**Positive weak solutions of elliptic Dirichlet problems** **with singularities in both the dependent and the**

**independent variables**

**Tomas Godoy**

^{B}

### and **Alfredo Guerin**

FaMAF, Universidad Nacional de Cordoba, Ciudad Universitaria, Cordoba, 5000, Argentina Received 29 April 2019, appeared 6 August 2019

Communicated by Maria Alessandra Ragusa

**Abstract.** We consider singular problems of the form −_{∆u} = k(·,u)−h(·,u) in Ω,
u = 0 on *∂Ω,* u > 0 in Ω, where Ω is a bounded C^{1,1} domain in **R**^{n}, n ≥ 2, h :
Ω×[0,∞) → [0,∞) and k : Ω×(0,∞) → [0,∞) are Carathéodory functions such
thath(x,·)is nondecreasing, andk(x,·)is nonincreasing and singular at the origina.e.

x ∈ Ω. Additionally, k(·,s) and h(·,s) are allowed to be singular on *∂Ω* for s > 0.

Under suitable additional hypothesis onhandk, we prove that the stated problem has
a unique weak solution u∈ H_{0}^{1}(_{Ω}), and thatubelongs to C Ω

. The behavior of the
solution near*∂Ω*is also addressed.

**Keywords:** singular elliptic problems, variational problems, sub-supersolutions
method, finite energy solutions, positive solutions.

**2010 Mathematics Subject Classification:** 35J75, 35D30, 35J20.

**1** **Introduction and statement of the main results**

Let Ω be a bounded domain in **R**^{n} with C^{1,1} boundary, and consider a singular semilinear
elliptic problem of the form

−_{∆}u=k(·_{,}u) in Ω,

u=0 on *∂Ω,*

u>0 in Ω,

(1.1)

where k : Ω×(0,∞) → [0,∞) is a Carathéodory function (i.e., k(·,s)is measurable for any
s ∈ (0,∞) and k(x,·) is continuous on (0,∞) a.e. x ∈ _{Ω), with} k = k(x,s) allowed to be
singular ats=0.

Singular problems like (1.1) arise, for instance, in the study of chemical catalysts process, non-Newtonian fluids, the temperature of some electrical conductors whose resistance de- pends on the temperature, thin films, and micro electro-mechanical systems (see e.g., [4,7,13, 15–17,26,33–35], and the references therein).

BCorresponding author. Email: godoy@mate.uncor.edu

Problem (1.1) was studied, in the case where k(x,s) = a(x)s^{−}* ^{α}*, under different sets of
assumptions onaand

*α, in [2,*8,11,13,16,23,36].

In [14] existence and nonexistence theorems were stated for Lane–Emden–Fowler equa- tions with convection and singular potential.

Recently, Chu, Gao and Gao [6], studied problems of the form −div(M(x)∇u) =
a(x)u^{−}^{α}^{(}^{x}^{)} in Ω, u = 0 on *∂Ω, where* a belongs to a suitable Lebesgue space. Among
other results, they found a very weak solution inH^{1}_{0}(_{Ω})(with test functions inC_{c}^{1}(_{Ω})) when
0<*α*<2 and 0<*α*∈ C Ω

.

Problems of the form (1.1), with k = k(x,s)singular at s = 0, and withk(·,s)allowed to
exhibit some kind of singularity on*∂*Ω, were studied in [1,24,28,30,31,37,38].

Diaz, Hernandez and Rakotoson [12] considered the problem

−_{∆u}= ad^{−}_{Ω}* ^{γ}*u

^{−}

*inΩ,*

^{β}u=0 on*∂*Ω,

u>0 inΩ,

(1.2)

where d_{Ω} := dist(·,*∂Ω*), *γ* < 2, and a ∈ L^{∞}(_{Ω}) satisfies inf_{Ω}a > 0. They studied the
existence of solutionsu ∈ L^{1}(_{Ω,}d_{Ω})(the d_{Ω}-weighted Lebesgue space) in the following very
weak sense:

−

Z

Ωu∆ϕ=

Z

Ωad^{−}_{Ω}^{γ}*ϕ* for any *ϕ*∈C^{2} Ω

such that *ϕ*=0 on *∂Ω.* (1.3)
Notice that the space of test functions involved in their notion of solution is strictly smaller
than the corresponding space in the present paper (as given in Definition1.1below). In Theo-
rem 2 they find, when*β*+*γ*<1, a very weak solution of problem (1.2), and prove that it be-
longs toW_{0}^{1}(_{Ω,}k · k_{N}_{(}_{r}_{)}_{,}_{∞})∩W_{loc}^{2,q}(_{Ω})for anyr ∈(0, 1)andq∈[_{1,}_{∞})_{, where}W_{0}^{1}(_{Ω,}k · k_{N}_{(}_{r}_{)}_{,}_{∞})
is the space of the functions w : Ω → ** _{R}** such that w ∈ W

^{1,1}(

_{Ω}) and |∇w| belongs to the Lorentz spaceL

^{N}

^{(}

^{r}

^{)}

^{,}

^{∞}(

_{Ω}), withN(r):=

_{n}

_{−}

^{n}

_{1}

_{+}

_{r}.

Regarding the case *β*+*γ* > 1, in theorem 1 they find a very weak solution of problem
(1.2) that belongs toC Ω

∩W_{loc}^{2,q}(_{Ω})for any q∈ [1,∞); and in Theorem 5, they prove that,
when *β*+*γ* > 1 and *γ* < 2, the solution that they found belongs to H^{1}_{0}(_{Ω}) if, and only if,
*β*+2γ<3. Additionally, in Theorem 4, they prove that, when*β*+*γ*>1, there exist positive
constantsc_{1}andc2such that the found solutionusatisfiesc_{1}d

2−*γ*
1+*β*

Ω ≤u≤c_{1}d

2−*γ*
1+*β*

Ω inΩ. However,
it is not obvious that, ifu∈ H_{0}^{1}(_{Ω}), then uis a weak solution of problem (1.2), i.e., that (1.3)
holds for any *ϕ*∈ H_{0}^{1}(_{Ω}).

The existence of classical solutionsu∈ C^{2}(_{Ω})∩C Ω

of problem (1.2) was addressed by
Mâagli [27] in the case when a ∈ C_{loc}* ^{σ}* (

_{Ω})for some

*σ*∈ (0, 1), and d

^{γ}_{Ω}a belongs to a suitable class related to the notion of Karamata classes. Our results heavily depend on those found in [27], which are summarized in Remark2.5below.

The interested reader can find an updated panoramic view of the area in the research books [19], [32], and in the survey article [18].

In this work we consider problem (1.1) when k : Ω×(0,∞) → [0,∞)is a Carathéodory function,k =k(x,s)is allowed to be singular ats =0, in the sense that

slim→0^{+}k(·,s) =_{∞} a.e. inΩ,
andk(·,s)is allowed to be singular on*∂Ω, in the sense that*

Ωlim3x→yk(x,s) =_{∞} for any (y,s)∈*∂*Ω×(0,∞).

Additionally, we allow the introduction of a second term, and consider the problem

−_{∆}u=k(·_{,}u)−h(·_{,}u) inΩ,

u=0 on *∂Ω,*

u>0 inΩ,

(1.4)

where h=h(·,s)is allowed to be singular on*∂Ω*for any s>0. Under some further assump-
tions on k and h, we prove existence and uniqueness results for weak solutions of problems
(1.1) and (1.4).

The notion of weak solution that we use in this work is the usual one, given by the follow- ing definition.

**Definition 1.1.** Let *ψ* : Ω → ** _{R}** be a measurable function such that

*ψϕ*∈ L

^{1}(

_{Ω}) for all

*ϕ*in H

_{0}

^{1}(

_{Ω}). We say thatu:Ω→

**is a weak solution of the problem**

_{R}−_{∆}u=*ψ* inΩ, u=0 on*∂*Ω, (1.5)

ifu ∈H_{0}^{1}(_{Ω})andR

Ωh∇u,∇*ϕ*i=R

Ω*ψϕ*for all *ϕ*in H_{0}^{1}(_{Ω}).

Similarly, for u ∈ H_{0}^{1}(_{Ω}), we say that u is a weak supersolution (respectively a weak
subsolution) of problem (1.5), and we write

(−_{∆u}≥*ψ* in Ω(resp. −_{∆u}≤ *ψ*inΩ),
u=0 on *∂Ω,*

to meanR

Ωh∇u,∇*ϕ*i ≥R

Ω*ψϕ*(resp. R

Ωh∇u,∇*ϕ*i ≤R

Ω*ψϕ) for all nonnegative* *ϕ*in H_{0}^{1}(_{Ω}).
**Definition 1.2.** Let d_{Ω} : Ω → ** _{R}** be the distance function d

_{Ω}:= dist(·,

*∂Ω*). For

*β*≥ 0 and

*γ*∈ [0, 2)define

*ϑ*

*:Ω→*

_{β,γ}**by:**

_{R}• *ϑ** _{β,γ}*:=d

_{Ω}if

*β*+

*γ*<1,

• *ϑ** _{β,γ}* := d

_{Ω}ln(

*ω*0d

^{−}

_{Ω}

^{1})if

*β*+

*γ*=1, where

*ω*0 is an arbitrary number, which we fix from now on, such that

*ω*

_{0}>diam(

_{Ω}),

• *ϑ** _{β,γ}*:=d

2−*γ*
1+*β*

Ω if *β*+*γ*>1.

We assume, from now on,n≥2. Let us state our main results.

**Theorem 1.3.** Let Ωbe a bounded domain in **R**^{n} with C^{1,1} boundary, and let k : Ω×(0,∞) → ** _{R}**
satisfy the following conditions:

k1) k is a nonnegative Carathéodory function;

k2) s→k(·_{,}s)is nonincreasing on(0,∞)a.e.x∈_{Ω};

k3) there exist*β*≥0,*γ*≥0,and B_{2} >0such that, for any s>0,k(·,s)≤B_{2}d^{−}_{Ω}* ^{γ}*s

^{−}

*a.e.inΩ;*

^{β}k4) there exist*δ*>0and B_{1} >0such that, for any s∈(0,*δ*),k(·,s)≥ B_{1}d^{−}_{Ω}* ^{γ}*s

^{−}

*a.e.inΩ.*

^{β}Let h:Ω×[0,∞)→** _{R}**satisfy the following conditions:

h1) h is a Carathéodory function;

h2) h(x,·)is nondecreasing on[0,∞),and h(·, 0) =0a.e.inΩ;

h3) h(·_{,}s) ≤ B_{3}d^{−}_{Ω}* ^{η}*s

^{p}a.e. inΩ for all s ∈ [0,∞), with B

_{3}>

_{0,}p >

_{1, 0}<

*η*<

*γ*+p+

*β*if

*β*+

*γ*≤1,and0<

*η*<

*γ*+ (p+

*β*)

^{2}

_{1}

^{−}

_{+}

^{γ}*β* if*β*+*γ*>1.

Then:

i) If *β*+2γ < 3, then problem (1.4) has a unique weak solution u ∈ H_{0}^{1}(_{Ω}). Moreover, u ∈
W_{loc}^{2,q}(_{Ω})∩C Ω

for any q∈[1,∞),and u satisfies cϑ* _{β,γ}* ≤ u≤c

^{0}

*ϑ*

*inΩ,for some positive constants c and c*

_{β,γ}^{0}.

ii) If problem(1.4)has a weak solution u∈ H^{1}_{0}(_{Ω})∩C Ω

,then *β*+2γ<3.

Note that, in particular, Theorem1.3says that−_{∆u}= d^{−}_{Ω}* ^{γ}* in Ω,u=0 on

*∂Ω,*u> 0 inΩ, has a weak solution if, and only if,

*γ*<

^{3}

_{2}.

The next theorem states that, when h is identically zero, the assertion i) of Theorem1.3 remains valid if the conditionk4)is replaced by the following milder condition:

k5) there exist *δ* > 0 and a measurable set E ⊂ _{Ω} such that |E| > _{0 and inf}_{E}_{×(}_{0,δ}_{)}_{k} > _{0,}
where inf stands for the essential infimum, and|E|denotes the Lebesgue measure ofE.

**Theorem 1.4.** Let Ωbe a bounded domain in **R**^{n} with C^{1,1} boundary, and let k : Ω×(0,∞) → ** _{R}**
satisfy the conditions k1)–k3) of Theorem1.3. Assume that

*β*+

*γ*<

^{3}

_{2}, and that the condition k5) holds. Then problem(1.1) has a unique weak solution u ∈ H

^{1}

_{0}(

_{Ω}),and u ∈ W

_{loc}

^{2,q}(

_{Ω})∩C Ω

for
any q∈ [1,∞),and cd_{Ω} ≤u≤c^{0}*ϑ** _{β,γ}* inΩ,for some positive constants c and c

^{0}.

Concerning the case whenhis nonidentically zero, our next result shows that the assertion i)of theorem1.3holds under a weaker condition thank4), at the expense of strengtheningh3).

**Theorem 1.5.** Let Ωbe a bounded domain in **R**^{n} with C^{1,1} boundary, and let k : Ω×(0,∞) → ** _{R}**
satisfy the conditions k1)–k3) of Theorem1.3, with

*β*and

*γ*satisfying

*β*>0,

*γ*∈ [0, 2),and

*β*+

*γ*<

3

2. Assume also the following condition:

k6) there exist*δ* >0and B_{1}>0such that, for any s∈(0,*δ*),k(·,s)≥B_{1}s^{−}* ^{β}* a.e.inΩ.

Let h: Ω×[0,∞)→** _{R}**satisfy the conditions h1) and h2) of Theorem1.3, and the following

h4) h(·,s)≤ B_{3}d^{−}_{Ω}* ^{η}*s

^{p}a.e.inΩfor all s∈[0,∞),with B

_{3}>0,p >1, 0<

*η*< p−1if

*β*+

*γ*≤1 and0<

*η*<(p−1)

^{2}

_{1}

^{−}

_{+}

^{γ}*β* if1< *β*+*γ*< ^{3}_{2}.

Then problem (1.4) has a unique weak solution u. Moreover, u∈W_{loc}^{2,q}(_{Ω})∩C Ω

for any q∈ [1,∞),
and there exist positive constants c and c^{0} such that c^{0}*ϑ** _{β,0}* ≤u≤cϑ

*inΩ.*

_{β,γ}**Remark 1.6.** Let us stress that the strength of the singularity, which is the theme in the back-
ground of the present work, needs to be limited if one expects weak solutions in H_{0}^{1}(_{Ω}).
Indeed, Lazer and McKenna [23] considered the problem −_{∆}u = au^{−}* ^{α}* in Ω, u =

_{0 on}

*∂*Ω, u > 0 in Ω, under the assumptions a ∈ C

*Ω*

^{γ}, min_{Ω}a > 0, *α* > 0, and Ωa bounded reg-
ular domain. They proved that there exists a unique solution u ∈ C^{2}(_{Ω})∩C Ω

; and that
u ∈ H_{0}^{1}(_{Ω})if, and only if, *α* < 3. A clear-cut simple condition like that is elusive when the
right hand side of the equation is not in the form au^{−}* ^{α}*; in [21] we addressed such a more
general situation, but still did not consider the case when a spatial singularity is added. This
latter situation is considered in the present work.

The paper is organized as follows: in Section 2, we collect some preliminary results.

Lemma2.4is an adaptation of Lemma 3.2 in [22] and states that, under suitable conditions, a
solution in the sense of distributions of an elliptic problem, is also a weak solution in H_{0}^{1}(_{Ω}).
Remark2.5recalls a result, due to Mâagli [27], about existence, uniqueness, and behavior near
the boundary, of positive classical solutions of problems of the form −_{∆u} = ad^{−}_{Ω}* ^{γ}*u

^{−}

*in Ω, u= 0 on*

^{β}*∂Ω*(for a suitable class of Hölder continuous functions a); and Remark 2.7recalls a sub-supersolution theorem for singular problems due to Loc and Schmitt [25]. In Section 3 we prove Theorems1.3,1.4and1.5, by combining the results of [27], [25], and Lemma2.4, jointly with some additional auxiliary results.

**2** **Preliminaries**

For w ∈ L^{1}_{loc}(_{Ω}), we write, as usual, w ∈ H_{0}^{1}(_{Ω})^{}^{0} to mean that wϕ ∈ L^{1}(_{Ω}) for any
*ϕ*∈ H_{0}^{1}(_{Ω}), and that the map *ϕ*→R

Ωwϕis continuous onH_{0}^{1}(_{Ω}).

**Remark 2.1.** Let us recall the Hardy inequality (as stated, e.g., in [29, Theorem 1.10.15], see
also [3, p. 313]): There exists a positive constant c such that

^{ϕ}

d_{Ω}

L^{2}(_{Ω}) ≤ ck∇*ϕ*k_{L}2(_{Ω}) for all
*ϕ*∈ H_{0}^{1}(_{Ω}).

**Remark 2.2.** If *ψ* ∈ L^{1}_{loc}(_{Ω}) and d_{Ω}*ψ* ∈ L^{2}(_{Ω}), then *ψϕ* ∈ L^{1}(_{Ω}) for any *ϕ* ∈ H_{0}^{1}(_{Ω}).
Moreover, *ψ* ∈ H_{0}^{1}(_{Ω})^{}^{0} and k*ψ*k_{(}_{H}1

0(_{Ω}))^{0} ≤ ckd_{Ω}*ψ*k_{2} with c independent of *ψ. Indeed, for*
*ϕ* ∈ H_{0}^{1}(_{Ω}), from the Hölder and the Hardy inequalities, R

Ω|*ψϕ*| ≤ kd_{Ω}*ψ*k_{2}kd^{−}_{Ω}^{1}*ϕ*k_{2} ≤
ckd_{Ω}*ψ*k_{2}k*ϕ*k_{H}1

0(_{Ω}), wherecis the constant in the Hardy inequality of Remark2.1.

**Remark 2.3.** (See e.g., [10])*λ*∈** _{R}**is called a principal eigenvalue for−

_{∆}inΩ, with homoge- neous Dirichlet condition and weight function b∈ L

^{∞}(

_{Ω}), if the problem−

_{∆}

*φ*=

*λbφ*in Ω,

*φ*=0 on

*∂Ω*has a solution

*ϕ*

_{1}(called a principal eigenfunction) such that

*ϕ*

_{1}>0 inΩ. It is a well known fact that, for any C

^{1,1}bounded domain Ω⊂

_{R}^{n}, b∈ L

^{∞}(

_{Ω}), and b

^{+}6≡0, there exists a unique positive principal eigenvalue

*λ*

_{1}(b), and its eigenspace V

_{λ}_{1}is a one dimen- sional subspace of C

^{1}Ω

. Moreover, for each positive *ϕ*_{1} ∈ V_{λ}_{1}, there are positive constants
c_{1},c_{2} such thatc_{1}d_{Ω} ≤ *ϕ*_{1} ≤c_{2}d_{Ω} in Ω. Consequently,|ln(*ϕ*_{1})| ∈L^{1}(_{Ω}); and *ϕ*_{1}^{t} ∈ L^{1}(_{Ω})if,
and only if,t >−1.

The following lemma is an adaptation of Lemma 3.2 in [22]

**Lemma 2.4.** Let *ψ* ∈ L^{∞}_{loc}(_{Ω})be such that |*ψ*| ∈ H^{1}_{0}(_{Ω})^{}^{0}, and let u ∈ W_{loc}^{1,2}(_{Ω})∩C(_{Ω}) be a
solution, in the sense of distributions, of the problem

−_{∆u}=*ψ* inΩ.

If there exist constants c >_{0} _{and r}> ^{1}_{2} _{such that}_{0}≤ u ≤ cd^{r}_{Ω} inΩ,then u ∈ H_{0}^{1}(_{Ω})∩C^{1}(_{Ω})∩
C Ω

,and u is a weak solution of−_{∆u}=*ψ*inΩ,u=0on*∂Ω.*

Proof. Let *ϕ*∈ H_{0}^{1}(_{Ω})such that supp(*ϕ*)⊂_{Ω. Then}R

Ωh∇u,∇*ϕ*i=R

Ω*ψϕ. Indeed, letδ*>0
be such that supp(*ϕ*)⊂ _{Ω}* _{δ}*, and let

*ϕ*_{j} _{j}_{∈}** _{N}**be a sequence inC

^{∞}

_{c}(

_{Ω})satisfying supp

*ϕ*

_{j}

⊂
Ω*δ* for allj, and such that

*ϕ*_{j} _{j}_{∈}** _{N}**converges to

*ϕ*in H

^{1}

_{0}(

_{Ω}

*). Now,∇u*

_{δ}_{|}

_{Ω}

*∈ L*

_{δ}^{2}(

_{Ω}

*,*

_{δ}**R**

^{n}), and so

*ζ*→ R

Ω*δ*h∇u,∇*ζ*iis continuous on H_{0}^{1}(_{Ω}* _{δ}*). Also, R

Ω

∇u,∇*ϕ*_{j}

= R

Ω*ψϕ*_{j} for all j. Then
R

Ωh∇u,∇*ϕ*i=_{lim}_{j}_{→}_{∞}R

Ω

∇u,∇*ϕ*_{j}

=_{lim}_{j}_{→}_{∞}R

Ω*ψϕ*_{j} =R

Ω*ψϕ.*

For eachj∈_{N, let}*µ*_{j} :**R**→** _{R}**be defined by

*µ*

_{j}(s):=0 ifs ≤

^{1}

_{j},

*µ*

_{j}(s):=−3j

^{2}s

^{3}+14js

^{2}− 19s+

^{8}

_{j}

_{if}

^{1}

_{j}< s<

^{2}

_{j}

_{, and}

*µ*

_{j}(s):= sif

^{2}

_{j}≤s. Then

*µ*

_{j}∈C

^{1}(

**),**

_{R}*µ*

_{j}(s) =0 fors<

^{1}

_{j}

_{,}

*µ*

^{0}

_{j}(s)≥

_{0}for

^{1}

_{j}<s<

^{2}

_{j}, and

*µ*

^{0}

_{j}(s) =1 for

^{2}

_{j}<s. Also, 0<

*µ*

_{j}(s)<sfor all s∈ 0,

^{2}

_{j}

.
Let*µ*_{j}(u):=*µ*_{j}◦u. Then, for allj,∇ *µ*_{j}(u)^{}= *µ*^{0}_{j}◦u

∇uin D^{0}(_{Ω}). Sinceu∈W_{loc}^{1,2}(_{Ω}),
it follows that*µ*_{j}(u)∈W_{loc}^{1,2}(_{Ω}). Since supp *µ*_{j}(u)^{}⊂_{Ω, we have}*µ*_{j}(u)∈ H^{1}_{0}(_{Ω}). Therefore,
for allj,R

Ω

∇u,∇ *µ*_{j}(u)^{}= R

Ω*ψµ*_{j}(u), i.e.,
Z

{u>0}

*µ*^{0}_{j}◦u

|∇u|^{2} =

Z

Ω*ψµ*_{j}(u). (2.1)

Now, *µ*^{0}_{j}◦u

|∇u|^{2} is nonnegative and lim_{j}_{→}_{∞} *µ*^{0}_{j}◦u

|∇u|^{2} = |∇u|^{2} a.e. inΩ, and so, from
(2.1) and Fatou’s lemma, we have

Z

Ω|∇u|^{2} ≤lim_{j}_{→}_{∞}
Z

Ω*ψµ*_{j}(u).

Let *ϕ*_{1} be the principal eigenfunction for −_{∆} in Ω with homogeneous Dirichlet condition,
and with weight function**1, normalized by** k*ϕ*_{1}k_{∞} = 1. Since, for some positive constantc^{0},
u≤ c^{0}*ϕ*^{r}_{1}in Ω, and*ϕ*^{r}_{1} ∈ H_{0}^{1}(_{Ω}), we have *ψu*∈ L^{1}(_{Ω}). Now, lim_{j}→_{∞}*ψµ*_{j}(u) = *ψu*inΩ, and,
for anyj∈_{N,}^{}*ψµ*_{j}(u)^{}≤ |*ψu*|inΩ. Then, Lebesgue’s dominated convergence theorem gives

jlim→_{∞}
Z

Ω*ψµ*_{j}(u) =

Z

Ω*ψu*<_{∞.}

ThusR

Ω|∇u|^{2} <_{∞}, and sou ∈H^{1}(_{Ω}). As−_{∆}u=*ψ*in D^{0}(_{Ω}),u∈ L^{∞}(_{Ω})and*ψ*∈ L^{∞}_{loc}(_{Ω}),
then the inner elliptic estimates (as stated e.g., in [5], Proposition 4.1.2, see also [20], Theorem
9.11) give that u ∈ C^{1}(_{Ω}). From 0 ≤ u ≤ cd^{r}_{Ω} in Ω, and u ∈ C(_{Ω}), we conclude that
u∈C Ω

. Sinceu∈ H^{1}(_{Ω}), u∈C Ω

, andu=0 on*∂*Ω, we getu∈ H_{0}^{1}(_{Ω}).
Asu∈ H_{0}^{1}(_{Ω}), we have that *ϕ*→R

Ωh∇u,∇*ϕ*iis continuous on H^{1}_{0}(_{Ω}). Therefore, since
C^{∞}_{c} (_{Ω})is dense inH^{1}_{0}(_{Ω}), and since, for any*ϕ*∈C_{c}^{∞}(_{Ω}),

Z

Ωh∇u,∇*ϕ*i=

Z

Ω*ψϕ,* (2.2)

we conclude that (2.2) holds for all *ϕ*∈ H_{0}^{1}(_{Ω}).

**Remark 2.5.** i) Let*ω*_{0}be as in Definition1.2,*α*<1, and *ρ*<2. Letz∈C([0,*ω*_{0}])be such that
z(0) = 0 and R*ω*_{0}

0 t^{1}^{−}* ^{ρ}*L

_{z}(t)dt <

_{∞}, where L

_{z}(t) := exp R

*ω*

_{0}t

z(s) s ds

. Let*σ* ∈ (0, 1), and let
a∈ C_{loc}* ^{σ}* (

_{Ω})satisfy, for some constantc>0,

1

cL_{z}◦d_{Ω} ≤d^{ρ}_{Ω}a≤cL_{z}◦d_{Ω} inΩ. (2.3)
Then, Theorem 1 in [27] says that the problem

−_{∆u}=au* ^{α}* inΩ,

u=0 on *∂Ω,*

u>0 inΩ

(2.4)

has a unique classical solutionu ∈ C^{2}(_{Ω})∩C Ω

; and that, for some positive constant c^{0}, u
satisfies,

c^{0}−1

*θ** _{ρ}*◦d

_{Ω}≤u≤c

^{0}

*θ*

*◦d*

_{ρ}_{Ω}inΩ,

where

*θ** _{ρ}*(t):=

_{Z}

_{ω}0

0

L_{z}(s)
s ds

_{1}_{−}^{1}

*α*

if *ρ* =2,
*θ**ρ*(t):=t^{2}^{1}^{−}^{−}^{α}* ^{ρ}* (L

_{z}(t))

^{1}

^{−}

^{1}

*if 1+*

^{α}*α*<

*ρ*<2,

*θ*

*(t):=t*

_{ρ}_{Z} _{ω}

0

t

L_{z}(s)
s ds

_{1}_{−}^{1}

*α*

if*ρ*=_{1}+* _{α,}*
and

*θ**ρ*(t):=t i f *ρ* <1+*α.*

ii) Let *β*≥ 0, and let*γ*< 2. If in i) we take*α*:=−*β,* z :=**0**(thenL_{z} =**1), and***ρ* :=*γ, we get*
that the problem

−_{∆}v=d^{−}_{Ω}* ^{γ}*v

^{−}

*inΩ,*

^{β}v=0 on*∂Ω,*

v>0 inΩ

(2.5)

has a unique classical solutionv* _{β,γ}* ∈C

^{2}(

_{Ω})∩C Ω

; and that there exists positive constants
c_{1} andc_{2} such that

c_{1}*ϑ** _{β,γ}* ≤v

*≤c*

_{β,γ}_{2}

*ϑ*

*inΩ, (2.6) where*

_{β,γ}*ϑ*

*is as in Definition1.2.*

_{β,γ}**Remark 2.6.** Let*β*≥0 and*γ*≥0 be such that *β*+2γ<3.

i) d^{1}_{Ω}^{−}^{γ}*ϑ*^{−}_{β,γ}* ^{β}* ∈ L

^{2}(

_{Ω}). Indeed, if

*β*+

*γ*< 1 then d

^{1}

_{Ω}

^{−}

^{γ}*ϑ*

^{−}

_{β,γ}*= d*

^{β}^{1}

_{Ω}

^{−}

^{β}^{−}

*∈ L*

^{γ}^{∞}(

_{Ω}) ⊂ L

^{2}(

_{Ω}). If

*β*+

*γ*=1, then d

^{1}

_{Ω}

^{−}

^{γ}*ϑ*

^{−}

_{β,γ}*= ln*

^{β}

^{ω}_{d}

^{0}

Ω

*β*

∈ L^{∞}(_{Ω})⊂ L^{2}(_{Ω}). If*β*+*γ* >1, then d^{1}_{Ω}^{−}^{γ}*ϑ*^{−}_{β,γ}* ^{β}* =
d

^{1}

^{−}

^{γ}^{−}

^{β}2−*γ*
1+*β*

Ω =d^{−}

1

*β*+1(*β*+*γ*−1)

Ω and, since*β*+2γ<3,− ^{2}

*β*+1(*β*+*γ*−1)>−1, and so, again in
this case,d^{1}_{Ω}^{−}^{γ}*ϑ*^{−}_{β,γ}* ^{β}* ∈ L

^{2}(

_{Ω})(because, forr∈

**d**

_{R,}^{r}

_{Ω}∈L

^{2}(

_{Ω})whenever 2r+1>0).

ii) There exist positive constantscand*τ*> ^{1}_{2} _{such that}*ϑ** _{β,γ}* ≤cd

^{τ}_{Ω}inΩ. Indeed, if

*β*+

*γ*<

_{1}then

*ϑ*

*=d*

_{β,γ}_{Ω}, if

*β*+

*γ*= 1 then

*ϑ*

*= d*

_{β,γ}_{Ω}ln

^{ω}_{d}

^{0}

Ω

and so, for any *ε*> 0,*ϑ** _{β,γ}* d

^{1}

_{Ω}

^{−}

*−1*

^{ε}=
d^{ε}_{Ω}ln ^{ω}_{d}^{0}

Ω

∈ L^{∞}(_{Ω}); and if*β*+*γ*>1, then*ϑ** _{β,γ}* =d

2−*γ*
1+*β*

Ω and, since*β*+2γ<3, ^{2}_{1}^{−}_{+}^{γ}* _{β}* >

^{1}

_{2}. iii) Letv

*be the solution of problem (2.5) given by Remark2.5. From i) and (2.6), it follows*

_{β,γ}thatd^{1}_{Ω}^{−}* ^{γ}*v

^{−}

_{β,γ}*∈ L*

^{β}^{2}(

_{Ω}).

iv) Letv* _{β,γ}* be as in iii). Then, by iii) and Remark 2.2,d

^{−}

_{Ω}

*v*

^{γ}^{−}

_{β,γ}*∈ H*

^{β}_{0}

^{1}(

_{Ω})

^{}

^{0}.

v) Letv* _{β,γ}* be as in iii). Sinced

^{−}

_{Ω}

*v*

^{γ}^{−}

_{β,γ}*∈ H*

^{β}_{0}

^{1}(

_{Ω})

^{}

^{0}and since, by Remark2.5,v

*∈C*

_{β,γ}^{2}(

_{Ω})∩ C Ω

, Lemma 2.4gives that v* _{β,γ}* ∈ H

_{0}

^{1}(

_{Ω}), and thatv

*is a weak solution of problem (2.5). Moreover, since v*

_{β,γ}*∈ L*

_{β,γ}^{∞}(

_{Ω}) andd

^{−}

_{Ω}

*v*

^{γ}^{−}

_{β,γ}*∈ L*

^{β}^{∞}

_{loc}(

_{Ω}), the inner elliptic estimates givev

*∈W*

_{β,γ}_{loc}

^{2,q}(

_{Ω})for anyq∈[1,∞).

**Remark 2.7.** Let g: Ω×(0,∞)→** _{R}**be a Carathéodory function. We say thatw ∈ L

^{1}

_{loc}(

_{Ω})is a subsolution (supersolution) of the problem

−_{∆}z =g(·_{,}z) inΩ (2.7)

in the sense of distributions, if, and only if: w>0 inΩ,g(·,w)∈L^{1}_{loc}(_{Ω}), andR

Ωh∇w,∇*ϕ*i ≤
(≥)R

Ωg(·,w)*ϕ* for all nonnegative *ϕ*∈C^{∞}_{c} (_{Ω}). We say thatz∈ L^{1}_{loc}(_{Ω})is a solution, in the
sense of distributions, of (2.7) if, and only if,z > 0 a.e. inΩ, and,R

Ωh∇z,∇*ϕ*i= R

Ωg(·,z)*ϕ*
for all *ϕ*∈ C_{c}^{∞}(_{Ω}).

According to Theorem 2.4 in [25], if (2.7) has a subsolutionzand a supersolution z(in the
sense of distributions), both inL^{∞}_{loc}(_{Ω})∩W_{loc}^{1,2}(_{Ω}), and such such that 0<z ≤z inΩ, and if
there exists*ψ* ∈ L^{∞}_{loc}(_{Ω})such that |g(x,s)| ≤ *ψ*(x) a.e. x ∈ _{Ω}for all s ∈ [z(x),z(x)]; then
(2.7) has a solutionzin the sense of distributions, which satisfies z≤z≤z inΩ.

**3** **Proof of the main results**

**Remark 3.1.** Let *ϕ*_{1} be a positive principal eigenfunction of −_{∆} in Ω, with homogeneous
Dirichlet boundary condition. If r > ^{1}_{2}, then *ϕ*^{r}_{1} ∈ H_{0}^{1}(_{Ω}). Indeed, *ϕ*^{r}_{1} ∈ L^{2}(_{Ω}). Also,
*ϕ*_{1}^{r}^{−}^{1} ∈ L^{2}(_{Ω})and|∇*ϕ*_{1}| ∈ L^{∞}(_{Ω}), thus∇(*ϕ*^{r}_{1})∈ L^{2}(_{Ω}).

**Lemma 3.2.** Let*η*and p be as in the condition h3) of Theorem1.3. Then:

i) d^{γ}_{Ω}^{−}^{η}*ϑ*_{β,γ}^{p}^{+}* ^{β}* ∈ L

^{∞}(

_{Ω}).

ii) If, in addition,*β*+2γ<3,then d^{1}_{Ω}^{−}^{η}*ϑ*_{β,γ}^{p} ∈L^{2}(_{Ω})and d^{1}_{Ω}^{−}^{γ}*ϑ*^{−}_{β,γ}* ^{β}* ∈ L

^{2}(

_{Ω}).

Proof. i) follows directly from the definition of *ϑ** _{β,γ}* and the facts that

*η*<

*γ*+p+

*β*when

*β*+

*γ*≤ 1, and that

*η*<

*γ*+ (p+

*β*)

^{2}

_{1}

^{−}

_{+}

^{γ}*when*

_{β}*β*+

*γ*> 1, and using, when

*β*+

*γ*= 1, that d

^{ε}_{Ω}ln

^{ω}_{d}

^{0}

Ω

∈ L^{∞}(_{Ω})for any*ε*>0.

To see the first assertion of ii)note that, byh3), 2(1−*η*+p) > 0 when *β*+*γ* ≤ 1. Now,
d^{1}_{Ω}^{−}^{η}*ϑ*_{β,γ}^{p} 2

= d^{2}_{Ω}^{(}^{1}^{−}^{η}^{+}^{p}^{)} when *β*+*γ* < 1, and (d^{1}_{Ω}^{−}^{η}*ϑ*_{β,γ}^{p} )^{2} = d^{2}_{Ω}^{(}^{1}^{−}^{η}^{+}^{p}^{)} ln ^{ω}_{d}^{0}

Ω

2p

when *β*+
*γ* = 1. Thus, in both cases, d^{1}_{Ω}^{−}^{η}*ϑ*_{β,γ}^{p} ∈ L^{∞}(_{Ω}) ⊂ L^{2}(_{Ω}). If *β*+*γ* > 1, then d^{1}_{Ω}^{−}^{η}*ϑ*^{p}* _{β,γ}* =
d

^{1}

^{−}

^{η}^{+}

^{p}

2−*γ*
1+*β*

Ω and, byh3), 2

1−*η*+p2−*γ*
1+*β*

+1>2

1−

*γ*+ (p+*β*)^{2}−*γ*
1+*β*

+p2−*γ*
1+*β*

+1

= ^{3}−*β*−2γ
*β*+1 >0.

Thus, again in this case,d^{1}_{Ω}^{−}^{η}*ϑ*^{p}* _{β,γ}* ∈ L

^{2}(

_{Ω}).

Finally, d^{1}_{Ω}^{−}^{γ}*ϑ*^{−}_{β,γ}* ^{β}* = d

^{1}

_{Ω}

^{−}

^{β}^{−}

*∈ L*

^{γ}^{∞}(

_{Ω}) ⊂ L

^{2}(

_{Ω}) when

*β*+

*γ*< 1, and d

^{1}

_{Ω}

^{−}

^{γ}*ϑ*

^{−}

_{β,γ}*= ln*

^{β}

^{ω}_{d}

^{0}

Ω

−*β*

∈ L^{∞}(_{Ω})⊂ L^{2}(_{Ω})when*β*+*γ*=1. If*β*+*γ*>1, thend^{1}_{Ω}^{−}^{γ}*ϑ*^{−}_{β,γ}* ^{β}* =d

^{1}

^{−}

^{γ}^{−}

^{β}2−*γ*
1+*β*

Ω and,
since 2 1−*γ*−*β*^{2}_{1}^{−}_{+}^{γ}_{β}

+1= ^{3}^{−}^{β}^{−}^{2γ}

*β*+1 >0, we have, again in this case,d^{1}_{Ω}^{−}^{γ}*ϑ*^{−}_{β,γ}* ^{β}* ∈ L

^{2}(

_{Ω}).

**Remark 3.3.**Assume the conditions k1), k3), h1), and h3) of Theorem1.3. Assume also that

*β*+2γ<3. Then, for any

*ε*>0,k(·,

*εv*

*)andh ·,*

_{β,γ}*εv*

_{β,γ}belong to H_{0}^{1}(_{Ω})^{}^{0}. Indeed, by k1)
and h1),k ·,*εv*_{β,γ}

andh ·,*εv*_{β,γ}

are measurable functions, and by k3) and h3),
d_{Ω}k ·,*εv*_{β,γ}

≤ *ε*^{−}* ^{β}*B2d

^{1}

_{Ω}

^{−}

*v*

^{γ}^{−}

_{β,γ}*≤*

^{β}*ε*

^{−}

*B2c*

^{β}^{−}

_{2}

*d*

^{β}^{1}

_{Ω}

^{−}

^{γ}*ϑ*

^{−}

_{β,γ}*a.e. inΩ, d*

^{β}_{Ω}h ·,

*εv*

_{β,γ}≤ *ε*^{p}B3d^{1}_{Ω}^{−}* ^{η}*v

^{p}

*≤*

_{β,γ}*ε*

^{p}B3c

_{3}

^{p}d

^{1}

_{Ω}

^{−}

^{η}*ϑ*

_{β,γ}^{p}a.e. in Ω.

Then, by Lemma3.2,d_{Ω}k ·,*εv*_{β,γ}

andd_{Ω}h ·,*εv*_{β,γ}

belong to L^{2}(_{Ω}), and so, by Remark2.2,
k ·_{,}*εv*_{β,γ}

andh ·_{,}*εv*_{β,γ}

belong to H_{0}^{1}(_{Ω})^{}^{0}.

**Lemma 3.4.** Assume the conditions k1), k3), k4), h1), and h3), of Theorem 1.3, and let v* _{β,γ}* be the
solution, given by Remark2.5, of problem(2.5). Then, for any

*ε*positive and small enough,

*εv*

*is a subsolution, in the sense of distributions, of problem (1.4) and, if in addition,*

_{β,γ}*β*+2γ < 3,then

*εv*

*is a weak subsolution of (1.4).*

_{β,γ}Proof. By Lemma3.2 i),there exists a positive constantc_{1} such thatd^{−}_{Ω}^{η}*ϑ*_{β,γ}^{p} ≤c_{1}d^{−}_{Ω}^{γ}*ϑ*^{−}_{β,γ}* ^{β}* in Ω,
and by Remark 2.5, there exist positive constants c2 and c3 such that c2

*ϑ*

*≤ v*

_{β,γ}*≤ c3*

_{β,γ}*ϑ*

*in Ω. Thus, for some positive constantc*

_{β,γ}_{4},d

^{−}

_{Ω}

*v*

^{η}

_{β,γ}^{p}≤ c

_{1}d

^{−}

_{Ω}

*v*

^{γ}^{−}

_{β,γ}*in Ω. Then, for any*

^{β}*ε*positive and small enough,

^{1}

_{2}

*ε*

^{−}

*B*

^{β}_{1}d

^{−}

_{Ω}

*v*

^{γ}^{−}

_{β,γ}*≥*

^{β}*ε*

^{p}B

_{3}d

^{−}

_{Ω}

*v*

^{η}^{p}

*in Ω. By diminishing*

_{β,γ}*ε*if necessary, we can assume that, in addition,

*ε*< min

1,_{k}_{v} ^{δ}

*β,γ*k_{∞},^{1}_{2}B_{1} . By Remark 2.5, v* _{β,γ}* satisfies, in the sense
of distributions,

−_{∆v}* _{β,γ}* = d

^{−}

_{Ω}

*v*

^{γ}^{−}

_{β,γ}*inΩ. (3.1) Then, in the sense of distributions,*

^{β}−_{∆} *εv*_{β,γ}

=*εd*^{−}_{Ω}* ^{γ}*v

^{−}

_{β,γ}*≤*

^{β}^{1}

2*ε*^{−}* ^{β}*B

_{1}d

^{−}

_{Ω}

*v*

^{γ}^{−}

_{β,γ}*(3.2)*

^{β}≤*ε*^{−}* ^{β}*B

_{1}d

^{−}

_{Ω}

*v*

^{γ}^{−}

_{β,γ}*−*

^{β}*ε*

^{p}B

_{3}d

^{−}

_{Ω}

*v*

^{η}^{p}

*≤ k ·,*

_{β,γ}*εv*

_{β,γ}−h ·,*εv*_{β,γ}

inΩ;

where, in the last inequality, we have used that, since *ε*< _{k} ^{δ}

v*β,γ*k_{∞} we have *εv** _{β,γ}* ≤

*δ*in Ω, and then, byk4),k ·,

*εv*

_{β,γ}≥*ε*^{−}* ^{β}*B

_{1}d

^{−}

_{Ω}

*v*

^{γ}^{−}

_{β,γ}*a.e. inΩ. Thus, for any nonnegative*

^{β}*ϕ*∈C

^{∞}

_{c}(

_{Ω}), Z

Ω

∇ *εv** _{β,γ}*
,∇

*ϕ*

≤

Z

Ω k ·,*εv*_{β,γ}

−h ·,*εv*_{β,γ}

*ϕ.* (3.3)

and so *εv** _{β,γ}* is a subsolution, in the sense of distributions, of problem (1.4). Now suppose

*β*+2γ < 3. By Remark2.6 v),

*εv*

*∈ H*

_{β,γ}_{0}

^{1}(

_{Ω}), and by Remark 3.3, k ·,

*εv*

_{β,γ}andh ·,*εv** _{β,γ}*
belong to H

_{0}

^{1}(

_{Ω})

^{}

^{0}. Thus k ·,

*εv*

_{β,γ}−h ·,*εv*_{β,γ}

*ϕ*∈L^{1}(_{Ω})for any *ϕ*∈ H_{0}^{1}(_{Ω})and, since
C_{c}^{∞}(_{Ω})is dense inH_{0}^{1}(_{Ω}), it follows that (3.3) holds for any nonnegative *ϕ*∈ H_{0}^{1}(_{Ω}).
**Remark 3.5.** Let us recall the following well known result: Let g : Ω×(0,∞) → ** _{R}** be a
Carathéodory function such thats→ g(x,s)is nonincreasing for a.e. x∈Ω, and consider the
problem

−_{∆u}= g(·,u) inΩ,

u=0 on*∂Ω,*

u>_{0} _{in}_{Ω.}

(3.4)

Let u ∈ H_{0}^{1}(_{Ω}) be a weak subsolution of problem (3.4) and let u ∈ H_{0}^{1}(_{Ω}) be a weak su-
persolution of the same problem. Then u ≤ u a.e. in Ω. Indeed, we have, in weak sense,

−_{∆}(u−u)≤ g(·,u)−g(·,u),u−u=0 on*∂Ω. Taking*(u−u)^{+}as test function, and noting
that (g(·,u)−g(·,u)) (u−u)^{+}≤0 a.e. inΩ, we conclude thatu≤u.

**Lemma 3.6.** Assume the conditions k1), k4), h1), and h3) of Theorem1.3. If problem(1.4)has a weak
solution u∈ H_{0}^{1}(_{Ω})∩C Ω

,then*γ*≤ ^{3}_{2}_{.}

Proof. Let u ∈ H_{0}^{1}(_{Ω})∩C Ω

be a weak solution of problem (1.4). For *ρ* > 0, let A* _{ρ}* :=
{x∈

_{Ω}:d

_{Ω}(x)≤

*ρ*}. Sinceu∈ C Ω

andu=0 on*∂*Ω, there exists*ρ*> 0 such thatu≤*δ* in
A*ρ*. Then, byk4),

k(·,u)≥B_{1}d^{−}_{Ω}* ^{γ}*u

^{−}

*a.e. in A*

^{β}*ρ*. (3.5) Let

*ϕ*

_{1}be a positive principal eigenfunction for−

_{∆}in Ω, with homogeneous Dirichlet condi- tion and weight function 1. Let

*ε*> 0, and let

*ϕ*:=

*ϕ*

1
2+*ε*

1 . Then *ϕ* ∈ H_{0}^{1}(_{Ω})∩L^{∞}(_{Ω}). Note
that, byk1) and h1), k(·,u)*ϕ* and h(·,u)*ϕ* are nonnegative measurable functions, and that,
byh3),

h(·,u)*ϕ*=d_{Ω}h(·,u)d^{−}_{Ω}^{1}*ϕ*≤ B_{3}d^{2}_{Ω}^{−}* ^{η}*kuk

^{p}

_{∞}

^{−}

^{1}

^{}

^{}

_{}d

^{−}

_{Ω}

^{1}u d

^{−}

_{Ω}

^{1}

*ϕ*

≤B_{3}kd_{Ω}k^{2}_{∞}^{−}* ^{η}*kuk

_{∞}

^{p}

^{−}

^{1}

^{}

^{}

_{}d

^{−}

_{Ω}

^{1}u d

^{−}

_{Ω}

^{1}

*ϕ*

a.e. inΩ;

and so, by the Hölder and the Hardy inequalities,h(·,u)*ϕ*∈ L^{1}(_{Ω}). Now, taking into account
(3.5) and thatkis nonnegative,

B_{1}kuk^{−}_{∞}^{β}

Z

A* _{ρ}*d

^{−}

_{Ω}

^{γ}*ϕ*≤B

_{1}Z

A* _{ρ}*d

^{−}

_{Ω}

*u*

^{γ}^{−}

^{β}*ϕ*≤

Z

A* _{ρ}*k(·

_{,}u)

*ϕ*(3.6)

≤

Z

Ωk(·,u)*ϕ*=

Z

Ωh∇u,∇*ϕ*i+

Z

Ωh(·,u)*ϕ*<_{∞.}

ThusR

Ωd^{−}_{Ω}^{γ}*ϕ* <∞, thereforeR

Ωd^{−}_{Ω}^{γ}^{+}^{1}^{2}^{+}* ^{ε}* <

_{∞. Then}−

*γ*+

^{1}

_{2}+

*ε*> −1, i.e.,

*γ*<

^{3}

_{2}+

*ε. Since*this holds for any

*ε*>0, the lemma follows.

**Remark 3.7.** Let us mention that, if the following condition k7) holds:

k7) There existsB_{1} >0 such that, for anys>0,k(·,s)≥ B_{1}d^{−}_{Ω}* ^{γ}*s

^{−}

*a.e. inΩ,*

^{β}then the conclusion of Lemma 3.6 remains valid when the assumption u ∈ H_{0}^{1}(_{Ω})∩C Ω
is weakened to u ∈ H_{0}^{1}(_{Ω})∩L^{∞}(_{Ω}). Indeed, define *ϕ* as in the proof of Lemma 3.6, and
observe that (3.6) holds with A* _{ρ}*replaced byΩ.

Proof of Theorem1.3.. We first provei). Letv* _{β,γ}* be the solution, given by Remark2.5, of prob-
lem (2.5). By Lemma 3.4, for

*ε*positive and small enough, z :=

*εv*

*is a weak subsolution of (1.4). Let z := B*

_{β,γ}1+1*β*

2 v* _{β,γ}*. By Remark 3.3, k(·,z)−h(·,z) ∈ H

_{0}

^{1}(

_{Ω})

^{}

^{0}and so, taking into account Remark2.6 v), z := B

1
1+*β*

2 v* _{β,γ}* is a weak supersolution of problem (1.4). In particular
z and z are a subsolution and a supersolution, respectively, in the sense of distributions, of
the problem−

_{∆u}= k(·,u)−h(·,u)in Ω. By diminishing

*ε*if necessary we can assume that z≤zinΩ. Moreover, by (2.6), there exist positive constantsc

_{1}andc

_{2}such thatz≥ c

_{1}

*ϑ*

*and z≤c*

_{β,γ}_{2}

*ϑ*

*inΩ. Thus, fora.e.x∈*

_{β,γ}_{Ω}and for s∈[z(x),z(x)],

|k(x,s)−h(x,s)| ≤k(x,z(x)) +h(x,z(x))

≤ B_{2}d^{−}_{Ω}* ^{γ}*(x)c

^{−}

_{1}

^{β}*ϑ*

^{−}

_{β,γ}*(x) +B*

^{β}_{3}d

^{−}

_{Ω}

*(x)c*

^{η}_{2}

^{p}

*ϑ*

_{β,γ}^{p}(x).

Also,d^{−}_{Ω}^{γ}*ϑ*^{−}_{β,γ}* ^{β}* andd

^{−}

_{Ω}

^{η}*ϑ*

^{p}

*belong toL*

_{β,γ}^{∞}

_{loc}(

_{Ω}). Then Remark2.7gives a solutionu, in the sense of distributions, of the problem−

_{∆u}=k(·,u)−h(·,u)inΩ, that satisfiesz≤ u≤z a.e. inΩ.

Consequently, for some positive constantsc_{1} andc_{2},

c_{1}*ϑ** _{β,γ}* ≤ u≤c

_{2}

*ϑ*

*a.e. inΩ. (3.7)*

_{β,γ}