Existence and multiplicity of positive solutions for a singular system via sub-supersolution method

Existence and multiplicity of positive solutions for a singular system via sub-supersolution method

and Mountain Pass Theorem

Suellen Cristina Q. Arruda

and Rubia G. Nascimento

1Faculdade de Ciências Exatas e Tecnologia, Universidade Federal do Pará, 68440-000 Abaetetuba, Pará, Brazil

2Instituto de Ciências Exatas e Naturais, Universidade Federal do Pará, 66075-110 Belém, Pará, Brazil

Received 3 April 2020, appeared 1 April 2021 Communicated by Gennaro Infante

Abstract: In this paper we show the existence and multiplicity of positive solutions using the sub-supersolution method and Mountain Pass Theorem in a general singular system which the operator is not homogeneous neither linear.

Keywords: p&q-Laplacian operator, sub-supersolution method, singular system, Mountain Pass theorem.

2020 Mathematics Subject Classification: Primary 35J20, 35J50; Secondary 58E05.

1 Introduction

In this paper we treat the question of the existence and multiplicity of positive solutions for the following class of singular systems of nonlinear elliptic equation















−div(a₁(|∇u|^p1)|∇u|^p1−2∇u) =h₁(x)u^−γ1 +F_u(x,u,v)inΩ,

−div(a2(|∇v|^p2)|∇v|^p2−2∇v) =h2(x)v^−γ2+Fv(x,u,v)inΩ, u,v>0 inΩ,

u =v=0 on∂Ω,

(1.1)

where Ω⊂_R^N is a bounded domain with smooth boundary, N≥3 and 2≤ p₁,p₂ < N. For i = 1, 2, γ_i > 0 is a fixed constant, a_i : R⁺ → _R⁺ is a C¹-function and h_i ≥ 0 is a nontrivial measurable function. More precisely, we suppose that the functions h_i and a_i satisfy the following assumptions:

(H) There exists 0<φ₀ ∈C¹₀(_Ω)_{such that}h_iφ⁻₀^γi ∈ L^∞(_Ω)_.

BCorresponding author. Email: scqarruda@ufpa.br

*Email: rubia@ufpa.br

(A₁) There exist constantsk₁,k₂,k₃,k₄ >0 and 2≤ p_i ≤q_i <N such that k₁t^pi+k₂t^qi ≤ a_i(t^pi)t^pi ≤ k₃t^pi+k₄t^qi, for allt ≥0.

(A₂) The functions

t7−→a_i(t^pi)t^pi−2 are increasing.

(A₃) The functions

t7−→ A_i(t^pi) are strictly convex, where A_i(t) =Rt

0 a_i(s)ds.

(A₄) There exist constantsµ_i, _q¹∗

1 <θ_s< _q¹

1 and _q¹∗

2 <θ_t < _q¹

2 such that 1

µ_ia_i(t)t≤ A_i(t), for allt ≥0, with ^q_p¹

1 ≤µ₁< ¹

θsp1 and ^q_p²

2 ≤µ₂< ¹

θtp2.

Notice that the functions a_i satisfy suitable monotonicity conditions which allow to con- sider a larger class of p&q type problems. In order to illustrate the degree of generality of the kind of problems studied here, in the following we present some examples of functionsa_i which are interesting from the mathematical point of view and have a wide range of applica- tions in physics and related sciences.

Example 1.1. If a_i ≡ 1, for eachi=1, 2, our operator is the p-Laplacian and so problem (1.1)

becomes 









−_∆p1u= h₁(x)u^−γ1+F_u(x,u,v)inΩ,

−_∆p2v= h2(x)v^−γ2+Fv(x,u,v)inΩ, u=v =0 on∂Ω,

withq_i = p_i,k₁+k₂ =1 andk₃+k₄=1.

Example 1.2. Ifa_i(t) =1+t^qi

−_pi

pi , for eachi=1, 2, we obtain











−_∆p1u−_∆q1u=h₁(x)u^−γ1+F_u(x,u,v)inΩ,

−_∆p2v−_∆q2v=h2(x)v^−γ2 +Fv(x,u,v)inΩ, u=v=0 on ∂Ω,

withk₁=k₂= k₃ =k₄=_1.

Example 1.3. Takinga_i(t) =1+ ¹

(1+t)^pi

−2 pi

, for eachi=1, 2, we get























−div



|∇u|^p1−2∇u+ |∇u|^p1−2∇u (1+|∇u|^p1)

p1−2 p1



= h₁(x)u^−γ1+F_u(x,u,v)inΩ,

−div



|∇v|^p2−2∇v+ |∇v|^p2−2∇v (1+|∇v|^p2)

p2−2 p2



= h₂(x)v^−γ2+F_v(x,u,v)inΩ, u=v=0 on∂Ω,

withq_i = p_i,k₁+k₂ =_{1 and}k₃+k₄=_2.

Example 1.4. If we considera_i(t) =1+t^qi

−_pi

pi + ¹

(1+t)^pi

−2 pi

, for eachi=1, 2, we obtain























−_∆p1u−_∆q1u−div





|∇u|^p1−2∇u (1+|∇u|^p1)

p1−2 p1



 =h₁(x)u^−γ1+F_u(x,u,v)inΩ,

−_∆p2v−_∆q2v−div





|∇v|^p2−2∇v (1+|∇v|^p)

p2−2 p2



 =h₂(x)v^−γ2+F_v(x,u,v)inΩ, u=v=0 on∂Ω,

wherek₁ =k₂=k₄=1 andk₃=2.

Remark 1.5. Note that by hypothesis(H)we haveh_i ∈ L^∞(_Ω)because

|h_i|=|h_iφ

−γi

0 φ

γi

0| ≤ kh_iφ

−γi

0 k_∞φ

γi

0. HereFis a function onΩ×_R² of classC¹ satisfying (F₁) There exists 0<δ< ¹₂ such that

−h₁(x)≤Fs(x,s,t)≤0 a.e. in Ω, for all 0≤ s≤δ and

−h₂(x)≤F_t(x,s,t)≤0 a.e. inΩ, for all 0≤t≤δ.

It is worthwhile to point out that, since p_i <q_i and by the boundedness ofΩ,W₀^1,pi(_Ω)∩ W₀^1,qi(_Ω) = W₀^1,qi(_Ω). Thus, in order to show the existence and multiplicity of solutions to system (1.1), we define the Sobolev spaceX=W₀^1,q1(_Ω)×W₀^1,q2(_Ω)endowed with the norm

k(u,v)k= kuk_1,q1+kvk_1,q2, where

kuk_1,qi =



 Z

Ω

|∇u|^qidx





1 qi

.

Moreover, we say that a pair (u,v) ∈ X is a positive weak solution of system (1.1) if u,v>0 inΩand it verifies

Z

Ω

a₁(|∇u|^p1)|∇u|^p1−2∇u∇φdx=

Z

Ω

h₁(x)u^−γ1φdx+

Z

Ω

F_u(x,u,v)φdx

and _Z

Ω

a2(|∇v|^p2)|∇v|^p2−2∇v∇ϕdx =

Z

Ω

h2(x)v^−γ2ϕdx+

Z

Ω

Fv(x,u,v)ϕdx, for all (φ,ϕ)∈X.

In our first theorem we apply the sub-supersolution method to establish the existence of a weak solution for system (1.1).

Theorem 1.6. Suppose that(H),(F₁)and(A₁)–(A₃)are satisfied. Then system(1.1) has a positive weak solution ifkh_ik_∞ is sufficiently small, for i=1, 2.

Furthermore, we assume the conditions below to prove the existence of two solutions for problem (1.1).

(F₂) Fori=1, 2, there exists 1<r <q^∗_i = _(N^Nq_−qⁱ

i) (q^∗_i =_∞ifq_i ≥ N)such that Fs(x,s,t)≤ h₁(x)(1+s^r−1+t^r−1) a.e. in Ω, for all s≥0 and

F_t(x,s,t)≤h₂(x)(1+s^r−1+t^r−1) a.e. in Ω, for allt≥0.

(F₃) There exists₀,t₀ >0 such that

0<F(x,s,t)≤θ_ssF_s(x,s,t) +θ_ttF_t(x,s,t) a.e. inΩ, for alls≥ s₀ andt ≥t₀, whereθ_s andθ_t appeared in(A₄).

Theorem 1.7. Suppose that(H), (F₁)–(F₃)and(A₁)–(A₄)are satisfied. Then system(1.1) has two positive weak solutions ifkh_ik_∞is sufficiently small, for i=1, 2.

Singular problems has been much studied in last years. We are going to cite some authors in last ten years. System (1.1) with Laplacian operator in both equations was studied in [9], where it was investigated the questions of existence, non-existence and uniqueness for solutions. The results in [9] were complemented in [16]. The general operator as we consider in this paper was studied in [5] using continuous unbounded of solutions. The cases with Laplacian operator involving weights were studied in [7] and [11].

In this paper we complement the results that can be found in [5], [7], [9], [11] and [16]

because we consider a general problem with singularity without restrictions in the exponents.

Moreover, we are considering the sub-supersolution method for a system that involves a non- linear and nonhomegeneous operator. The reader can see the generality of the operator in [5].

We would like to highlight that our theorems can be applied for the model nonlinearity F(x,s,t) =h₁(x)

s^r

r −sδ^r−1

+h2(x) t^r

r −tδ^r−1

.

This paper is organized in the following way. Section 2 is devoted to some preliminary results in order to prove the main results. The first theorem is proved in the Section 3 and the second theorem in the Section 4.

2 Preliminary results

The next lemma provides the uniqueness of solution to the linear problem. The proof can be found in [5, Lemma 1]. However, for the convenience of the reader, we also prove it here.

Lemma 2.1. Assume that the conditions (A₁)and(A₂)hold. Then, there exists an unique solution u_i ∈W₀^1,qi(_Ω)of the linear problem

(−div(a_i(|∇u_i|^pi)|∇u_i|^pi−2∇u_i) =h_i(x)inΩ, u_i =₀on ∂Ω,

where h_i ∈(W₀^1,qi(_Ω))⁰, for all i=_{1, 2}and2≤ p_i ≤q_i < N.

Proof. Consider the operatorT_i :W₀^1,qi(_Ω)−→(W₀^1,qi(_Ω))⁰ given by hT_iu_i,φ_ii=

Z

Ω

a_i(|∇u_i|^pi)|∇u_i|^pi−2∇u_i∇φ_i dx.

In virtue of hypothesis (A₁), we can show that the operator T_i is well defined and it is con- tinuous. Furthermore, by considering the hypothesis(A₂), we argument as [8, Lemma 2.4] to obtain the following inequality

C_i|u_i−v_i|^pi ≤ ha_i(|u_i|^pi)|u_i|^pi−2u_i−a_i(|v_i|^pi)|v_i|^pi−2v_i,u_i−v_ii, for someC_i >0 and for alli=1, 2. Therefore,

hT_iu_i−T_iv_i,u_i−v_ii>0, for allu_i,v_i ∈W₀^1,qi(_Ω)withu_i 6=v_i, which implies thatT_i is monotone. Moreover, using(A₁)again we get

hT_iu_i,u_ii

ku_ik_1,qi ≥k₂ku_ik^q_1,qⁱ⁻¹

i

and hence

kuiklim_1,qi→_∞

hT_iu_i,u_ii

ku_ik_1,qi = +_∞,

which shows that T_i is coercive. Thus, applying the Minty–Browder Theorem [2, Theo- rem 5.15] there exists an unique u_i ∈W₀^1,qi(_Ω)such thatT_iu_i = h_i(x).

Our approach in the study of system (1.1) rests heavily on the following Weak Comparison Principle for the p&q-Laplacian operator. The proof of the result below for the scalar case can be found in [6, Lemma 2.1].

Lemma 2.2. LetΩa bounded domain and consider u_i,v_i ∈W₀^1,qi(_Ω)satisfying

(−div(a_i(|∇u_i|^pi)|∇u_i|^pi−2∇u_i)≤ −div(a_i(|∇v_i|^pi)|∇v_i|^pi−2∇v_i)inΩ, u_i ≤v_i on∂Ω,

then u_i ≤v_i a.e. inΩ, for all i=1, 2and2≤ p_i ≤q_i < N.

Proof. Using the test functionφ_i = (u_i−v_i)⁺ _:=_max{u_i−v_i, 0} ∈W₀^1,qi(_Ω)_{, we get}

Z

Ω∩{u_i>v_i}

ha_i(|∇u_i|^pi)|∇u_i|^pi−2∇u_i−a_i(|∇v_i|^pi)|∇v_i|^pi−2∇v_i,∇u_i− ∇v_iidx≤0.

From Lemma 2.1,k(u_i−v_i)⁺k ≤0, which implies thatu_i ≤ v_i a.e. in Ω.

Now, using Lemma2.2, it is possible to repeat the same arguments of [13, Hopf’s Lemma]

to obtain the next result

Lemma 2.3. Let Ω ⊂ _R^N be a bounded domain with smooth boundary and i = 1, 2. If u_i ∈ C¹(_Ω)∩W₀^1,qi(_Ω), with2≤ p_i ≤q_i < N, and











−_div(a_i(|∇u_i|^pi)|∇u_i|^pi−2∇u_i)≥₀inΩ, u_i >0inΩ,

u_i =0on∂Ω.

Then, ^∂u_∂ηⁱ <0on∂Ω, where ηis the outwards normal to∂Ω.

We enunciate an iteration lemma due to Stampacchia that we will use to prove the L^∞- regularity of the solutions for this class of p&qtype problems.

Lemma 2.4(See [14]). Assume that φ:[0,∞)→[0,∞)is a nonincreasing function such that if h>

k > k₀, for some α> 0, β > 1, φ(h) ≤ ^C₍^(φ(k))β

h−k)^α . Then, φ(k₀+d) =0, where d^α = C2^βαβ−1φ(k₀)^β−1 and C is positive constant.

Lemma 2.5. Let u_i ∈W₀^1,qi(_Ω)be solution to problem

(−div(a_i(|∇u_i|^pi)|∇u_i|^pi−2∇u_i) = f_i inΩ,

u_i =0on∂Ω, (2.1)

such that f_i ∈ L^ri(_Ω)with r_i > _q∗^q∗i

i−qi. Then, u_i ∈ L^∞. In particular, ifkf_ik_ri is small, then alsoku_ik_∞ is small, for all i=1, 2and2≤ p_i ≤q_i < N.

Proof. Sinceu_i is the weak solution to (2.1) we can write Z

Ω

a_i(|∇u_i|^pi)|∇u_i|^pi−2∇u_i∇φ_i dx =

Z

Ω

f_iφ_i dx, ∀φ_i ∈W₀^1,qi(_Ω). Fork>0, we define the test function

v_i =sign(u_i)(|u_i| −k) =











u−k, ifu>k, 0, ifu= k, u+k, ifu<k.

Then, u_i = _vi+_{k sign}(_ui) _and ^∂u_∂xⁱ

j = ^∂v_∂xⁱ

j in the set A(_k) = {x ∈ _Ω;|u(_x)| > _k}, v_i = ₀ in Ω−A(k) and v_i ∈ W₀^1,qi(_Ω). By considering the test function v_i and using the Hölder inequality, we get

Z

A(k)

a_i(|∇v_i|^pi)|∇v_i|^pidx=

Z

Ω

f_iv_i dx≤





 Z

A(k)

|v_i|^q∗idx







1 q∗

i





 Z

A(k)

|f_i|^ridx







1 ri

|A(k)|¹⁻

1 q∗

i

+¹

ri

,

where |A(k)|denotes the Lebesgue measure of A(k). Moreover, applying (A₁)and Sobolev inequality we obtain

k₂S





 Z

A(k)

|v_i|^q∗idx







qi−1 q∗

i

≤





 Z

A(k)

|f_i|^ridx







1 ri

|A(k)|¹⁻

1 q∗

i

+¹

ri

, (2.2)

whereSis the best constant in the Sobolev inclusion.

Note that if 0<k<h, then A(h)⊂ A(k)and

|A(k)|

1 q∗

i (h−k) =





 Z

A(h)

(h−k)^q∗idx







1 q∗

i

≤





 Z

A(h)

|v_i|^q∗idx







1 q∗

i

≤





 Z

A(k)

|v_i|^q∗idx







1 q∗

i

. (2.3)

It follows from (2.2) and (2.3) that

|A(k)| ≤ ¹ (h−k)^q∗i

1 (k₂S)

q∗ qi−i1

kf_ik

q∗ qi−i1

r_i |A(k)|

q∗ qi−i1

1−

1 q∗ i

+¹

ri

.

Sincer_i > _q∗^qi∗

i−qi we have β:= _q^q∗i

i−1

1− _q¹∗

i +_r¹

i

>1. Therefore, if we define

φ(h) =|A(h)|, α= q^∗_i, β:= ^q

∗i

q_i−1

1− 1

q_i^∗ + ¹ r_i

, k0=0, we obtain that φis a nonincreasing function and

φ(h)≤ ^C(φ(k))^β

(h−k)^{α , for all h}>k>0.

By Lemma2.4, we conclude thatφ(d) =0 ford= C^kfik

qi1−1 ri

(k2S)

qi1−1

|_Ω|^β−α1 and hence,

ku_ik_∞ ≤C kf_ik

1 qi−1

ri

(k₂S)^qi1−1

|_Ω|^β−α1_, where β,α,SandCare constants that do not depend on f_i andu_i.

Regarding the regularity of the solution of (2.1) the next result hold and the proof can be done repeating the same arguments of [10, Theorem 1].

Lemma 2.6. Fix h_i ∈ L^∞(_Ω), for all i=1, 2, and consider u_i ∈ W₀^1,qi(_Ω)∩L^∞(_Ω), with2 ≤ p_i ≤ q_i < N, satisfying the problem

(−div(a_i(|∇u_i|^pi)|∇u_i|^pi−2∇u_i) =h_i inΩ, u_i =0on∂Ω,

Then, u_i ∈C^1,α(_Ω)_,for someα∈(_{0, 1}).

The following result can be found in [12, Lemma 2.6]. The proof is presented for the completeness of the paper.

Lemma 2.7. Let φ,ω > 0 be any functions on C₀¹(_Ω). If ^∂φ_∂ν > 0 in ∂Ω, where ν is the inwards normal to∂Ω, then there exists C>0such that

φ(x)

ω(_x) ≥C>0, for all x∈_Ω. Proof. Forδ >0 sufficiently small, we consider the following set

Ωδ = {x∈_{Ω; dist}(x,∂Ω)<δ}.

Sinceφ,ω >0 inΩandΩ\_Ωδ is compact, there existsm>0 such that φ(x)

ω(x) ≥m, for all x∈ _Ω\_Ωδ. (2.4)

It follows from ^∂φ_∂ν >0 in∂Ωthat ^∂φ_∂η < 0, whereηis the outwards normal to ∂Ω. Further- more, since Ω ⊂ _Rⁿ is bounded domain, then ∂Ωis a compact set and consequently, there existsC1 <0 satisfying

∂φ(x)

∂η

≤C₁, for allx ∈_Ωδ. Sinceω∈C₀¹(_Ω), there existsC2>0 such that

∂ω(x)

∂η

≤C₂, for allx ∈_Ωδ. Consider K₀ = _infΩ

δ

∂ω

∂η <0 and define the function H(x) = αω(x)−φ(x)_{, for all}x ∈ _Ωδ andα∈_Rto be chosen later. Since 0<α< ^C_K¹

0 we obtain

∂H(x)

∂η =α∂ω(x)

∂η − ^∂φ(x)

∂η ≥αK₀−C₁>0, for allx ∈_Ωδ. Now, fix x∈_Ωδ and consider the function

f(x) = H(x+sη), for alls∈_R.

For everyx ∈ _Ωδ, we choose an uniquexe∈ _Ωδ so that there existsbs > 0 such that x+_bsη= xe∈ ∂Ω. Hence, since H(∂Ω)≡0 we have

f(_bs) = H(x+_bsη) =H(x_e) =0.

Applying the Mean Value Theorem, there existsξ ∈(0,bs)such that f(_bs)− f(0) = f⁰(ξ)(_bs−0),

which implies that

−H(x) = ^∂H

∂η(x+ξη)_bs>0 in Ωδ. Therefore, H(x)≤0 for all x∈_Ωδ and hence,

αω(x)−φ(x)≤0, for all x∈_Ωδ, which result in

αω(x)≤φ(x), for all x∈_Ωδ. Thus,

φ(x)

ω(x) ≥ α>0, for allx ∈_Ωδ. (2.5) By virtue of (2.4) and (2.5), we conclude that there existsC>0 so that

φ(x)

ω(x) ≥C, for allx ∈_Ω.

3 Proof of Theorem 1.6

In the proof of Theorem 1.6, we combine the sub-supersolution method with minimization arguments. Before this, we need of the following definition.

We say that (u,v),(u,v) ∈ X form a pair of sub and supersolution for system (1.1) if u,u,v,v∈ L^∞(_Ω)with

(a) u≤ u,v≤v inΩandu=0≤u,v=0≤von∂Ω, (b) _Given(_φ,ϕ)∈X, withφ,ϕ≥_{0, we have}













 Z

Ω

a1(|∇u|^p1)|∇u|^p1−2∇u∇φdx≤

Z

Ω

h1(_x)_u^−γ1φdx+

Z

Ω

Fu(_x,u,w)φdx, for allw∈ [_v,v]_,

Z

Ω

a₂(|∇v|^p2)|∇v|^p2−2∇v∇ϕdx≤

Z

Ω

h₂(x)v^−γ2ϕdx+

Z

Ω

F_v(x,w,v)ϕdx, for allw∈ [u,u]

and













 Z

Ω

a₁(|∇u|^p1)|∇u|^p1−2∇u∇φdx≥

Z

Ω

h₁(x)u^−γ1φdx+

Z

Ω

F_u(x,u,w)φdx, for allw∈[v,v]

Z

Ω

a2(|∇v|^p2)|∇v|^p2−2∇v ∇ϕdx≥

Z

Ω

h2(x)v^−γ2ϕ, dx+

Z

Ω

Fv(x,w,v)ϕdx, for allw∈[u,u].

The next result is essential to provide the existence of a subsolution and a supersolution for system (1.1) whenever we fix the value ofkh_ik_∞with i=1, 2.

Lemma 3.1. Suppose that(H),(F₁)and(A₁)–(A₂)are satisfied. Ifkh_ik_∞ is small, for i=1, 2, then there exist u,u,v,v∈C^1,α(_Ω), for someα∈ (0, 1), such that

i) h₁u^−γ1,h2v^−γ2 ∈ L^∞(_Ω), kuk_∞ ≤ δ andkvk_∞ ≤ δ, where δ is the constant that appeared in the hypothesis(F₁);

ii) kuk_∞ ≤δandkvk_∞ ≤δ, where δis the constant that appeared in the hypothesis(F₁); iii) 0<u(x)≤ u(x)a.e. inΩand0<v(x)≤v(x)a.e. inΩ;

iv) (u,v)is a subsolution and(u,v)is a supersolution for system(_1.1).

Proof. From Lemma2.1 and maximum principle, there exists an unique positive solution 0<

u∈W₀^1,q1(_Ω)satisfying the problem below

(−div(a₁(|∇u|^p1)|∇u|^p1−2∇u) =h₁(x)in Ω,

u=0 on∂Ω. (3.1)

Similary, there exists an unique positive solution 0<v∈W₀^1,q2(_Ω)satisfying (−_div(a₂(|∇v|^p2)|∇v|^p2−2∇v) =h₂(x)_inΩ,

v=0 on ∂Ω. (3.2)

Sinceh₁,h₂ ∈L^∞(_Ω), it follows from Lemma2.5thatu,v∈ L^∞(_Ω)and there existC₁,C₂>

0 such that

kuk_∞≤ C₁kh₁k

p11−1

∞ and kvk_∞ ≤C₂kh₂k

1 p2−1

∞ ,

whereC₁andC₂are constants that does not depend onh_i,uandv. Therefore, we may choose kh_ik_∞ sufficiently small, withi=1, 2, so that

kuk_∞ ≤δ< ¹

2 and kvk_∞ ≤δ < ¹ 2.

Moreover, from Lemma 2.6 we have u,v ∈ C^1,α(_Ω), for some α ∈ (0, 1). Thus, by virtue of Lemmas2.3and2.7, there existC₃,C₄ >0 such that

u(x)^−γ1

φ₀(x)^−γ1 ≤ C₃^−γ1 and v(x)^−γ2

φ₀(x)^−γ2 ≤C⁻₄^γ2, for all x∈_Ω.

Therefore, by(H)we get

|h₁u^−γ1| ≤C₃^−γ1kh₁φ₀^−γ1k_∞ and |h2v^−γ2| ≤C⁻₄^γ2kh2φ⁻₀^γ2k_∞, (3.3) implying thath₁u^−γ1,h₂v^−γ2 ∈ L^∞(_Ω), which ends the proof of condition (i).

In order to prove (ii), we invoke Lemma 2.1 and maximum principle once again to claim that there exists an unique positive solution 0<u∈W₀^1,q1(_Ω)satisfying

(−div(a₁(|∇u|^p1)|∇u|^p1−2∇u) =h₁(x)u^−γ1 in Ω,

u =0 on∂Ω, (3.4)

and there exists an unique positive solution 0< v∈W₀^1,q2(_Ω)satisfying (−div(a₂(|∇v|^p2)|∇v|^p2−2∇v) =h₂(x)v^−γ2 in Ω,

v=0 on ∂Ω. (3.5)

Sinceh₁u^−γ1,h₂v^−γ2 ∈ L^∞(_Ω), we use Lemma2.5 to obtainu,v ∈ L^∞(_Ω)and hence, from Lemma2.6 we obtain u,v ∈ C^1,α(_Ω), for some α ∈ (0, 1). Furthermore, note that using (3.3) we have

kuk_∞ ≤C^∗₁kh₁u^−γ1k

1 p1−1

∞ ≤C₁^∗kh₁k

1 p1−1

∞ C^−γ1

1

p1−1

3 kφ₀k^−γ1

1

p1−1

∞

and

kvk_∞ ≤C^∗₂kh2v^−γ2k

p21−1

∞ ≤C₂^∗kh2k

p21−1

∞ C^−γ2

1

p2−1

4 kφ0k^−γ1

1

p1−1

∞ .

So, choosingkh_ik_∞ sufficiently small, withi=1, 2, we obtain kuk_∞ ≤δ< ¹

2 and kvk_∞ ≤δ < ¹ 2.

Now, sincekuk_∞ andkvk_∞are small it follows from (3.1), (3.2), (3.4) and (3.5) that

−div(a₁(|∇u|^p1)|∇u|^p1−2∇u) =h₁(x)u^−γ1 ≥h₁(x)kuk⁻_∞^γ1 ≥h₁(x)

=−div(a₁(|∇u|^p1)|∇u|^p1−2∇u)

and

−div(a₂(|∇v|^p2)|∇v|^p2−2∇v) =h₂(x)v^−γ2 ≥h₂(x)kvk⁻_∞^γ2 ≥ h₂(x)

=−div(a₂(|∇v|^p2)|∇v|^p2−2∇v).

Therefore, applying the Weak Comparison Principle for the p&q-Laplacian operator we con- clude that

0<u(x)≤ u(x) a.e. in Ω and 0<v(x)≤v(x) a.e. inΩ, which proves (iii).

Our final task is to check that the condition(iv)holds. Indeed, we invoke (F₁), (i), (3.1) and (3.2) to obtain

−div(a₁(|∇u|^p1)|∇u|^p1−2∇u)−h₁(x)u^−γ1−F_u(x,u,v)

≤2h₁(x)−h₁(x)u^−γ1 ≤ h₁(x)(2− kuk⁻_∞^γ1)≤0 and

−div(a₂(|∇v|^p2)|∇v|^p2−2∇v)−h₂(x)v^−γ2 −Fv(x,u,v),

≤2h₂(x)−h₂(x)v^−γ2 ≤h₂(x)(2− kvk⁻_∞^γ2)≤0, which implies that (u,v)is a subsolution for system (1.1). Finally, we use(F₁), (ii), (iii), (3.4) and (3.5) to get

−div(a₁(|∇u|^p1)|∇u|^p1−2∇u)−h₁(x)u^−γ1−F_u(x,u,v)≥h₁(x)(u^−γ1−u^−γ1)≥0 and

−div(a₂(|∇v|^p2)|∇v|^p2−2∇v−h₂(x)v^−γ2−F_v(x,u,v)≥ h₂(x)(v^−γ2−v^−γ2)≥0, which shows that(u,v)is a supersolution for system (1.1).

Following the same idea in [4] (see also [3]), we introduce the truncation operators T : W₀^1,q1(_Ω)→ L^∞(_Ω)andS:W₀^1,q2(_Ω)→ L^∞(_Ω)given by

Tu(x) =











u(x), ifu(x)> u(x)

u(x), ifu(x)≤ u(x)≤u(x) u(x), ifu(x)< u(x)

(3.6)

and

Sv(x) =











v(x), ifv(x)>v(x)

v(x), ifv(x)≤v(x)≤ v(x) v(x), ifv(x)<v(x).

(3.7)

It is well that the truncation operatorsTandSare continuous and bounded. Now, we consider the following functions

G_u(x,u,v) =h₁(x)(Tu)^−γ1+F_u(x,Tu,Sv) (3.8) and

G_v(x,u,v) =h₂(x)(Sv)^−γ2 +F_v(x,Tu,Sv) _(3.9)

and the auxiliary problem















−div(a₁(|∇u|^p1)|∇u|^p1−2∇u) =G_u(x,u,v)in Ω,

−div(a₂(|∇v|^p2)|∇v|^p2−2∇v) =Gv(x,u,v)inΩ, u,v>_{0 inΩ,}

u=v=0 on∂Ω.

(3.10)

Define the energy functionalΦ:X→_Rassociated with problem (3.10) by Φ(u,v) = ¹

p₁ Z

Ω

A₁(|∇u|^p1)dx+ ¹ p₂

Z

Ω

A₂(|∇v|^p2)dx−

Z

Ω

G(x,u,v)dx, ∀(u,v)∈ X,

whereG(x,s,t) =Rs

0 G_ξ(x,ξ,t)dξ+Rt

0G_ξ(x,s,ξ)dξ.

It follows from Lemma3.1(i)–(iii), (3.8), (3.9) and(F₁)_that

|Gu(x,u,v)| ≤K₁ a.e. in Ω, for someK₁ >0,∀(u,v)∈ X. (3.11) Similarly,

|G_v(x,u,v)| ≤K₂ a.e. inΩ, for someK₂>_0,∀(u,v)∈X. (3.12) Consequently, we use(A₁)to show that the functional Φis well defined and it is of classC¹ on Sobolev spaceXwith

Φ⁰(u,v)(φ,ϕ) =

Z

Ω

a₁(|∇u|^p1)|∇u|^p1−2∇u∇φ+a₂(|∇v|^p2)|∇v|^p2−2∇v∇ϕ dx

−

Z

Ω

G_u(x,u,v)φdx−

Z

Ω

G_v(x,u,v)ϕdx,∀(u,v),(φ,ϕ)∈ X.

Next, consider

M ={(u,v)∈ X;u≤u≤ ua.e. inΩandv≤ v≤va.e. in Ω}.

We claim that Φ is bounded from below in M. Indeed, for all (u,v) ∈ X, we use (A₁)_, (_3.11)_,(_3.12)and continuous embeddingW₀^1,qi(_Ω),→ L^1,qi(_Ω)_{, fori}=1, 2, to obtain thatΦis coercive inM. Moreover, since(A₃)holds andGu,Gv∈ L^∞(_Ω)we have thatΦis weak lower semi-continuous on M. Thus, as M is closed and convex in X, we use [15, Theorem 1.2] to conclude thatΦis bounded from below in Mand attains it is infimum at a point(u,v)∈ M.

Using the same arguments as in the proof of [15, Theorem 2.4], we see that this minimum point (u,v) is a weak solution of problem (3.10). Indeed, for all φ,ϕ ∈ C^∞₀ (_Ω)andε > 0, let the functionsuε,vε ∈ M be given by

u_ε(x) =











u(x)_, u(x) +εφ(x)>u(x)

u(x) +εφ(x), u(x)≤u(x) +εφ(x)≤u(x) u(x), u(x) +εφ(x)<u(x)

and

v_ε(x) =











v(x), v(x) +εϕ(x)>v(x)

v(x) +εϕ(x), v(x)≤v(x) +εϕ(x)≤ v(x) v(x), v(x) +εϕ(x)<v(x).