1 Introduction and statement of the main results

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

independent variables

Tomas Godoy

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 Rⁿ, 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∈ H₀¹(_Ω), and thatubelongs to C Ω

. The behavior of the solution near∂Ωis 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 Rⁿ 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 onaandα, 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 inH¹₀(_Ω)(with test functions inC_c¹(_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^γu^−β inΩ,

u=0 on∂Ω,

u>0 inΩ,

(1.2)

where d_Ω := dist(·,∂Ω), γ < 2, and a ∈ L^∞(_Ω) satisfies inf_Ωa > 0. They studied the existence of solutionsu ∈ L¹(_Ω,d_Ω)(the d_Ω-weighted Lebesgue space) in the following very weak sense:

−

Z

Ωu∆ϕ=

Z

Ωad⁻_Ω^γϕ for any ϕ∈C² 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, whenβ+γ<1, a very weak solution of problem (1.2), and prove that it be- longs toW₀¹(_Ω,k · k_N(r),∞)∩W_loc^2,q(_Ω)for anyr ∈(0, 1)andq∈[_1,∞)_{, where}W₀¹(_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 H¹₀(_Ω) if, and only if, β+2γ<3. Additionally, in Theorem 4, they prove that, whenβ+γ>1, there exist positive constantsc₁andc2such that the found solutionusatisfiesc₁d

2−γ 1+β

Ω ≤u≤c₁d

2−γ 1+β

Ω inΩ. However, it is not obvious that, ifu∈ H₀¹(_Ω), then uis a weak solution of problem (1.2), i.e., that (1.3) holds for any ϕ∈ H₀¹(_Ω).

The existence of classical solutionsu∈ C²(_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_Ωa 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∂Ωfor 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 ∈ L¹(_Ω) for all ϕin H₀¹(_Ω). We say thatu:Ω→_Ris a weak solution of the problem

−_∆u=ψ inΩ, u=0 on∂Ω, (1.5)

ifu ∈H₀¹(_Ω)andR

Ωh∇u,∇ϕi=R

Ωψϕfor all ϕin H₀¹(_Ω).

Similarly, for u ∈ H₀¹(_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≤ ψinΩ), u=0 on ∂Ω,

to meanR

Ωh∇u,∇ϕi ≥R

Ωψϕ(resp. R

Ωh∇u,∇ϕi ≤R

Ωψϕ) for all nonnegative ϕin H₀¹(_Ω). Definition 1.2. Let d_Ω : Ω → _R be the distance function d_Ω := dist(·,∂Ω). For β ≥ 0 and γ∈ [0, 2)defineϑ_β,γ :Ω→_Rby:

• ϑ_β,γ:=d_Ωif β+γ<1,

• ϑ_β,γ := d_Ωln(ω0d⁻_Ω¹)if β+γ =1, where ω0 is an arbitrary number, which we fix from now on, such that ω₀>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 Rⁿ 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 existβ≥0,γ≥0,and B₂ >0such that, for any s>0,k(·,s)≤B₂d⁻_Ω^γs^−β a.e.inΩ;

k4) there existδ>0and B₁ >0such that, for any s∈(0,δ),k(·,s)≥ B₁d⁻_Ω^γs^−β a.e.inΩ.

Let h:Ω×[0,∞)→_Rsatisfy 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) ≤ B₃d⁻_Ω^ηs^p a.e. inΩ for all s ∈ [0,∞), with B₃ > _0, p > _{1, 0} < η < γ+p+β if β+γ≤1,and0<η< γ+ (p+β)²₁⁻₊^γ

β ifβ+γ>1.

Then:

i) If β+2γ < 3, then problem (1.4) has a unique weak solution u ∈ H₀¹(_Ω). Moreover, u ∈ W_loc^2,q(_Ω)∩C Ω

for any q∈[1,∞),and u satisfies cϑ_β,γ ≤ u≤c⁰ϑ_β,γ inΩ,for some positive constants c and c⁰.

ii) If problem(1.4)has a weak solution u∈ H¹₀(_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< ³₂.

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 infE×(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 Rⁿ with C^1,1 boundary, and let k : Ω×(0,∞) → _R satisfy the conditions k1)–k3) of Theorem1.3. Assume that β+γ < ³₂, and that the condition k5) holds. Then problem(1.1) has a unique weak solution u ∈ H¹₀(_Ω),and u ∈ W_loc^2,q(_Ω)∩C Ω

for any q∈ [1,∞),and cd_Ω ≤u≤c⁰ϑ_β,γ inΩ,for some positive constants c and c⁰.

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 Rⁿ with C^1,1 boundary, and let k : Ω×(0,∞) → _R satisfy the conditions k1)–k3) of Theorem1.3, withβandγsatisfyingβ>0,γ∈ [0, 2),andβ+γ<

3

2. Assume also the following condition:

k6) there existδ >0and B₁>0such that, for any s∈(0,δ),k(·,s)≥B₁s^−β a.e.inΩ.

Let h: Ω×[0,∞)→_Rsatisfy the conditions h1) and h2) of Theorem1.3, and the following

h4) h(·,s)≤ B₃d⁻_Ω^ηs^p a.e.inΩfor all s∈[0,∞),with B₃>0,p >1, 0<η< p−1ifβ+γ≤1 and0<η<(p−1)²₁⁻₊^γ

β if1< β+γ< ³₂.

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 c⁰ such that c⁰ϑ_β,0 ≤u≤cϑ_β,γinΩ.

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 H₀¹(_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^γ Ω

, min_Ωa > 0, α > 0, and Ωa bounded reg- ular domain. They proved that there exists a unique solution u ∈ C²(_Ω)∩C Ω

; and that u ∈ H₀¹(_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 H₀¹(_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^γu^−β in Ω, u= 0 on∂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 ∈ L¹_loc(_Ω), we write, as usual, w ∈ H₀¹(_Ω)⁰ to mean that wϕ ∈ L¹(_Ω) for any ϕ∈ H₀¹(_Ω), and that the map ϕ→R

Ωwϕis continuous onH₀¹(_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

L²(_Ω) ≤ ck∇ϕk_L2(_Ω) for all ϕ∈ H₀¹(_Ω).

Remark 2.2. If ψ ∈ L¹_loc(_Ω) and d_Ωψ ∈ L²(_Ω), then ψϕ ∈ L¹(_Ω) for any ϕ ∈ H₀¹(_Ω). Moreover, ψ ∈ H₀¹(_Ω)⁰ and kψk_(H1

0(_Ω))⁰ ≤ ckd_Ωψk₂ with c independent of ψ. Indeed, for ϕ ∈ H₀¹(_Ω), from the Hölder and the Hardy inequalities, R

Ω|ψϕ| ≤ kd_Ωψk₂kd⁻_Ω¹ϕk₂ ≤ ckd_Ωψk₂kϕk_H1

0(_Ω), wherecis the constant in the Hardy inequality of Remark2.1.

Remark 2.3. (See e.g., [10])λ∈_Ris called a principal eigenvalue for−_∆inΩ, with homoge- neous Dirichlet condition and weight function b∈ L^∞(_Ω), if the problem−_∆φ = λbφin Ω, φ=0 on ∂Ωhas a solutionϕ₁ (called a principal eigenfunction) such thatϕ₁ >0 inΩ. It is a well known fact that, for any C^1,1 bounded domain Ω⊂ _Rⁿ, b∈ L^∞(_Ω), and b⁺ 6≡0, there exists a unique positive principal eigenvalue λ₁(b), and its eigenspace V_λ1 is a one dimen- sional subspace of C¹ Ω

. Moreover, for each positive ϕ₁ ∈ V_λ1, there are positive constants c₁,c₂ such thatc₁d_Ω ≤ ϕ₁ ≤c₂d_Ω in Ω. Consequently,|ln(ϕ₁)| ∈L¹(_Ω); and ϕ₁^t ∈ L¹(_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| ∈ H¹₀(_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 that0}≤ u ≤ cd^r_Ω inΩ,then u ∈ H₀¹(_Ω)∩C¹(_Ω)∩ C Ω

,and u is a weak solution of−_∆u=ψinΩ,u=0on∂Ω.

Proof. Let ϕ∈ H₀¹(_Ω)such that supp(ϕ)⊂_{Ω. Then}R

Ωh∇u,∇ϕi=R

Ωψϕ. Indeed, letδ>0 be such that supp(ϕ)⊂ _Ωδ, and let

ϕ_{j j∈N}be a sequence inC^∞_c (_Ω)satisfying supp ϕ_j

⊂ Ωδ for allj, and such that

ϕ_{j j∈N}converges toϕin H¹₀(_Ωδ). Now,∇u_|Ωδ ∈ L²(_Ωδ,Rⁿ), and so ζ → R

Ωδh∇u,∇ζiis continuous on H₀¹(_Ωδ). Also, R

Ω

∇u,∇ϕ_j

= R

Ωψϕ_j for all j. Then R

Ωh∇u,∇ϕi=_limj→∞R

Ω

∇u,∇ϕ_j

=_limj→∞R

Ωψϕ_j =R

Ωψϕ.

For eachj∈_{N, let}µ_j :R→_Rbe defined byµ_j(s):=0 ifs ≤ ¹_j,µ_j(s):=−3j²s³+14js²− 19s+ ⁸_{j if} ¹_j < s< ²_{j, and}µ_j(s):= sif ²_j ≤s. Thenµ_j ∈C¹(_R),µ_j(s) =0 fors< ¹_j,µ⁰_j(s)≥₀ for ¹_j <s< ²_j, andµ⁰_j(s) =1 for ²_j <s. Also, 0<µ_j(s)<sfor all s∈ 0,²_j

. Letµ_j(u):=µ_j◦u. Then, for allj,∇ µ_j(u)= µ⁰_j◦u

∇uin D⁰(_Ω). Sinceu∈W_loc^1,2(_Ω), it follows thatµ_j(u)∈W_loc^1,2(_Ω). Since supp µ_j(u)⊂_{Ω, we have}µ_j(u)∈ H¹₀(_Ω). Therefore, for allj,R

Ω

∇u,∇ µ_j(u)= R

Ωψµ_j(u), i.e., Z

{u>0}

µ⁰_j◦u

|∇u|² =

Z

Ωψµ_j(u). (2.1)

Now, µ⁰_j◦u

|∇u|² is nonnegative and lim_j→∞ µ⁰_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

Ωψµ_j(u).

Let ϕ₁ be the principal eigenfunction for −_∆ in Ω with homogeneous Dirichlet condition, and with weight function1, normalized by kϕ₁k_∞ = 1. Since, for some positive constantc⁰, u≤ c⁰ϕ^r₁in Ω, andϕ^r₁ ∈ H₀¹(_Ω), we have ψu∈ L¹(_Ω). Now, lim_j→_∞ψµ_j(u) = ψuinΩ, and, for anyj∈_N,ψµ_j(u)≤ |ψu|inΩ. Then, Lebesgue’s dominated convergence theorem gives

jlim→_∞ Z

Ωψµ_j(u) =

Z

Ωψu<_∞.

ThusR

Ω|∇u|² <_∞, and sou ∈H¹(_Ω). As−_∆u=ψin D⁰(_Ω),u∈ L^∞(_Ω)andψ∈ 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 ∈ C¹(_Ω). From 0 ≤ u ≤ cd^r_Ω in Ω, and u ∈ C(_Ω), we conclude that u∈C Ω

. Sinceu∈ H¹(_Ω), u∈C Ω

, andu=0 on∂Ω, we getu∈ H₀¹(_Ω). Asu∈ H₀¹(_Ω), we have that ϕ→R

Ωh∇u,∇ϕiis continuous on H¹₀(_Ω). Therefore, since C^∞_c (_Ω)is dense inH¹₀(_Ω), and since, for anyϕ∈C_c^∞(_Ω),

Z

Ωh∇u,∇ϕi=

Z

Ωψϕ, (2.2)

we conclude that (2.2) holds for all ϕ∈ H₀¹(_Ω).

Remark 2.5. i) Letω₀be as in Definition1.2,α<1, and ρ<2. Letz∈C([0,ω₀])be such that z(0) = 0 and Rω₀

0 t^1−ρL_z(t)dt < _∞, where L_z(t) := exp Rω₀ t

z(s) s ds

. Letσ ∈ (0, 1), and let a∈ C_loc^σ (_Ω)satisfy, for some constantc>0,

1

cL_z◦d_Ω ≤d^ρ_Ωa≤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 ∈ C²(_Ω)∩C Ω

; and that, for some positive constant c⁰, u satisfies,

c⁰−1

θ_ρ◦d_Ω ≤u≤c⁰θ_ρ◦d_Ω inΩ,

where

θ_ρ(t):= _{Z ω}

0

0

L_z(s) s ds

₁₋¹

α

if ρ =2, θρ(t):=t^21−−αρ (L_z(t))^1−1α if 1+α<ρ<2, θ_ρ(t):=t

_{Z ω}

0

t

L_z(s) s ds

₁₋¹

α

ifρ=₁+_α, and

θρ(t):=t i f ρ <1+α.

ii) Let β≥ 0, and letγ< 2. If in i) we takeα:=−β, z :=0(thenL_z =1), andρ :=γ, we get that the problem











−_∆v=d⁻_Ω^γv^−β inΩ,

v=0 on∂Ω,

v>0 inΩ

(2.5)

has a unique classical solutionv_β,γ ∈C²(_Ω)∩C Ω

; and that there exists positive constants c₁ andc₂ such that

c₁ϑ_β,γ ≤v_β,γ ≤c₂ϑ_β,γ inΩ, (2.6) whereϑ_β,γ is as in Definition1.2.

Remark 2.6. Letβ≥0 andγ≥0 be such that β+2γ<3.

i) d¹_Ω^−γϑ⁻_β,γ^β ∈ L²(_Ω). Indeed, if β+γ < 1 then d¹_Ω^−γϑ⁻_β,γ^β = d¹_Ω^−β−γ ∈ L^∞(_Ω) ⊂ L²(_Ω). If β+γ=1, then d¹_Ω^−γϑ⁻_β,γ^β = ln ^ω_d⁰

Ω

β

∈ L^∞(_Ω)⊂ L²(_Ω). Ifβ+γ >1, then d¹_Ω^−γϑ⁻_β,γ^β = d^1−γ−β

2−γ 1+β

Ω =d⁻

1

β+1(β+γ−1)

Ω and, sinceβ+2γ<3,− ²

β+1(β+γ−1)>−1, and so, again in this case,d¹_Ω^−γϑ⁻_β,γ^β ∈ L²(_Ω)(because, forr∈_R,d^r_Ω ∈L²(_Ω)whenever 2r+1>0).

ii) There exist positive constantscandτ> ¹_{2 such that}ϑ_β,γ ≤cd^τ_ΩinΩ. Indeed, if β+γ<₁ thenϑ_β,γ =d_Ω, if β+γ= 1 then ϑ_β,γ = d_Ωln ^ω_d⁰

Ω

and so, for any ε> 0,ϑ_β,γ d¹_Ω^−ε−1

= d^ε_Ωln ^ω_d⁰

Ω

∈ L^∞(_Ω); and ifβ+γ>1, thenϑ_β,γ =d

2−γ 1+β

Ω and, sinceβ+2γ<3, ²₁⁻₊^γ_β > ¹₂. iii) Letv_β,γ be the solution of problem (2.5) given by Remark2.5. From i) and (2.6), it follows

thatd¹_Ω^−γv⁻_β,γ^β ∈ L²(_Ω).

iv) Letv_β,γ be as in iii). Then, by iii) and Remark 2.2,d⁻_Ω^γv⁻_β,γ^β ∈ H₀¹(_Ω)⁰.

v) Letv_β,γ be as in iii). Sinced⁻_Ω^γv⁻_β,γ^β ∈ H₀¹(_Ω)⁰ and since, by Remark2.5,v_β,γ ∈C²(_Ω)∩ C Ω

, Lemma 2.4gives that v_β,γ ∈ H₀¹(_Ω), and thatv_β,γ is a weak solution of problem (2.5). Moreover, since v_β,γ ∈ L^∞(_Ω) andd⁻_Ω^γv⁻_β,γ^β ∈ L^∞_loc(_Ω), the inner elliptic estimates givev_β,γ∈W_loc^2,q(_Ω)for anyq∈[1,∞).

Remark 2.7. Let g: Ω×(0,∞)→_Rbe a Carathéodory function. We say thatw ∈ L¹_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)∈L¹_loc(_Ω), andR

Ωh∇w,∇ϕi ≤ (≥)R

Ωg(·,w)ϕ for all nonnegative ϕ∈C^∞_c (_Ω). We say thatz∈ L¹_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,∇ϕi= 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 existsψ ∈ L^∞_loc(_Ω)such that |g(x,s)| ≤ ψ(x) a.e. x ∈ _Ωfor 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 ϕ₁ be a positive principal eigenfunction of −_∆ in Ω, with homogeneous Dirichlet boundary condition. If r > ¹₂, then ϕ^r₁ ∈ H₀¹(_Ω). Indeed, ϕ^r₁ ∈ L²(_Ω). Also, ϕ₁^r−1 ∈ L²(_Ω)and|∇ϕ₁| ∈ L^∞(_Ω), thus∇(ϕ^r₁)∈ L²(_Ω).

Lemma 3.2. Letηand p be as in the condition h3) of Theorem1.3. Then:

i) d^γ_Ω^−ηϑ_β,γ^p+β ∈ L^∞(_Ω).

ii) If, in addition,β+2γ<3,then d¹_Ω^−ηϑ_β,γ^p ∈L²(_Ω)and d¹_Ω^−γϑ⁻_β,γ^β ∈ L²(_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)²₁⁻₊^γ_β whenβ+γ > 1, and using, when β+γ = 1, that d^ε_Ωln ^ω_d⁰

Ω

∈ L^∞(_Ω)for anyε>0.

To see the first assertion of ii)note that, byh3), 2(1−η+p) > 0 when β+γ ≤ 1. Now, d¹_Ω^−ηϑ_β,γ^p 2

= d²_Ω^(1−η+p) when β+γ < 1, and (d¹_Ω^−ηϑ_β,γ^p )² = d²_Ω^(1−η+p) ln ^ω_d⁰

Ω

2p

when β+ γ = 1. Thus, in both cases, d¹_Ω^−ηϑ_β,γ^p ∈ L^∞(_Ω) ⊂ L²(_Ω). If β+γ > 1, then d¹_Ω^−ηϑ^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,d¹_Ω^−ηϑ^p_β,γ ∈ L²(_Ω).

Finally, d¹_Ω^−γϑ⁻_β,γ^β = d¹_Ω^−β−γ ∈ L^∞(_Ω) ⊂ L²(_Ω) when β+γ < 1, and d¹_Ω^−γϑ⁻_β,γ^β = ln ^ω_d⁰

Ω

−β

∈ L^∞(_Ω)⊂ L²(_Ω)whenβ+γ=1. Ifβ+γ>1, thend¹_Ω^−γϑ⁻_β,γ^β =d^1−γ−β

2−γ 1+β

Ω and, since 2 1−γ−β²₁⁻₊^γ_β

+1= ^3−β−2γ

β+1 >0, we have, again in this case,d¹_Ω^−γϑ⁻_β,γ^β ∈ L²(_Ω). Remark 3.3. Assume the conditions k1), k3), h1), and h3) of Theorem1.3. Assume also that β+2γ<3. Then, for anyε>0,k(·,εv_β,γ)andh ·,εv_β,γ

belong to H₀¹(_Ω)⁰. Indeed, by k1) and h1),k ·,εv_β,γ

andh ·,εv_β,γ

are measurable functions, and by k3) and h3), d_Ωk ·,εv_β,γ

≤ ε^−βB2d¹_Ω^−γv⁻_β,γ^β ≤ ε^−βB2c⁻₂^βd¹_Ω^−γϑ⁻_β,γ^β a.e. inΩ, d_Ωh ·,εv_β,γ

≤ ε^pB3d¹_Ω^−ηv^p_β,γ ≤ε^pB3c₃^pd¹_Ω^−ηϑ_β,γ^p a.e. in Ω.

Then, by Lemma3.2,d_Ωk ·,εv_β,γ

andd_Ωh ·,εv_β,γ

belong to L²(_Ω), and so, by Remark2.2, k ·_,εv_β,γ

andh ·_,εv_β,γ

belong to H₀¹(_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 anyεpositive and small enough, εv_β,γ is a subsolution, in the sense of distributions, of problem (1.4) and, if in addition, β+2γ < 3,thenεv_β,γ is a weak subsolution of (1.4).

Proof. By Lemma3.2 i),there exists a positive constantc₁ such thatd⁻_Ω^ηϑ_β,γ^p ≤c₁d⁻_Ω^γϑ⁻_β,γ^β in Ω, and by Remark 2.5, there exist positive constants c2 and c3 such that c2ϑ_β,γ ≤ v_β,γ ≤ c3ϑ_β,γ in Ω. Thus, for some positive constantc₄,d⁻_Ω^ηv_β,γ^p ≤ c₁d⁻_Ω^γv⁻_β,γ^β in Ω. Then, for anyε positive and small enough, ¹₂ε^−βB₁d⁻_Ω^γv⁻_β,γ^β ≥ ε^pB₃d⁻_Ω^ηv^p_β,γ in Ω. By diminishingε if necessary, we can assume that, in addition, ε < min

1,_kv ^δ

β,γk_∞,¹₂B₁ . By Remark 2.5, v_β,γ satisfies, in the sense of distributions,

−_∆vβ,γ = d⁻_Ω^γv⁻_β,γ^β inΩ. (3.1) Then, in the sense of distributions,

−_∆ εv_β,γ

=εd⁻_Ω^γv⁻_β,γ^β ≤ ¹

2ε^−βB₁d⁻_Ω^γv⁻_β,γ^β (3.2)

≤ε^−βB₁d⁻_Ω^γv⁻_β,γ^β−ε^pB₃d⁻_Ω^ηv^p_β,γ ≤ k ·,εv_β,γ

−h ·,εv_β,γ

inΩ;

where, in the last inequality, we have used that, since ε< _k ^δ

vβ,γk_∞ we have εv_β,γ ≤ δin Ω, and then, byk4),k ·,εv_β,γ

≥ε^−βB₁d⁻_Ω^γv⁻_β,γ^β a.e. inΩ. Thus, for any nonnegativeϕ∈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_β,γ ∈ H₀¹(_Ω), and by Remark 3.3, k ·,εv_β,γ

andh ·,εv_β,γ belong to H₀¹(_Ω)⁰. Thus k ·,εv_β,γ

−h ·,εv_β,γ

ϕ∈L¹(_Ω)for any ϕ∈ H₀¹(_Ω)and, since C_c^∞(_Ω)is dense inH₀¹(_Ω), it follows that (3.3) holds for any nonnegative ϕ∈ H₀¹(_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 inΩ.}

(3.4)

Let u ∈ H₀¹(_Ω) be a weak subsolution of problem (3.4) and let u ∈ H₀¹(_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∈ H₀¹(_Ω)∩C Ω

,thenγ≤ ³_2.

Proof. Let u ∈ H₀¹(_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 existsρ> 0 such thatu≤δ in Aρ. Then, byk4),

k(·,u)≥B₁d⁻_Ω^γu^−β a.e. in Aρ. (3.5) Let ϕ₁be a positive principal eigenfunction for−_∆in Ω, with homogeneous Dirichlet condi- tion and weight function 1. Letε > 0, and let ϕ := ϕ

1 2+ε

1 . Then ϕ ∈ H₀¹(_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_Ωh(·,u)d⁻_Ω¹ϕ≤ B₃d²_Ω^−ηkuk^p_∞⁻¹d⁻_Ω¹u d⁻_Ω¹ϕ

≤B₃kd_Ωk²_∞^−ηkuk_∞^p−1d⁻_Ω¹u d⁻_Ω¹ϕ

a.e. inΩ;

and so, by the Hölder and the Hardy inequalities,h(·,u)ϕ∈ L¹(_Ω). Now, taking into account (3.5) and thatkis nonnegative,

B₁kuk⁻_∞^β

Z

A_ρd⁻_Ω^γϕ≤B₁ Z

A_ρd⁻_Ω^γu^−βϕ≤

Z

A_ρk(·_,u)ϕ (3.6)

≤

Z

Ωk(·,u)ϕ=

Z

Ωh∇u,∇ϕi+

Z

Ωh(·,u)ϕ<_∞.

ThusR

Ωd⁻_Ω^γϕ <∞, thereforeR

Ωd⁻_Ω^γ+12+ε < _{∞. Then} −γ+ ¹₂+ε > −1, i.e., γ < ³₂+ε. Since this holds for anyε>0, the lemma follows.

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

k7) There existsB₁ >0 such that, for anys>0,k(·,s)≥ B₁d⁻_Ω^γs^−β a.e. inΩ,

then the conclusion of Lemma 3.6 remains valid when the assumption u ∈ H₀¹(_Ω)∩C Ω is weakened to u ∈ H₀¹(_Ω)∩L^∞(_Ω). Indeed, define ϕ as in the proof of Lemma 3.6, and observe that (3.6) holds with A_ρreplaced 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_β,γ is a weak subsolution of (1.4). Let z := B

1+1β

2 v_β,γ. By Remark 3.3, k(·,z)−h(·,z) ∈ H₀¹(_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 εif necessary we can assume that z≤zinΩ. Moreover, by (2.6), there exist positive constantsc₁andc₂such thatz≥ c₁ϑ_β,γand z≤c₂ϑ_β,γ inΩ. Thus, fora.e.x∈ _Ωand for s∈[z(x),z(x)],

|k(x,s)−h(x,s)| ≤k(x,z(x)) +h(x,z(x))

≤ B₂d⁻_Ω^γ(x)c⁻₁^βϑ⁻_β,γ^β(x) +B₃d⁻_Ω^η(x)c₂^pϑ_β,γ^p (x).

Also,d⁻_Ω^γϑ⁻_β,γ^β andd⁻_Ω^ηϑ^p_β,γ belong toL^∞_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 constantsc₁ andc₂,

c₁ϑ_β,γ ≤ u≤c₂ϑ_β,γ a.e. inΩ. (3.7)