The Role of Short-term Consolidation in Memory Persistence

1.   Introduction and statement of the main results

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

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

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

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

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,

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

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

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

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

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.

2.   Preliminaries

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

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

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

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,

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

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

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),

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, $

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

and

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

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

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.,

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,

Also,

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,

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,

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

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

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),

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

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

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}, $

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

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

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

the Lebesgue dominated convergence theorem gives

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),

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

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,

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

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], $

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, $

$ \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

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

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,

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

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^{+}. $

3.   Proof of the main results

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

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

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}, $

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

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

(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,

and so

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

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

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), $

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

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}, $

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

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

Acknowledgment

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.

Conflict of interest

The author declare no conflicts of interest in this paper