }g\left( ., \mathbf{u}\right) \varphi-\int_{\left\{ \mathbf{u} = 0\right\} } \theta^{\ast}\varphi. \end{align} (3.9)

    for any nonnegative \varphi\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right), and by writing \varphi = \varphi ^{+}-\varphi^{-} it follows that (3.9) holds also for any \varphi\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right).

    Let \Omega_{0} be as in h3). If \Omega_{0} = \varnothing then \mathbf{u} > 0 a.e. in \Omega (because \mathbf{u} > 0 a.e. in \left\{ a > 0\right\} ) and thus, by (3.9), \mathbf{u} is a solution, in the sense of Definition 1.1, of problem (1.2). Consider now the case when \Omega_{0}\neq\varnothing . We claim that, in this case, \mathbf{u}\in W_{loc}^{2, p}\left(\Omega_{0}\right) for any p\in\left[1, \infty \right). Indeed, let \Omega_{0}^{\prime} be a an arbitrary C^{1, 1} subdomain of \Omega_{0} such that \overline{\Omega_{0}^{\prime}} \subset\Omega_{0}. We have \chi_{\left\{ \mathbf{u} > 0\right\} } a\mathbf{u}^{-\alpha} = 0 on \Omega_{0}, and so, from (3.9), -\Delta\mathbf{u} = -\chi_{\left\{ \mathbf{u} > 0\right\} }g\left(., \mathbf{u}\right) -\theta^{\ast} in D^{\prime}\left(\Omega_{0}\right). Also, the restrictions to \Omega_{0} of \mathbf{u} and \theta^{\ast} belong to L^{\infty}\left(\Omega_{0}\right) and so, from the inner elliptic estimates (as stated e.g., in [20], Theorem 8.24), \mathbf{u}\in W^{2, p}\left(\Omega_{0}^{\prime}\right) . Then \mathbf{u}\in W_{loc}^{2, p}\left(\Omega_{0}\right) for any p\in\left[1, \infty\right). Thus, for any p\in\left[1, \infty\right) , \mathbf{u} is a strong solution in W_{loc}^{2, p}\left(\Omega_{0}\right) of -\Delta\mathbf{u} = -\chi_{\left\{ \mathbf{u} > 0\right\} }g\left(., \mathbf{u}\right) -\theta^{\ast} in \Omega_{0}.

    Taking into account (3.9), in order to complete the proof of the theorem it is enough to see that the set E: = \left\{ \mathbf{u} = 0\right\} \cap\left\{ \theta^{\ast} > 0\right\} has zero measure. Suppose that \left\vert E\right\vert > 0. Since \mathbf{u} > 0 a.e. in \left\{ a > 0\right\}, from h5) it follows that E\subset\Omega_{0}\cup V, for some measurable V\subset\Omega such that \left\vert V\right\vert = 0. Since \left\vert E\right\vert > 0, there exists a subdomain \Omega^{\prime }, with \overline{\Omega^{\prime}}\subset\Omega_{0}, and such that E^{\prime}: = E\cap\Omega^{\prime} has positive measure. Since \mathbf{u} = 0 a.e. in E^{\prime} and \mathbf{u}\in W^{1, p}\left(\Omega^{\prime }\right) we have \nabla\mathbf{u} = 0 a.e. in E^{\prime} (see [20], Lemma 7.7). Thus \frac{\partial\mathbf{u} }{\partial x_{i}} = 0 a.e. in E^{\prime} for each i = 1, 2, ..., n; and since \frac{\partial\mathbf{u}}{\partial x_{i}}\in W^{1, p}\left(\Omega _{0}^{\prime}\right) the same argument gives that also the second order derivatives \frac{\partial^{2}\mathbf{u}}{\partial x_{i}\partial x_{j}} vanish a.e. in E^{\prime}. Then \Delta\mathbf{u} = 0 a.e. in E^{\prime }, which, taking into account that g\left(., \mathbf{u}\right) is nonnegative and \theta^{\ast} > 0 in E^{\prime} , contradicts the fact that -\Delta\mathbf{u} = -\chi_{\left\{ \mathbf{u} > 0\right\} }g\left(., \mathbf{u}\right) -\theta^{\ast} a.e. in \Omega_{0}.

    Proof of Theorem 1.3. Notice that the condition h4') is stronger than h4) and so Theorem 1.2 gives a weak solution \boldsymbol{u} , in the sense of definition 1.1, of problem (1.2) which satisfies \boldsymbol{u} > 0 a.e. in \left\{ a > 0\right\}, and so, since a > 0 a.e. in \Omega, by Lemma 2.18, we have \boldsymbol{u} > 0 a.e. in \Omega. Thus \boldsymbol{u} is a weak solution, in the sense of Definition 1.1, of the problem

    \left\{ \begin{array} [c]{c} -\Delta\boldsymbol{u} = a\boldsymbol{u}^{-\alpha}-g\left( ., \boldsymbol{u} \right) \text{ in }\Omega, \\ \boldsymbol{u} = 0\text{ on }\partial\Omega. \end{array} \right.

    Let a_{0}: = \boldsymbol{u}^{-1}g\left(., \boldsymbol{u}\right). Since g\geq0 and \boldsymbol{u}\in L^{\infty}\left(\Omega\right), h4') gives 0\leq a_{0}\in L^{\infty}\left(\Omega\right). Now, in the sense of Definition 1.1, -\Delta\boldsymbol{u} +a_{0}\boldsymbol{u} = a\boldsymbol{u}^{-\alpha} in \Omega, \boldsymbol{u} = 0 on \partial\Omega, and \boldsymbol{u} > 0 a.e. in \Omega; Then, for some \eta > 0 and some measurable set E\subset\Omega with \left\vert E\right\vert > 0, we have \chi_{\left\{ u > 0\right\} }a\boldsymbol{u} ^{-\alpha}\geq\eta\chi_{E} a.e. in \Omega. Let \psi\in\cap_{1\leq q < \infty}W^{2, , q}\left(\Omega\right) \cap W_{0}^{1, , q}\left(\Omega\right) be the solution of the problem -\Delta\psi+a_{0}\psi = \eta\chi_{E} in \Omega, \psi = 0 on \partial\Omega. By the Hopf maximum principle (as stated, e.g., in [34], Theorem 1.1) there exists a positive constant c_{1} such that \psi\geq c_{1}d_{\Omega} in \Omega; and, from (1.7) we have -\Delta\boldsymbol{u} +a_{0}\boldsymbol{u}\geq\eta\chi_{E} in D^{\prime}\left(\Omega\right). Then, by the weak maximum principle (as stated, e.g., in [20], Theorem 8.1), \boldsymbol{u}\geq\psi a.e. in \Omega. Therefore, \boldsymbol{u}\geq c_{1}d_{\Omega} a.e. in \Omega. Thus, for some positive constant c^{\prime}, a\boldsymbol{u}^{-\alpha}\leq c^{\prime }d_{\Omega}^{-\alpha} a.e. in \Omega. Also, g\left(., \boldsymbol{u} \right) \in L^{\infty}\left(\Omega\right) and so, for a larger c^{\prime} if necessary, we have \left\vert a\boldsymbol{u}^{-\alpha }-g\left(., \boldsymbol{u}\right) \right\vert \leq c^{\prime}d_{\Omega }^{-\alpha} a.e. in \Omega. Then, taking into account that \alpha\leq1, the Hardy inequality gives, for any \varphi\in H_{0}^{1}\left(\Omega\right),

    \int_{\Omega}\left\vert \left( a\boldsymbol{u}^{-\alpha}-g\left( ., \boldsymbol{u}\right) \right) \varphi\right\vert \leq\int_{\Omega }c^{\prime}d_{\Omega}^{1-\alpha}\left\vert d_{\Omega}^{-1}\varphi\right\vert \leq c^{\prime\prime}\left\Vert \varphi\right\Vert _{H_{0}^{1}\left( \Omega\right) }.

    with c^{\prime\prime} a positive constant independent of \varphi. Thus a\boldsymbol{u}^{-\alpha}-g\left(., \boldsymbol{u}\right) \in\left(H_{0}^{1}\left(\Omega\right) \right) ^{\prime}. Let z be as in Lemma 2.5. Since \boldsymbol{u}\leq u_{\varepsilon_{j}}\leq z, Lemma 2.5 gives that \boldsymbol{u}\leq c^{\prime\prime\prime} d_{\Omega}^{\tau} for some positive constants c^{\prime\prime\prime} and \tau. Therefore, by Lemma 2.13, \boldsymbol{u} is a weak solution, in the usual sense of H_{0}^{1}\left(\Omega\right), of problem (1.2). Moreover, since

    \begin{equation} cd_{\Omega}\leq\boldsymbol{u}\leq c^{\prime\prime\prime}d_{\Omega}^{\tau }\text{ }a.e.\text{ in }\Omega, \end{equation} (3.10)

    then a\boldsymbol{u}^{-\alpha}-g\left(., \boldsymbol{u}\right) \in L_{loc}^{\infty}\left(\Omega\right), also \boldsymbol{u}\in L^{\infty }\left(\Omega\right) and then, by the inner elliptic estimates, \boldsymbol{u}\in W_{loc}^{2, p}\left(\Omega\right) for any p\in\left[1, \infty\right). Thus \boldsymbol{u}\in C\left(\Omega\right) and from (3.10), u is also continuous at \partial\Omega. Thus u\in C\left(\overline{\Omega}\right).

    Proof of Theorem 1.4. Suppose that 0 < \alpha < \frac{1}{2} +\frac{1}{n} when n > 2, that and 0 < \alpha\leq1 when n\leq2. Assume also that g\left(., s\right) = 0 on \Omega^{+} and that h1)-h4) and h5) hold. Let z be as in Remark 2.4, let \left\{ \varepsilon_{j}\right\} _{j\in\mathbb{N}}\subset\left(0, 1\right) be a nonincreasing sequence such that \lim_{j\rightarrow\infty}\varepsilon_{j} = 0, and let \left\{ u_{\varepsilon_{j}}\right\} _{j\in\mathbb{N}} be as in Theorem 1.2. Let \boldsymbol{u}: = \lim_{j\rightarrow\infty }u_{\varepsilon_{j}} . By Lemma 2.5 we have, u_{\varepsilon _{j}}\leq z in \Omega for all j\in\mathbb{N}, and so \boldsymbol{u}\leq z a.e. in \Omega. Thus, by Remark 2.4, there exist positive constants c and \tau such that \boldsymbol{u}\leq cd_{\Omega }^{\tau} a.e. in \Omega. Let \Omega^{+} as given by h6), and let \zeta:\Omega^{+}\rightarrow\mathbb{R} be as given by Remark 2.19. Thus, by Remark 2.19 ii), there exists a positive constant c^{\prime} such that \zeta\geq c^{\prime }d_{\Omega^{+}} in \Omega^{+}, and by Remark 2.20, u_{\varepsilon_{j}}\geq\zeta in \Omega^{+} for all j\in\mathbb{N} . Then u_{\varepsilon_{j}}\geq c^{\prime}d_{\Omega^{+}} in \Omega^{+} for all j, and so \boldsymbol{u}\geq cd_{\Omega^{+}} a.e. in \Omega^{+}.

    Let \varphi\in H_{0}^{1}\left(\Omega\right) and, for k\in\mathbb{N}, let \varphi_{k}:\Omega\rightarrow\mathbb{R} be defined by \varphi_{k}\left(x\right) = \varphi\left(x\right) if \left\vert \varphi\left(x\right) \right\vert \leq k, \varphi_{k}\left(x\right) = k if \varphi\left(x\right) > k and \varphi_{k}\left(x\right) = -k if \varphi\left(x\right) < -k. Thus \varphi_{k}\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right) and \left\{ \varphi_{k}\right\} _{k\in\mathbb{N}} converges to \varphi in H_{0} ^{1}\left(\Omega\right). By Theorem 1.2, u is a weak solution, in the sense of definition 1.1, of problem (1.2). Then, for all k\in\mathbb{N},

    \begin{align} \int_{\Omega}\left\langle \nabla\boldsymbol{u}, \nabla\varphi_{k}\right\rangle & = \int_{\Omega}\chi_{\left\{ \boldsymbol{u} \gt 0\right\} }\left( a\boldsymbol{u}^{-\alpha}-g\left( ., \boldsymbol{u}\right) \right) \varphi_{k} \\ & = \int_{\Omega}\left( a\boldsymbol{u}^{-\alpha}-\chi_{\left\{ \boldsymbol{u} \gt 0\right\} }g\left( ., \boldsymbol{u}\right) \right) \varphi_{k}\\ & = \int_{\Omega}\left( \chi_{\left\{ a \gt 0\right\} }a\boldsymbol{u} ^{-\alpha}-\chi_{\left\{ \boldsymbol{u} \gt 0\right\} }g\left( ., \boldsymbol{u} \right) \right) \varphi_{k}. \end{align} (3.11)

    Note that \chi_{\left\{ a > 0\right\} }a\boldsymbol{u}^{-\alpha} -\chi_{\left\{ \boldsymbol{u} > 0\right\} }g\left(., \boldsymbol{u}\right) \in\left(H_{0}^{1}\left(\Omega\right) \right) ^{\prime}. Indeed, by h4), \chi_{\left\{ \boldsymbol{u} > 0\right\} }g\left(., \boldsymbol{u}\right) \in L^{\infty}\left(\Omega\right) \subset\left(H_{0}^{1}\left(\Omega\right) \right) ^{\prime}, and, since \boldsymbol{u}\geq cd_{\Omega^{+}} a.e. in \Omega^{+} and a = 0 a.e. in \Omega\setminus\Omega^{+}, we have \chi_{\left\{ a > 0\right\} }a\boldsymbol{u}^{-\alpha}\in L^{\left(2^{\ast}\right) ^{\prime}}\left(\Omega\right) \subset\left(H_{0}^{1}\left(\Omega\right) \right) ^{\prime} when n > 2 (because \ 0 < \alpha < \frac{1}{2}+\frac{1}{n} if n > 2 ), and, in the case n\leq2, \chi_{\left\{ a > 0\right\} }a\boldsymbol{u} ^{-\alpha}\in L^{\frac{1}{\alpha}-\eta}\left(\Omega\right) \subset\left(H_{0}^{1}\left(\Omega\right) \right) ^{\prime} for \eta positive and small enough, (because 0 < \alpha\leq1 if n\leq2 ). Now, we take \lim_{k\rightarrow\infty} in (3.11), to obtain

    \begin{array}{l} \int_{\Omega}\left\langle \nabla\boldsymbol{u}, \nabla\varphi\right\rangle & = \int_{\Omega}\left( \chi_{\left\{ a \gt 0\right\} }a\boldsymbol{u}^{-\alpha }-\chi_{\left\{ \boldsymbol{u} \gt 0\right\} }g\left( ., \boldsymbol{u}\right) \right) \varphi\\ & = \int_{\Omega}\chi_{\left\{ \boldsymbol{u} \gt 0\right\} }\left( a\boldsymbol{u}^{-\alpha}-g\left( ., \boldsymbol{u}\right) \right) \varphi, \end{array}

    the last equality because u > 0 a.e. in \left\{ a > 0\right\}.

    The author wish to thank an anonymous referee for his/her helpful suggestions and critical comments, which led to a substantial improvement of the paper.

    The author declare no conflicts of interest in this paper



    [1] F. Busetti, A. M. R. Taylor, Tests of stationarity against a change in persistence, J. Econom., 123 (2004), 33–66. https://doi.org/10.1016/j.jeconom.2003.10.028 doi: 10.1016/j.jeconom.2003.10.028
    [2] M. Caporin, R. Gupta, Time-varying persistence in US inflation, Empir. Econ., 53 (2017), 423–439. https://doi.org/10.1007/s00181-016-1144-y doi: 10.1007/s00181-016-1144-y
    [3] Z. S. Chen, Z. Jin, Z. Tian, P. Y. Qi, Bootstrap testing multiple changes in persistence for a heavy-tailed sequence, Comput. Stat. Data Anal., 56 (2012), 2303–2316. https://doi.org/10.1016/j.csda.2012.01.011 doi: 10.1016/j.csda.2012.01.011
    [4] Z. Chen, Z. Tian, Y. Xing, Sieve bootstrap monitoring persistence change in long memory process, Stat. Interface, 9 (2016), 37–45. https://doi.org/10.4310/SII.2016.v9.n1.a4 doi: 10.4310/SII.2016.v9.n1.a4
    [5] Z. Chen, Y. Xing, Z. A. Chen, Y. H. Xing, F. X. Li, Sieve bootstrap monitoring for change from short to long memory, Econ. Lett., 140 (2016), 53–56. https://doi.org/10.1016/j.econlet.2015.12.023 doi: 10.1016/j.econlet.2015.12.023
    [6] Z. S. Chen, F. X. Li, L. Zhu, Y. H. Xing, Monitoring mean and variance change-points in long-memory time series, J. Syst. Sci. Complex., 2021. https://doi.org/10.1007/s11424-021-0222-1 doi: 10.1007/s11424-021-0222-1
    [7] Z. S. Chen, Y. T. Xiao, F. X. Li, Monitoring memory parameter change-points in long-memory time series, Empir. Econ., 60 (2021), 2365–2389. https://doi.org/10.1007/s00181-020-01840-4 doi: 10.1007/s00181-020-01840-4
    [8] H. Dette, J. Gösmann, A likelihood ratio approach to sequential change point detection for a general class of parameters, J. Am. Stat. Assoc., 115 (2019), 1361–1377. https://doi.org/10.1080/01621459.2019.1630562 doi: 10.1080/01621459.2019.1630562
    [9] J. Gösmann, T. Kley, H. Dette, A new approach for open-end sequential change point monitoring, J. Time Ser. Anal., 42 (2021), 63–84. https://doi.org/10.1111/jtsa.12555 doi: 10.1111/jtsa.12555
    [10] D. I. Harvey, S. J. Leybourne, A. M. R. Taylor, Modified tests for a change in persistence, J. Econom., 134 (2006), 441–469. https://doi.org/10.1016/j.jeconom.2005.07.002 doi: 10.1016/j.jeconom.2005.07.002
    [11] U. Hassler, J. Scheithauer, Detecting changes from short to long memory, Stat. Papers, 52 (2011), 847–870. https://doi.org/10.1007/s00362-009-0292-y doi: 10.1007/s00362-009-0292-y
    [12] U. Hassler, B. Meller, Detecting multiple breaks in long memory the case of U.S. inflation, Empir. Econ., 46 (2014), 653–680. https://doi.org/10.1007/s00181-013-0691-8 doi: 10.1007/s00181-013-0691-8
    [13] F. Iacone, Š. Lazarová, Semiparametric detection of changes in long range dependence, J. Time Ser. Anal., 40 (2019), 693–706. https://doi.org/10.1111/jtsa.12448 doi: 10.1111/jtsa.12448
    [14] M. Kejriwal, P. Perron, J. Zhou, Wald tests for detecting multiple structural changes in persistence, Economet. Theor., 29 (2013), 289–323. https://doi.org/10.1017/S0266466612000357 doi: 10.1017/S0266466612000357
    [15] J. Y. Kim, Detection of change in persistence of a linear time series, J. Econ., 95 (2000), 97–116. https://doi.org/10.1016/S0304-4076(99)00031-7 doi: 10.1016/S0304-4076(99)00031-7
    [16] C. Kirch, S. Weber, Modified sequential change point procedures based on estimating functions, Electron. J. Stat., 12 (2018), 1579–1613. https://doi.org/10.1214/18-EJS1431 doi: 10.1214/18-EJS1431
    [17] F. Lavancier, R. Leipus, A. Philippe, D. Surgailis, Detection of nonconstant long memory parameter, Economet. Theor., 29 (2013), 1009–1056. https://doi.org/10.1017/S0266466613000303 doi: 10.1017/S0266466613000303
    [18] S. Leybourne, R. Taylor, T. H. Kim, CUSUM of squares-based tests for a change in persistence, J. Time Ser. Anal., 28 (2007), 408–433. https://doi.org/10.1111/j.1467-9892.2006.00517.x doi: 10.1111/j.1467-9892.2006.00517.x
    [19] F. X. Li, Z. S. Chen, Y. T. Xiao, Sequential change-point detection in a multinomial logistic regression model, Open Math., 18 (2020), 807–819. https://doi.org/10.1515/math-2020-0037 doi: 10.1515/math-2020-0037
    [20] R. B. Qin, Y. Liu, Block bootstrap testing for changes in persistence with heavy-tailed innovations, Commun. Stat.-Theor. M., 47 (2018), 1104–1116. https://doi.org/10.1080/03610926.2017.1316398 doi: 10.1080/03610926.2017.1316398
    [21] P. Sibbertsen, R. Kruse, Testing for a break in persistence under long-range dependences, J. Time Ser. Anal., 30 (2009), 263–285. https://doi.org/10.1111/j.1467-9892.2009.00611.x doi: 10.1111/j.1467-9892.2009.00611.x
    [22] M. S. Taqqu, Weak convergence to fractional Brownian motion and to the Rosenblatt process, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 31 (1975), 287–302. https://doi.org/10.1007/BF00532868 doi: 10.1007/BF00532868
    [23] W. Z. Zhao, Y. X. Xue, X. Liu, Monitoring parameter change in linear regression model based on the efficient score vector, Physica A, 527 (2019), 121135. https://doi.org/10.1016/j.physa.2019.121135 doi: 10.1016/j.physa.2019.121135
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1994) PDF downloads(42) Cited by(2)

Figures and Tables

Figures(1)  /  Tables(3)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog