New criteria on global Mittag-Leffler synchronization for Caputo-type delayed Cohen-Grossberg Inertial Neural Networks

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