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Natural limestone discolouration triggered by microbial activity—a contribution

  • Received: 30 April 2018 Accepted: 13 July 2018 Published: 10 August 2018
  • Colour is a major argument that drives the decision of an architect in a specific architecture project and one of the most important characteristics and perceptible aspects of natural building stones. “Blue” limestones are building rocks, with different geological ages, typically used in several countries, and are known for their vulnerability to alteration, which causes colour change and the occurrence of unaesthetic patterns. Owing to this vulnerability, the conservation-restoration works in monuments, or new buildings constructed with “blue” limestone is extremely costly. Considering that the main limitation of this lithological variation is the chromatic change, a multidisciplinary approach was envisaged in this study to allow a closer insight into the chemical and mineralogical alterations and the microbial communities. Results obtained suggest that the inorganic alteration in the “blue” limestone may create favourable conditions for microbial growth and could lead to an increment in deterioration process.

    Citation: Luís Dias, Tânia Rosado, Ana Coelho, Pedro Barrulas, Luís Lopes, Patrícia Moita, António Candeias, José Mirão, Ana Teresa Caldeira. Natural limestone discolouration triggered by microbial activity—a contribution[J]. AIMS Microbiology, 2018, 4(4): 594-607. doi: 10.3934/microbiol.2018.4.594

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  • Colour is a major argument that drives the decision of an architect in a specific architecture project and one of the most important characteristics and perceptible aspects of natural building stones. “Blue” limestones are building rocks, with different geological ages, typically used in several countries, and are known for their vulnerability to alteration, which causes colour change and the occurrence of unaesthetic patterns. Owing to this vulnerability, the conservation-restoration works in monuments, or new buildings constructed with “blue” limestone is extremely costly. Considering that the main limitation of this lithological variation is the chromatic change, a multidisciplinary approach was envisaged in this study to allow a closer insight into the chemical and mineralogical alterations and the microbial communities. Results obtained suggest that the inorganic alteration in the “blue” limestone may create favourable conditions for microbial growth and could lead to an increment in deterioration process.


    Let $ \Omega $ be a bounded and regular enough domain in $ \mathbb{R}^{n}, $ let $ \alpha > 0, $ and let $ a:\Omega\rightarrow\mathbb{R} $ 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 $ -\Delta u = au^{-\alpha}+f\left(., \lambda u\right) $ in $ \Omega, $ $ u = 0 $ on $ \partial\Omega, $ $ u > 0 $ in $ \Omega, $ were studied by Coclite and Palmieri [4], under the assumptions $ a\in C^{1}\left(\overline{\Omega}\right), $ $ a > 0 $ in $ \overline{\Omega}, $ $ f\in C^{1}\left(\overline{\Omega}\times\left[0, \infty\right) \right) $ and $ \lambda > 0. $ Problems of the form $ -\Delta u = Ku^{-\alpha}+\lambda s^{p} $ in $ \Omega, $ $ u = 0 $ on $ \partial\Omega, $ $ u > 0 $ in $ \Omega, $ were studied by 35 [35], when $ p\in\left(0, 1\right), $ $ K $ is a regular enough function that may change sign, and $ \lambda\in\mathbb{R} $. Ghergu and Rădulescu [19] addressed multi-parameter singular bifurcation problems of the form $ -\Delta u = g\left(u\right) +\lambda\left\vert \nabla u\right\vert ^{p}+\mu f\left(., u\right) $ in $ \Omega, $ $ u = 0 $ on $ \partial\Omega, $ $ u > 0 $ in $ \Omega, $ where $ g $ is Hölder continuous, nonincreasingt and positive on $ \left(0, \infty\right), $ and singular at the origin; $ f:\overline {\Omega}\times\left[0, \infty\right) \rightarrow\left[0, \infty\right) $ is Hölder continuous, positive on $ \overline{\Omega}\times\left(0, \infty\right), $ and such that $ f\left(x, s\right) $ is nondecreasing with respect to $ s, $ $ 0 < p\leq2, \; $and $ \lambda > 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 $ -\Delta u = a(x)h\left(u\right) +\lambda f\left(u\right) $ in $ \Omega, $ $ u = 0 $ on $ \partial\Omega, $ $ u > 0 $ in $ \Omega $, in the case when $ \Omega $ is a regular enough domain, $ f $ and $ h $ are positive Hölder continuous functions on $ \left[0, \infty\right) $ and $ \left(0, \infty\right) $ respectively satisfying some monotonicity assumptions, $ h $ singular at the origin, and $ h\left(s\right) \leq cs^{-\alpha} $ for some positive constant $ c $ and some $ \alpha\in\left(0, 1\right). $

    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\left(x\right) s^{-\alpha}, $ with $ 0\leq a\in L^{\infty}\left(\Omega\right), $ $ a\not \equiv 0 $ in $ \Omega, $ and $ \alpha $ positive$. $

    Recently, problem (1.1) has been addressed by Chu, Gao and Gao [8], under the assumption that $ \alpha = \alpha\left(x\right) $ (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

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

    where $ 0 < \alpha < 1, $ $ \lambda > 0, $ and $ f:\Omega\times\left[0, \infty\right) \rightarrow\left[0, \infty\right) $ is a Carathéodory function $ f $ such that, for $ a.e. $ $ x\in\Omega, $ $ f\left(x, s\right) $ is nondecreasing and concave in $ s, $ and satisfies $ \lim_{s\rightarrow\infty}f\left(x, s\right) /s = 0 $ uniformly on $ x\in\Omega. $ and where, for $ h:\Omega\times\left(0, \infty\right) \rightarrow\mathbb{R}, $ $ \chi_{\left\{ s > 0\right\} }h\left(x, s\right) $ stands for the function defined on $ \Omega \times\left[0, \infty\right) $ by $ \chi_{\left\{ s > 0\right\} }h\left(x, s\right) : = h\left(x, s\right) $ if $ s > 0, $ and $ \chi_{\left\{ s > 0\right\} }h\left(x, s\right) : = 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 $ \Omega $ is a bounded domain in $ \mathbb{R}^{n} $ with $ C^{1, 1} $ boundary, $ \alpha\in\left(0, 1\right], $ $ a:\Omega\rightarrow\mathbb{R} $, and $ g:\Omega\times\left[0, \infty\right) \rightarrow\mathbb{R} $, with $ a $ and $ g $ satisfying the following conditions h1)-h4):

    h1) $ 0\leq a\in L^{\infty}\left(\Omega\right) $ and $ a\not \equiv 0, $

    h2) $ \left\{ x\in\Omega:a\left(x\right) = 0\right\} = \Omega_{0}\cup N $ for some (possibly empty) open set $ \Omega _{0}\subset\Omega $ and some measurable set $ N\subset\Omega $ such that $ \left\vert N\right\vert = 0, $

    h3) $ g $ is a nonnegative Carathéodory function on $ \Omega\times\left[0, \infty\right), $ i.e., $ g\left(., s\right) $ is measurable for any $ s\in\left[0, \infty\right), $ and $ g\left(x, .\right) $ is continuous on $ \left[0, \infty\right) $ for $ a.e. $ $ x\in\Omega $,

    h4) $ \sup_{0\leq s\leq M}g\left(., s\right) \in L^{\infty}\left(\Omega\right) $ for any $ M > 0. $

    Here and below, $ \chi_{\left\{ u > 0\right\} }\left(au^{-\alpha}-g\left(., u\right) \right) $ stands for the function $ h:\Omega\rightarrow\mathbb{R} $ defined by $ h\left(x\right) : = a\left(x\right) u^{-\alpha}\left(x\right) -g\left(x, u\left(x\right) \right) $ if $ u\left(x\right) \neq0, $ and $ h\left(x\right) : = 0 $ otherwise; $ u\not \equiv 0 $ in $ \Omega $ means $ \left\vert \left\{ x\in\Omega:u\left(x\right) \neq0\right\} \right\vert > 0 $ and, by a weak solution of (1.2), we mean a solution in the sense of the following:

    Definition 1.1. Let $ h:\Omega\rightarrow\mathbb{R} $ be a measurable function such that $ h\varphi\in L^{1}\left(\Omega\right) $ for all $ \varphi $ in $ H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right). $ We say that $ u:\Omega\rightarrow\mathbb{R} $ is a weak solution to the problem

    $ {Δu=h in Ω,u=0 on Ω $ (1.3)

    if $ u\in H_{0}^{1}\left(\Omega\right), $ and $ \int_{\Omega}\left\langle \nabla u, \nabla\varphi\right\rangle = \int_{\Omega}h\varphi $ for all $ \varphi $ in $ H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right). $

    We will say that, in weak sense,

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

    if $ u\in H_{0}^{1}\left(\Omega\right), $ and $ \int_{\Omega}\left\langle \nabla u, \nabla\varphi\right\rangle \leq\int_{\Omega}h\varphi $ (respectively $ \int_{\Omega}\left\langle \nabla u, \nabla\varphi\right\rangle \geq \int_{\Omega}h\varphi) $ for all nonnegative $ \varphi $ in $ H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right). $

    Our first result reads as follows:

    Theorem 1.2. Let $ \Omega $ be a bounded domain in $ \mathbb{R}^{n} $ with $ C^{1, 1} $ boundary. Let $ \alpha\in\left(0, 1\right] $, let $ a:\Omega \rightarrow\left[0, \infty\right) $ and let $ g:\Omega\times\left(0, \infty\right) \rightarrow\mathbb{R} $; and assume that $ a $ and $ g $ satisfy the conditions h1)-h4). Then there exists a nonnegative weak solution $ u\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right), $ in the sense of Definition 1.1, to problem (1.2), and such that $ u > 0 $ $ a.e. $ in $ \left\{ a > 0\right\}. $ In particular, $ \chi_{\left\{ u > 0\right\} }\left(au^{-\alpha}-g\left(., u\right) \right) \not \equiv 0 $ in $ \Omega $ and $ \chi_{\left\{ u > 0\right\} }\left(au^{-\alpha}-g\left(., u\right) \right)\varphi \in L^{1}\left(\Omega\right) $ for any $ \varphi\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right) $).

    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 $ 0\leq a\in L^{\infty}\left(\Omega\right), $ $ a\not \equiv 0, $ $ 0 < \alpha < 1 $, and $ g\left(., u\right) = -bu^{p} $, with $ 0 < p < \frac{n+2}{n-2} $, and $ 0\leq b\in L^{r}\left(\Omega\right) $ 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:\Omega\times\left[0, \infty\right) \rightarrow\left[0, \infty\right) \left(x, s\right), $ in the case when $ h $ is a Carathéodory function on $ \Omega\times\left[0, \infty\right), $ which satisfies $ h\left(., 0\right) \leq0 $ 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\left(., 0\right) \leq0 $ is not required and $ \chi_{\left\{ s > 0\right\} }g\left(., s\right) $ is not, in general, a Carathéodory function on $ \Omega\times\left[0, \infty\right) $ (except when $ g\left(., 0\right) \equiv0 $ in $ \Omega $).

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

    h4') $ a > 0 $ $ a.e. $ in $ \Omega $ and $ \sup_{0 < s\leq M} s^{-1}g\left(., s\right) \in L^{\infty}\left(\Omega\right) $ for any $ M > 0, $

    then the solution $ u, $ given by Theorem 1.2, is positive $ a.e. $ in $ \Omega $ and is a weak solution in the usual sense of $ H_{0}^{1}\left(\Omega\right). $

    Theorem 1.3. Let $ \Omega, $ $ \alpha, $ and $ a $ be as in Theorem 1.2, and let $ g:\Omega\times\left(0, \infty\right) \rightarrow\mathbb{R} $. Assume the conditions h1)-h3) and h4'). Then the solution $ u $ of (1.2), given by Theorem 1.2, belongs to $ C\left(\overline{\Omega}\right) \cap W_{loc}^{2, p}\left(\Omega\right) $ for any $ p\in\left[1, \infty\right), $ there exist positive constants $ c, $ $ c^{\prime} $ and $ \tau $ such that $ cd_{\Omega}\leq u\leq c^{\prime}d_{\Omega}^{\tau} $ in $ \Omega, $ and $ u $ is a weak solution, in the usual $ H_{0}^{1}\left(\Omega\right) $ sense, of the problem

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

    i.e., for any $ \varphi\in H_{0}^{1}\left(\Omega\right), $ $ \left(au^{-\alpha}-g\left(., u\right) \right) \varphi\in L^{1}\left(\Omega\right) $ and $ \int_{\Omega}\left\langle \nabla u, \nabla\varphi \right\rangle = \int_{\Omega}\left(au^{-\alpha}-g\left(., u\right) \right) \varphi. $

    Finally, our last result says that, if in addition to h1)-h4), $ \alpha $ is sufficiently small, the set where $ a > 0 $ is nice enough and, for any $ s\geq0, $ $ g\left(., s\right) = 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 $ H_{0}^{1}\left(\Omega\right), $ and that it is positive on some subset of $ \Omega $:

    Theorem 1.4. Let $ \Omega $ be a bounded domain in $ \mathbb{R}^{n} $ with $ C^{1, 1} $ boundary. Assume the hypothesis h1)-h4) of Theorem 1.2 and that $ 0 < \alpha < \frac{1}{2}+\frac{1}{n} $ when $ n > 2, $ and $ \alpha\in\left(0, 1\right] $ when $ n\leq2. $ Let $ A^{+}: = \left\{ x\in\Omega:a\left(x\right) > 0\right\} $ and assume, in addition, the following two conditions:

    h5) $ g\left(., s\right) = 0 $ $ a.e. $ in $ A^{+} $ for any $ s\geq0. $

    h6) $ A^{+} = \Omega^{+}\cup N^{+} $ for some open set $ \Omega^{+} $ and a measurable set $ N^{+} $ such that $ \left\vert N^{+} \right\vert = 0, $ and with $ \Omega^{+} $ such that $ \Omega^{+} $ has a finite number of connected components $ \left\{ \Omega_{l}^{+}\right\} _{1\leq l\leq N} $ and each $ \Omega_{l}^{+} $ is a $ C^{1, 1} $ domain.

    Then the solution $ u $ of problem (1.2), given by Theorem 1.2, is a weak solution, in the usual $ H_{0}^{1}\left(\Omega\right) $ sense, to the same problem, and there exist positive constants $ c, $ $ c^{\prime} $ and $ \tau $ such that $ u\geq cd_{\Omega^{+}} $ $ a.e. $ in $ \Omega^{+}, $ and $ u\leq c^{\prime}d_{\Omega}^{\tau} $ $ a.e. $ in $ \Omega. $

    The article is organized as follows: In Section 2 we study, for $ \varepsilon\in\left(0, 1\right], $ 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 $ \Omega $ is a bounded domain in $ \mathbb{R}^{n} $ with $ C^{1, 1} $ boundary, $ \alpha\in\left(0, 1\right], $ $ a:\Omega\rightarrow\left[0, \infty\right) $ is a nonnegative function in $ L^{\infty}\left(\Omega\right) $ such that $ \left\vert \left\{ x\in\Omega:a\left(x\right) > 0\right\} \right\vert > 0, $ and $ \left\{ g_{\varepsilon}\right\} _{\varepsilon\in\left(0, 1\right] } $ is a family of real valued functions defined on $ \Omega\times\left[0, \infty\right) $ satisfying the following conditions h7)-h9):

    h7) $ g_{\varepsilon} $ is a nonnegative Carathéodory function on $ \Omega\times\left[0, \infty\right) $ for any $ \varepsilon\in\left(0, 1\right]. $

    h8) $ \sup_{0 < s\leq M}s^{-1}g_{\varepsilon}\left(., s\right) \in L^{\infty}\left(\Omega\right) $ for any $ \varepsilon\in\left(0, 1\right] $ and $ M > 0. $

    h9) The map $ \varepsilon\rightarrow g_{\varepsilon}\left(x, s\right) $ is nonincreasing on $ \left(0, 1\right] $ for any $ \left(x, s\right) \in \Omega\times\left[0, \infty\right). $

    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 $ \left\{ a > 0\right\}; $ 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 $ \Omega, $ belongs to $ C\left(\overline{\Omega}\right), $ and is also a weak solution in the usual sense of $ H_{0}^{1}\left(\Omega\right) $. By using a sub-supersolution theorem of [28] as well as an adaptation of arguments of [27], Lemma 2.15 shows that, for any $ \varepsilon\in\left(0, 1\right], $ problem (1.6) has a solution $ u_{\varepsilon}\in H_{0} ^{1}\left(\Omega\right), $ which is a weak solution in the usual sense of $ H_{0}^{1}\left(\Omega\right), $ and is maximal in the sense that, if $ v $ is a solution, in the sense of Definition 1.1, of problem (1.6) then $ v\leq u_{\varepsilon}. $ Lemma 2.16 states that $ \varepsilon\rightarrow u_{\varepsilon} $ is nondecreasing, Lemma 2.17 says that $ \left\{ u_{\varepsilon }\right\} _{\varepsilon\in\left(0, 1\right] } $ is bounded in $ H_{0} ^{1}\left(\Omega\right), $ and Lemma 2.18 says that the function $ \boldsymbol{u}: = \lim_{\varepsilon\rightarrow0^{+}}u_{\varepsilon} $ belong to $ H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right) $ and is positive in $ \left\{ a > 0\right\}. $

    To prove Theorems 1.2–1.4 we consider, in Section 3, the family $ \left\{ g_{\varepsilon}\right\} _{\varepsilon \in\left(0, 1\right] } $ defined by $ g_{\varepsilon}\left(., s\right) : = s\left(s+\varepsilon\right) ^{-1}g\left(., s\right) $ and we show that, in each case, the corresponding function $ \boldsymbol{u} $ defined above is a solution of problem (1.2) with the desired properties.

    We assume, from now on, that $ \Omega $ is a bounded domain in $ \mathbb{R}^{n} $ with $ C^{1, 1} $ boundary, $ \alpha\in\left(0, 1\right] $ and $ a:\Omega\rightarrow\left[0, \infty\right) $ is a nonnegative function in $ L^{\infty}\left(\Omega\right) $ such that $ \left\vert \left\{ x\in \Omega:a\left(x\right) > 0\right\} \right\vert > 0, $ and additional conditions will be explicitely impossed on $ a $ and $ \alpha $ when necessary, at some steps of the paper. Our aim in this section is to study, for $ \varepsilon\in\left(0, 1\right], $ the existence of weak solutions to problem (1.6), in the case when $ \left\{ g_{\varepsilon}\right\} _{\varepsilon\in\left(0, 1\right] } $ 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 $ b\in L^{\infty}\left(\Omega\right) $ such that $ b\not \equiv 0, $ we say that $ \lambda\in\mathbb{R} $ is a principal eigenvalue for $ -\Delta $ on $ \Omega, $ with weight function $ b $ and homogeneous Dirichlet boundary condition, if the problem $ -\Delta u = \lambda bu $ in $ \Omega, $ $ u = 0 $ on $ \partial\Omega $ has a solution $ u $ wich is positive in $ \Omega $. If $ b\in L^{\infty}\left(\Omega\right) $ and $ b^{+}\not \equiv 0 $, it is well known that there exists a unique positive principal eigenvalue for the above problem, which we wiill denote by $ \lambda _{1}\left(b\right) $. 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 $ \beta\in\left(0, 3\right), $ $ \widetilde{a}:\Omega\rightarrow\mathbb{R} $ and $ f:\Omega \times\left[0, \infty\right) \rightarrow\mathbb{R} $; and assume the following conditions H1)-H6):

    H1) $ 0\leq\widetilde{a}\in L^{\infty}\left(\Omega\right), $ and $ \widetilde{a}\not \equiv 0, $

    H2) $ f $ is a Carathéodory function on $ \Omega \times\left[0, \infty\right), $

    H3) $ \sup_{0\leq s\leq M}\left\vert f\left(., s\right) \right\vert \in L^{1}\left(\Omega\right) $ for any $ M > 0, $

    H4) One of the two following conditions holds:

    H4') $ \sup_{s > 0}\frac{f\left(., s\right) }{s}\leq b $ for some $ b\in L^{\infty}\left(\Omega\right) $ such that $ b^{+}\not \equiv 0, $ and $ \lambda_{1}\left(b\right) > m $ for some integer $ m\geq\max\left\{ 2, 1+\beta\right\}, $

    H4") $ f\in L^{\infty}\left(\Omega\times\left(0, \sigma\right) \right) $ for all $ \sigma > 0, $ and $ \overline{\lim}_{s\rightarrow\infty}\frac{f\left(., s\right) }{s}\leq0 $ uniformly on $ \Omega, $ i.e., for any $ \varepsilon > 0 $ there exists $ s_{0} > 0 $ such that $ \sup_{s\geq s_{0}}\frac{f\left(., s\right) }{s}\leq\varepsilon, $ $ a.e. $ in $ \Omega, $

    H5) $ f\left(., 0\right) \geq0. $

    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) $ u\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right) $ such that $ u > 0 $ $ a.e. $ in $ \left\{ \widetilde{a} > 0\right\}. $

    Lemma 2.2. Let $ a\in L^{\infty}\left(\Omega\right) $ be such that $ a\geq0 $ in $ \Omega $ and $ a\not \equiv 0, $ let $ \alpha\in\left(0, 1\right] $, and let $ \left\{ g_{\varepsilon }\right\} _{\varepsilon\in\left(0, 1\right] } $ be a family of functions defined on $ \Omega\times\left[0, \infty\right) $ satisfying the conditions h7)-h9) stated at the introduction. Then, for any $ \varepsilon\in\left(0, 1\right], $ problem (1.7) has at least a weak solution $ u\in H_{0} ^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right) $, in the sense of Definition 1.1, such that $ u > 0 $ $ a.e. $ in $ \left\{ a > 0\right\}. $

    Proof. Notice that, since $ g_{\varepsilon} $ is a Carathéodory function, we have $ g_{\varepsilon}\left(., 0\right) = \lim_{s\rightarrow0^{+}}g_{\varepsilon }\left(., s\right) = \lim_{s\rightarrow0^{+}}\left(ss^{-1}g_{\varepsilon }\left(., s\right) \right) = 0, $ the last inequality by h8). Thus $ g_{\varepsilon}\left(., 0\right) = 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 $ 0\leq h\in L^{\infty}\left(\Omega\right); $ and let $ v\in\cap_{1\leq p < \infty }\left(W^{2, p}\left(\Omega\right) \cap W_{0}^{1, p}\left(\Omega\right) \right) $ be the strong solution of $ -\Delta v = h $ in $ \Omega, $ $ v = 0 $ on $ \partial\Omega. $ Then $ v\geq cd_{\Omega}\int_{\Omega}hd_{\Omega} $ $ a.e. $ in $ \Omega, $ where $ d_{\Omega}: = dist\left(., \partial\Omega\right), $ and $ c $ is a positive constant depending only on $ \Omega. $

    ⅱ) (see e.g., [25], Remark 8) Let $ \Psi $ be a nonnegative function in $ L_{loc}^{1}\left(\Omega\right), $ and let $ v $ be a function in $ H_{0} ^{1}\left(\Omega\right) $ such that $ -\Delta v\geq\Psi $ on $ \Omega $ in the sense of distributions. Then

    $ v(x)cdΩΩΨdΩa.e. in Ω, $ (2.2)

    where $ c $ is a positive constant depending only on $ \Omega. $

    Remark 2.4. (See, e.g., [23], Lemmas 2.9, 2.10 and 2.12) Let $ a\in L^{\infty}\left(\Omega\right) $ be such that $ a\geq0 $ in $ \Omega $ and $ a\not \equiv 0, $ and let let $ \alpha\in\left(0, 1\right] $. 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, $ z\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right). $ Moreover:

    ⅰ) $ z\in C\left(\overline{\Omega}\right). $

    ⅱ) There exists positive constants $ c_{1}, $ $ c_{2} $ and $ \tau > 0 $ such that $ c_{1}d_{\Omega}\leq z\leq c_{2}d_{\Omega}^{\tau} $ in $ \Omega. $

    ⅲ) $ z $ is a solution of problem (2.3) in the usual weak sense, i.e., for any $ \varphi\in H_{0}^{1}\left(\Omega\right), $ $ az^{-\alpha}\varphi\in L^{1}\left(\Omega\right) $ and $ \int_{\Omega }\left\langle \nabla z, \nabla\varphi\right\rangle = \int_{\Omega}az^{-\alpha }\varphi. $

    Lemma 2.5. Let $ a $, $ \alpha, $ and $ \left\{ g_{\varepsilon }\right\} _{\varepsilon\in\left(0, 1\right] } $ be as in Lemma 2.2, let $ z $ be as given in Remark 2.4; and let $ \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, of problem (1.7), then $ u\leq z $ $ a.e. $ in $ \Omega. $

    Proof. By h5), $ g_{\varepsilon}\left(., u\right) \geq0 $ and so, from Lemma 2.2 and Remark 2.4, we have, in the sense of Definition 1.1,

    $ -\Delta\left( u-z\right) = au^{-\alpha}-g_{\varepsilon}\left( ., u\right) -az^{-\alpha}\leq a\left( u^{-\alpha}-z^{-\alpha}\right) \text{ in }\Omega, $

    Thus, taking $ \left(u-z\right) ^{+} $ as a test function, we get

    $ \int_{\Omega}\left\vert \nabla\left( u-z\right) ^{+}\right\vert ^{2}\leq \int_{\Omega}a\left( u^{-\alpha}-z^{-\alpha}\right) \left( u-z\right) ^{+}\leq0 $

    which implies $ u\leq z $$ a.e. $ in $ \Omega. $

    Lemma 2.6. Let $ a $, $ \alpha, $ and $ \left\{ g_{\varepsilon }\right\} _{\varepsilon\in\left(0, 1\right] } $ be as in Lemma 2.2. If $ \varepsilon\in\left(0, 1\right] $ and $ 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, of problem (1.7), then:

    i) There exists a positive constant $ c_{1} $ (which may depend on $ \varepsilon $) and constants $ c_{2} $ and $ \tau $ such that $ c_{1}d_{\Omega}\leq u\leq c_{2}d_{\Omega}^{\tau} $ $ a.e. $ in $ \Omega $ (and so, in particular, $ u > 0 $ in $ \Omega $).

    ii) For any $ \varphi\in H_{0}^{1}\left(\Omega\right) $ we have $ \left(au^{-\alpha}-g_{\varepsilon}\left(., u\right) \right) \varphi\in L^{1}\left(\Omega\right) $ and

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

    i.e., $ u $ is a weak solution, in the usual sense of $ H_{0}^{1}\left(\Omega\right) $, to the problem $ -\Delta u = au^{-\alpha}-g_{\varepsilon }\left(., u\right) $ in $ \Omega, $ $ u = 0 $ on $ \partial\Omega. $

    Proof. We have, in the weak sense of Definition 1.1, $ -\Delta u = \chi_{\left\{ u > 0\right\} }au^{-\alpha}-g_{\varepsilon}\left(., u\right) $ in $ \Omega, $ $ u = 0 $ on $ \partial\Omega. $ Also, $ u\geq0 $ in $ \Omega $ and $ u\not \equiv 0 $ in $ \Omega. $ Let $ a_{0}:\Omega\rightarrow\mathbb{R} $ be defined by $ a_{0}\left(x\right) = u^{-1}\left(x\right) g_{\varepsilon }\left(x, u\left(x\right) \right) $ if $ u\left(x\right) \neq0 $ and by $ a_{0}\left(x\right) = 0 $ otherwise. Since $ u\in L^{\infty}\left(\Omega\right) $ and taking into account h7) and h8), we have $ 0\leq a_{0}\in L^{\infty}\left(\Omega\right), $ and from the definition of $ a_{0} $ we have $ g_{\varepsilon}\left(., u\right) = a_{0}u $ $ a.e. $ in $ \Omega. $ Therefore $ u $ satisfies, in the sense of Definition 1.1, $ -\Delta u+a_{0}u = \chi_{\left\{ u > 0\right\} }au^{-\alpha} $ in $ \Omega, $ $ u = 0 $ on $ \partial\Omega. $ Thus, since $ u $ is nonidentically zero, it follows that $ \chi_{\left\{ u > 0\right\} }au^{-\alpha} $ is nonidentically zero on $ \Omega. $ Then there exist $ \eta > 0, $ and a measurable set $ E\subset\Omega, $ such that $ \left\vert E\right\vert > 0 $ and $ \chi_{\left\{ u > 0\right\} }au^{-\alpha}\geq\eta\chi_{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 u+a_{0}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), $ u\geq\psi $ in $ \Omega. $ Hence $ u\geq c_{1}d_{\Omega} $ in $ \Omega. $ Also, by Lemma 2.5, $ u\leq z $ $ a.e. $ in $ \Omega, $ and so Remark 2.4 gives positive constants $ c_{2} $ and $ \tau $ (both independent of $ \varepsilon $) such that $ u\leq c_{2}d_{\Omega}^{\tau} $ in $ \Omega. $ Thus i) holds.

    To see ii), consider an arbitrary function $ \varphi\in H_{0}^{1}\left(\Omega\right), $ and for $ k\in\mathbb{N} $, let $ \varphi _{k}^{+}: = \max\left\{ k, \varphi^{+}\right\}. $ Thus $ \varphi_{k}^{+}\in H_{0}^{1}\left(\Omega\right) \cap L^{\infty}\left(\Omega\right), $ $ \left\{ \varphi_{k}^{+}\right\} _{k\in\mathbb{N}} $ converges to $ \varphi^{+} $ in $ H_{0}^{1}\left(\Omega\right) $ and, after pass to some subsequence if necessary, we can assume also that $ \left\{ \varphi_{k} ^{+}\right\} _{k\in\mathbb{N}} $ converges to $ \varphi^{+} $ $ a.e. $ in $ \Omega. $ Since $ u $ is a weak solution, in the sense of Definition 1.1, of (1.7) and $ u > 0 $ $ a.e. $ in $ \Omega, $ we have, for all $ k\in\mathbb{N}, $ $ \left(au^{-\alpha }-g_{\varepsilon}\left(., u\right) \right) \varphi_{k}^{+}\in L^{1}\left(\Omega\right), $ and, by h6), $ g_{\varepsilon}\left(., u\right) \in L^{\infty}\left(\Omega\right). $ Thus $ g_{\varepsilon}\left(., u\right) \varphi_{k}^{+}\in L^{1}\left(\Omega\right). $ Then $ au^{-\alpha}\varphi_{k}^{+}\in L^{1}\left(\Omega\right). $

    From (1.7),

    $ Ωu,φ+k+Ωgε(.,u)φ+k=Ωauαφ+k. $ (2.4)

    Now, $ \lim_{k\rightarrow\infty}\int_{\Omega}\left\langle \nabla u, \nabla \varphi_{k}^{+}\right\rangle = \int_{\Omega}\left\langle \nabla u, \nabla \varphi^{+}\right\rangle. $ Also, for any $ k, $

    $ 0\leq g_{\varepsilon}\left( ., u\right) \varphi_{k}^{+}\leq\sup\limits_{s\in\left[ 0, \left\Vert u\right\Vert _{\infty}\right] }g_{\varepsilon}\left( ., s\right) \varphi^{+}\in L^{1}\left( \Omega\right) , $

    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

    $ Ωu,φ++Ωgε(.,u)φ+=Ωauαφ+. $ (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

    $ Δw=f(.,w) in Ω. $ (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

    $ Δu=auαgε(.,u) in Ω, $ (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.,

    $ {u>0}hj(u)|u|2=Ω(auαgε(.,u))hj(u). $ (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),

    $ Δvχ{v1>v2}fε(.,v1)+χ{v2>v1}fε(.,v2)+χ{v1=v2}12(fε(.,v1)+fε(.,v2))=fε(.,v) in D(Ω) $

    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

    $ Δu=auαgε(.,u) in Ω. $ (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

    $ Ωwk,φΩ(awαkgε(.,wk))φ, $ (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}, $

    $ Ω|wk|2+Ωgε(.,wk)wk{a>0}aw1αk $ (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

    $ Ωwk,φ+Ωgε(.,wk)φΩawαkφ. $ (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

    $ Δu=auαgη(.,u) in Ω. $ (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, $

    $ Ωuεj,φ+Ωgεj(.,uεj)φ=Ωauαεjφ. $ (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

    $ {a>0}auαφ=limjΩauαεjφ=limj(Ωuεj,φ+Ωgεj(.,uεj)φ)¯limjΩuεj,φ+¯limjΩgεj(.,uεj)φΩu,φ+Ωsup0szg(.,s)φ<. $

    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

    $ {Δζ=aζα in Ω+,ζ=0 on Ω+,ζ>0 in Ω+, $ (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

    $ gε(.,s):=s(s+ε)1g(.,s), $ (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

    $ limmΩlFmγl,mθlg(.,uεl)φ=Ωθφ, $ (3.2)
    $ limmΩlFmγl,muεl,φ=Ωu,φ $ (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}, $

    $ auαεLmφalFmγl,muαεlφauαεLmφ, $ (3.4)

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

    $ limjΩauαεjφ=limj{a>0}auαεjφ={a>0}auαφ=Ωχ{u>0}auαφ, $ (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

    $ limm{a>0}alFmγl,muαεlφ=Ωχ{u>0}auαφ. $ (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,

    $ {Δ(lFmγl,muεl)=alFmγl,muαεllFmγl,mθlg(.,uεl) in Ω,lFmγl,muεl=0 on Ω $ (3.7)

    and so

    $ ΩlFmγl,muεl,φ=ΩalFmγl,muαεlφΩlFmγl,mθlg(.,uεl)φ. $ (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

    $ Ωu,φ=Ωχ{u>0}auαφΩθφ=Ωχ{u>0}auαφΩχ{u>0}g(.,u)φ{u=0}θφ. $ (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\{ Δu=auαg(.,u) in Ω,u=0 on Ω. \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

    $ cdΩucdτΩ a.e. in Ω, $ (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.$in$Ω.$Thus,byRemark2.4,thereexistpositiveconstants$c$and$τ$suchthat$ucdτΩ 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}, $

    $ Ωu,φk=Ωχ{u>0}(auαg(.,u))φk=Ω(auαχ{u>0}g(.,u))φk=Ω(χ{a>0}auαχ{u>0}g(.,u))φk. $ (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

    $ Ωu,φ=Ω(χ{a>0}auαχ{u>0}g(.,u))φ=Ωχ{u>0}(auαg(.,u))φ, $

    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

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