On some nonlinear anisotropic elliptic equations in anisotropic Orlicz space

Abstract

Purpose

In the present paper, the authors will discuss the solvability of a class of nonlinear anisotropic elliptic problems (P), with the presence of a lower-order term and a non-polynomial growth which does not satisfy any sign condition which is described by an N-uplet of N-functions satisfying the Δ₂-condition, within the fulfilling of anisotropic Sobolev-Orlicz space. In addition, the resulting analysis requires the development of some new aspects of the theory in this field. The source term is merely integrable.

Design/methodology/approach

An approximation procedure and some priori estimates are used to solve the problem.

Findings

The authors prove the existence of entropy solutions to unilateral problem in the framework of anisotropic Sobolev-Orlicz space with bounded domain. The resulting analysis requires the development of some new aspects of the theory in this field.

Originality/value

To the best of the authors’ knowledge, this is the first paper that investigates the existence of entropy solutions to unilateral problem in the framework of anisotropic Sobolev-Orlicz space with bounded domain.

Citation

Benslimane, O., Aberqi, A. and Bennouna, J. (2023), "On some nonlinear anisotropic elliptic equations in anisotropic Orlicz space", Arab Journal of Mathematical Sciences, Vol. 29 No. 1, pp. 29-51. https://doi.org/10.1108/AJMS-12-2020-0133

1. Introduction

Let Ω be a bounded domain of RN(N≥2). The aim behind this paper is the study of boundary value problems for a class of nonlinear anisotropic elliptic equations. More specifically, we consider the unilateral elliptical operators whose nonlinearity is given by a vector of N-functions like

(P)A(u)+∑i=1Nbi(x,u,∇u)=finΩ,u≥ζa.e  inΩ,

where A(u)=∑i=1N(σi(x,u,∇u))xi is a Leray–Lions operator defined in W∘M1(Ω) (defined as the adherence space C0∞(Ω)) into its dual (see assumptions (19), (20), (21) in Section 3); M(t) = (M₁(t), …, M_N(t)) are N-uplet Orlicz functions that satisfy Δ₂−condition; the obstacle ζ is a measurable function that belongs to L∞(Ω)∩W∘M1(Ω); and for the i=1,…,N,bi(x,s,ξ):Ω×R×RN→R are Carathéodory functions (measurable with respect to x in Ω for every (s, ξ) in R×RN, and continuous with respect to (s, ξ) in R×RN for almost every x in Ω) that does not satisfy any sign condition and the growth which is described by the vector N-function (M₁(t), …, M_N(t)) (see assumption (22)). As well as f ∈ L¹(Ω).

For several years great effort has been devoted to the study of nonlinear elliptic equations with an operator which was described by polynomial growth. For example, in the classical Sobolev space, Boccardo and Gallouêt in [1], proved the existence of a weak solution of (P) in the case φ ≡ g ≡ 0. Bénilan in [2] presented the idea of entropy solutions which were adjusted to the Boltzmann condition. For a deeper comprehension of these types of equations in this field, we refer the reader to [3–10] and references therein.

Next, in the Orlicz space, Benkirane and Bennouna in [11] demonstrated the existence of entropy solutions to the following nonlinear elliptic problem:

−diva(x,u,∇u)+div(φ(u))=f,

where φ∈(C0(R))N and f ∈ L¹(Ω). For more results, we refer the reader to [12–24] and references therein.

And in the anisotropic Sobolev-Orlicz space, there are few results dealing with this topic. We will mention recent papers, and we are starting by the pertinent works of Korolev and Cianchi [25, 26] who proved the embeddings of this space. Then, Benslimane, Aberqi and Bennouna in [27] studied the existence and uniqueness of the solution in the following problem in an unbounded domain

(P)A(u)+∑i=1Nbi(x,u,∇u)=f(x)inΩ,u=0on∂Ω

where A(u)=∑i=1N(ai(x,u,∇u))xi is a Leray-Lions operator defined from W∘B1(Ω) into its dual, B(θ) = ( B₁(θ), …, B_N(θ)) are N-uplet Orlicz functions that satisfy the Δ₂−condition, and for i=1,…,N,bi(x,u,∇u):Ω×R×RN→R are the Carathéodory functions that do not satisfy any sign condition and the growth described by the vector N-function B(θ). After that, Kozhevnikova in [28] established the existence of entropy solutions in an unbounded domain to the following problem:

∑i=1N(ai(x,∇u))xi=a0(x,u)inΩ,u(x)=ψ(x)on∂Ω,

where Ω is an arbitrary domain in RN,N≥2,

a0(x,s0)=a0(x,ψ)+b(x,s0),

with a₀(x, ψ) ∈ L¹(Ω), the function b(x, s₀) satisfies the Carathéodory condition and decreases in s0∈R,b(x,ψ)=0 for all x in Ω; therefore, for any x∈Ω,s0∈R

b(x,s0)(s0−ψ)>0.

The author supposed two other conditions: the first one is

sup|s0|≤k|b(x,s0)|=Gk(x)∈L1,loc(Ω),

the second one, δ₀ > 0 such as

b(x,ψ±δ0)∈L1(Ω).

For more results we refer the reader to [29–32] and the references therein.

This type of operator arises in a quite natural way in many different contexts, such as the study of fluid filtration in porous media, constrained heating, elasticity, electro-rheological fluids, optimal control, financial mathematics and other domains, see [33–36] and the references therein.

As far we know, no previous research has investigated the existence of entropy solutions to unilateral problem (P) with the second term as an operator with growth described by an n-uplet of N-functions satisfying the Δ₂−condition, within the fulfilling of anisotropic Sobolev-Orlicz space with bounded domain, the function bi(x,u,∇u) does not satisfy any sign condition and the source f is merely integrable. Hence, motived by the aforementioned papers, our main work is to obtain the existence Theorem for unilateral problems corresponding to (P) via an approximation procedure and some priori estimates.

The rest of this paper is organized as follows: In Section 2, we give some definitions and fundamental properties of anisotropic Sobolev-Orlicz spaces. In Section 3, we give our assumptions on data and the definition of entropy solutions to (P). In Section 4, we will show the existence of entropy solutions, with the functions bi(x,u,∇u),i=1,…,N which does not satisfy any sign condition. Finally, in the Appendix.

2. Definitions and preliminary tools

In this section, we recall the most important and relevant properties and notations about of anisotropic Sobolev-Orlicz space which we will need in our analysis of the problem (P). A comprehensive presentation of Sobolev- Orlicz anisotropic space can be found in the work of M.A Krasnoselskii and Ja. B. Rutickii [25, 37].

Definition 1.

We say that M:R+→R+ is a N-function if M is continuous, convex, with M(t) > 0 for t > 0, M(t)t→0whent→0 and M(t)t→∞whent→∞.

This N-function M admits the following representation: M(t)=∫0tb(s)ds, with b:R+→R+ which is an increasing function on the right with b(0) = 0 in the case t > 0 and b(t) → ∞ when t → ∞.

Its conjugate is noted by M¯;(t)=∫0tq(s)ds with q also satisfies all the properties already quoted from b, with

(1)M¯(t)=supμ≥0(μt−M(μ)),t>0.

The Young’s inequality is given as follows:

(2)∀t,μ>0tμ≤M(μ)+M¯;(t).

Definition 2.

The N-function M(t) satisfies the Δ₂−condition if ∃c > 0, t₀ > 0 such as

(3) M(2t)≤cM(t)t≥t0.

This definition is equivalent to ∀k > 1, ∃ c(k) > 0 such as

(4) M(kt)≤c(k)M(t)fort≥t0.

Definition 3.

The N-function M(t) satisfies the Δ₂−condition as long as there exist positive numbers c > 1 and t₀ ≥ 0 such as for t ≥ t₀ we have

(5) tb(t)≤cM(t).

Also, each N-function M(t) satisfies the inequality

(6) M(μ+t)≤cM(t)+cM(μ)t,μ≥0.

We consider the Orlicz space L_M(Ω) provided with the norm of Luxemburg given by

(7)‖u‖M,Ω=infk>0/∫ΩM|u(x)|kdx≤1.

According to [37] we obtain the inequalities

(8)∫ΩM|u(x)|‖u‖M,Ωdx≤1,

and

(9)‖u‖M,Ω≤∫ΩM(|u|)dx+1.

Moreover, the Hölder’s inequality holds and we have for all u ∈ L_M(Ω) and v∈LM¯(Ω)

(10)∫Ωu(x)v(x)dx≤2‖u‖M,Ω.‖v‖M¯,Ω.

In [25, 37], if P(t) and M(t) are two N-functions such as P(t) ≪ M(t) and meas Ω < ∞, then L_M(Ω) ⊂ L_P(Ω). Furthermore,

(11)‖u‖P,Ω≤A0(measΩ)‖u‖M,Ωu∈LM(Ω).

And for all N-functions M(t), if meas Ω < ∞, then L_∞(Ω) ⊂ L_M(Ω) with

(12)‖u‖M,Ω≤A1(measΩ)‖u‖∞,Ωu∈LM(Ω).

Also for all N-functions M(t), if meas Ω < ∞, then L_M(Ω) ⊂ L¹(Ω) with

(13)‖u‖1,Ω≤A2‖u‖M,Ωu∈LM(Ω).

We define for all N-functions M₁(t), …, M_N(t) the space of Sobolev-Orlicz anisotropic W∘M1(Ω) as the adherence space C0∞(Ω) under the norm

(14)‖u‖W∘M1(Ω)=∑i=1N‖uxi‖Mi,Ω.

Definition 4.

A sequence { u_m } is said to converge modularly to u in W∘M1(Ω) if for some k > 0 we have

(15) ∫ΩM|um−u|kdx→0asm→∞.

Remark 1.

Since M satisfies the Δ₂−condition, then the modular convergence coincides with the norm convergence.

Proposition 1.

(16) θM′(t)=M¯(M′(t))+M(t)t>0,

with M′ is the right derivative of the N-function M(t).

Proof.

By (2) we take μ = M′(t), then we obtain

M′(t)t≤M(t)+M¯(M′(t)),

and by Ch. I [37] we get the result. □

Since Ω is a bounded domain in RN. The following Lemmas are true:

Lemma 1.

[8] For all u∈W∘LM1(Ω) with meas Ω < ∞, we have

∫ΩM|u|λdx≤∫ΩM(|∇u|)dx,

where λ = diam(Ω) is the diameter of Ω.

Note by h(t)=∏i=1NBi−1(t)t1N and we assume that ∫01h(t)tdt converge. So, we consider the N-functions M*(z) defined by (M*)−1(z)=∫0|z|h(t)tdt.

Lemma 2.

[26] Let u∈W∘M1(Ω). If

(17) ∫1∞h(t)tdt=∞,

then, W∘M1(Ω)⊂LM*(Ω) and ‖u‖M*,Ω≤N−1N‖u‖W∘M1(Ω).

If

∫1∞h(t)tdt≤∞,

then, W∘M1(Ω)⊂L∞(Ω) and ‖u‖∞,Ω≤β‖u‖W∘M1(Ω), with β=∫0∞h(t)tdt.

In the following, we will assume that for each N-function Mi(z)=∫0|z|bi(t)dt obeys the further condition:

(18)limα→∞infθ>0bi(αθ)bi(α)=∞i=1,…,N.

Remark 2.

The Following function:

M(z)=|z|b(|ln|z||+1),

with b > 1 check the Δ₂-condition and (18).

3. Assumptions on data and definition of solution

Statement of the problem: Suppose they have non-negative measurable functions φ, ϕ ∈ L¹(Ω) and positive constants a¯ and ã such as:

(19)∑i=1Nσi(x,s,ξ)−σi(x,s,ξ′).(ξi−ξi′)>0,

(20)∑i=1Nσi(x,s,ξ).ξi≥a¯∑i=1NMi(|ξi|)−φ(x),

(21)∑i=1N|σi(x,s,ξ)|≤ã∑i=1NM¯i−1Mi(|ξ|)+ϕ(x),

and

(22)∑i=1N|bi(x,s,ξ)|≤h(x)+l(s).∑i=1NMi(|ξ|),

with M¯(t) the complementary of M(t), h(x) ∈ L¹(Ω) and l:R→R+ a positive continuous function such as: l∈L1(R)∩L∞(R).

3.1 Definition of entropy solutions:

Definition 5.

A measurable function u is said to be an entropy solution for the problem (P), if u∈W∘M1(Ω) such that u ≥ ζ a.e. in Ω and

∑i=1N∫Ωσi(x,u,∇u).∇(u−v)dx+∑i=1N∫Ωbi(x,u,∇u).(u−v)dx+∫Ωm.Tm(u−ζ)−.sg1m(u).(u−v)dx≤∫Ωf(x).(u−v)dx∀v∈Kζ∩L∞(Ω),

where, Kζ={u∈W∘M1(Ω)such asu≥ζa.e. inΩ},form∈N*sgm(s)=Tm(s)m. We define the truncation at height m, Tm(u):R→R by

Tm(u)=uif|u|≤m,mif|u|>m.

4 . Main result

In this section, we will show the existence of our problem (P). We will assume that f^m → f in L¹(Ω), m → ∞, | f^m(x) | ≤ | f(x) | and for i=1,…,N,σim(x,um,∇um):(W∘M1(Ω))N→(W∘M¯−1(Ω))N being Carathéodory functions with

σim(x,u,∇u)=σi(x,Tm(u),∇u),

and bim(x,um,∇um):Ω×R×RN→R again being Carathéodory functions not satisfying any sign condition, with

bm(x,u,∇u)=b(x,u,∇u)1+1m|b(x,u,∇u)|,

and

(23)|bm(x,u,∇u)|=|b(x,Tm(u),∇u)|≤mfor all m∈N*,

Consider the penalized equations:

Pm:∑i=1N∫Ωσimx,um,∇um.∇(um−v)dx+∑i=1N∫Ωbim(x,um,∇um).(um−v)dx+∫Ωm.Tmum−ζ−.sg1m(um).(um−v)dx=∫Ωfm(x).(um−v)dx∀v∈W∘M1(Ω).

Theorem 1.

Let’s assume that conditions (19)–(22) and (18) hold true, then there exists at least one solution of the approximate problem (Pm).

Proof.

See Appendix. □

Now, we will show some results in the form of propositions that will be useful for the demonstration of existence Theorem 2, see below.

Proposition 2.

(see [27]) Suppose that conditions (19)–(22) are satisfied, and let (um)m∈N be a sequence in W∘B1(Ω) such as.

(a)um⇀u in W∘M1(Ω),

(b)σm(x,um,∇um) is bounded in LM¯(Ω),

(c)∫Ωσm(x,um,∇um)−σm(x,um,∇uχK).∇(um−uχK)dx→0 as K → + ∞ (χ_K the characteristic function of Ω_K = {x ∈Ω; | ∇u | ≤ K }).

Then:

M(|∇um|)→M(|∇u|) inL1(Ω).

Proposition 3.

(see [31]) Let’s assume that conditions (19)–(22) and (18) hold true, then the generalized solution of the problems (Pm) satisfies the following estimate:

∫ΩM(|∇TK(um)|)≤c=c(K),K>0.

Proposition 4.

(see [31]) Suppose that conditions (19)–(22) and (18) are satisfied, and let (um)m∈N be a solution of the problem (Pm), then there exists a measurable function u such as ∀ K > 0, we have for all subsequence noted again u_n,

(a) u_m → u a.e in Ω,

(b)TK(um)⇀TK(u) weakly  inW∘M1(Ω),

(c)TK(um)→TK(u) strongly  inW∘M¯−1(Ω).

Proposition 5.

Suppose that conditions (19)–(22) and (18) are satisfied, and let (um)m∈N be a solution of the problem (Pm), then for any K > 0, we have

(1) σm(x,TK(um),∇TK(um)) is  bounded  inW∘M¯1(Ω),

(2) M( | ∇T_K(u_m) |) → M( | ∇T_K(u) |) is strongly in L¹(Ω),

Proof.

1

‖σm(x,TK(um),∇TK(um))‖M¯,Ω=∑i=1N‖σim(x,TK(um),∇TK(um))‖Mi¯,Ω≤∑i=1N∫ΩMi(|∇TK(um)|)dx+‖ϕ‖1+N,

from Proposition 3 we obtain:

‖σm(x,TK(um),∇TK(um))‖M¯,Ω≤c(K)+‖ϕ‖1+N.

Hence, σ^m(x, T_K(u_m), ∇T_K(u_m)) is bounded in W∘M¯1(Ω).

2 Showing that M(|∇T_K(u_m)|) → M(|∇T_K(u)|) strongly in L¹(Ω) that’s why, let’s introduce the following functions of a variable K defined as hj(K)=1if|K|≤j,0if|K|⩾j+1,j+1+|K| ifj<|K|<j+1, with j as a non-negative real parameter, Ω_K = 

{ x ∈ Ω: | ∇T_K(u(x)) | ≤ K } and we note that χ_K is a characteristic function of Ω_K. It’s clear that Ω_K ⊂ Ω_K+1 and meas(Ω \Ω_K) → 0 since K → ∞ shows that the following assertions are true.

Assertion 1.

limj→∞limm→∞∫{Ω:j<|K|<j+1}σm(x,um,∇um).∇umdx=0.

Assertion 2.

TK(um)→TK(u)modular convergence inW∘M1(Ω).

Proof of assertion 1.

Let

v=um+exp(G(|um|)).T1(um−Tj(um)),

with G(s)=∫0sl(t)a¯dt as a test function in (Pm) then we get:

∑i=1N∫Ωσim(x,um,∇um).∇(exp(G(|um|)).T1(um−Tj(um)))dx+∑i=1N∫Ωbim(x,um,∇um).exp(G(|um|)).T1(um−Tj(um))dx+∫Ωm.Tm(um−ζ)−.sg1m(um).exp(G(|um|)).T1(um−Tj(um))dx=∫Ωfm(x).exp(G(|um|)).T1(um−Tj(um))dx,

by (20) and (22) we obtain:

∑i=1N∫Ωσim(x,um,∇um).exp(G(|um|)).∇(T1(um−Tj(um)))dx+∫Ωm.Tm(um−ζ)−.sg1m(um).exp(G(|um|)).T1(um−Tj(um))dx≤∫Ωfm(x)+h(x)+φ(x).l(um)a¯.exp(G(|um|)).T1(um−Tj(um))dx,

since f, φ ∈ L¹(Ω), l∈L1(R)∩L∞(R), exp(G(±∞))≤exp‖l(um)‖L1(R)a¯ and by proposition 4 we obtain

limm→∞ limj→∞∫Ωfm(x)+h(x)+φ(x).l(um)a¯.exp(G(|um|)).T1(um−Tj(um))dx=0.

Hence,

limm→∞ limj→∞∑i=1N∫{Ω:j<|um|<j+1}σim(x,um,∇um).∇umdx=0,

and

limm→∞ limj→∞∫Ωm.Tm(um−ζ)−.sg1m(um).exp(G(|um|)).T1(um−Tj(um))dx=0.

Proof of assertion 2.

Let j ≥ K > 0, we consider

v=um+exp(G(|um|)).(TK(um)−TK(u)).hj(um),

as a test function in (Pm) we obtain:

∑i=1N∫Ωσim(x,um,∇um).∇(exp(G(|um|)).(TK(um)−TK(u)).hj(um))dx+∑i=1N∫Ωbim(x,um,∇um).exp(G(|um|)).(TK(um)−TK(u)).hj(um)dx+∫Ωm.Tm(um−ζ)−.sg1m(um).exp(G(|um|)).(TK(um)−TK(u)).hj(um)dx=∫Ωfm(x).exp(G(|um|)).(TK(um)−TK(u)).hj(um)dx,

by (20) and (22) we have:

(24) ∑i=1N∫{|um|≤K}σim(x,TK(um),∇TK(um)).(∇TK(um)−∇TK(u)).exp(G(|um|))dx+∫Ωm.Tm(um−ζ)−.sg1m(um)exp(G(|um|)).(TK(um)−TK(u)).hj(um)dx≤∫Ωfm(x)+h(x)+φ(x).l(um)a¯.exp(G(|um|)).(TK(um)−TK(u)).hj(um)dx+∑i=1N∫{Ω:K<|um|<j+1}|σim(x,Tj+1(um),∇Tj+1(um))|.|∇TK(um)|.exp(G(|um|))dx+∑i=1N∫{Ω:K<|um|<j+1}σim(x,um,∇um).∇um.|TK(um)−TK(u)|.exp(G(|um|))dx,

and since T_K(u_m) ⇀ T_K(u) is weakly in W∘M1(Ω), we have:

∫Ωfm(x)+h(x)+φ(x).l(um)a¯.exp(G(|um|)).(TK(um)−TK(u)).hj(um)dx→0,

and

∫Ωm.Tm(um−ζ)−.sg1m(um)exp(G(|um|)).(TK(um)−TK(u)).hj(um)dx→0,

since |σim(x,Tj+1(um),∇Tj(um))| is bounded in LM¯(Ω), then there exist σ̃m∈LM¯(Ω) such that

(25) |σim(x,Tj+1(um),∇Tj+1(um))|⇀σ̃minLM¯(Ω),

(26) ∑i=1N∫{Ω:K<|um|<j+1}|σim(x,Tj+1(um),∇Tj+1(um))|.|∇TK(um)|.exp(G(|um|))dx≤exp‖l‖L1(R)a¯.∑i=1N∫{Ω:K<|um|<j+1}|σim(x,Tj+1(um),∇Tj+1(um))|.|∇TK(um)|dx→exp‖l‖L1(R)a¯.∑i=1N∫{Ω:K<|um|<j+1}σ̃m.|∇TK(u)|dx=0withm→∞,

according to Assertion 1, we get:

(27) ∑i=1N∫{Ω:K<|um|<j+1}σim(x,um,∇um).∇um.|TK(um)−TK(u)|.exp(G(|um|))dx≤2K.exp‖l‖L1(R)a¯.∑i=1N∫{Ω:K<|um|<j+1}σim(x,um,∇um).∇umdx→0withj→∞,

combine (24)–(27) we obtain:

(28) ∑i=1N∫Ωσim(x,TK(um),∇TK(um))−σim(x,TK(um),∇TK(u)).(∇TK(um)−∇TK(u))dx→0withm→∞.

According to Proposition 2 we conclude that

M(|∇TK(um)|)→M(|∇TK(u)|)inL1(Ω).

Proposition 6.

(See [31]) Suppose that the conditions (19)–(22) and (18) are true, and uj,u∈W∘M1(Ω)

(29) ‖uj‖W∘M1(Ω)≤c,j=1,…,∞

(30) uj→u in LM(Ω),

with M(z) is a N-function. Let’s assume the following functions:

Aj(x)=∑i=1Nσim(x,uj,∇uj)−σim(x,u,∇u)∇(uj−u)+∑i=1Nbim(x,uj,∇uj)−bim(x,u,∇u)(uj−u),

j = 1, ⋯ satisfying the condition

(31) ∫ΩAj(x)dx→0,j→∞.

Then, there exists a sequence of natural numbers J⊂N such that as j → ∞, j ∈ J

(32)σim(x,uj,∇uj)⇀σim(x,u,∇u) in LM¯i(Ω)i=1,…,N.

Theorem 2.

Under assumptions (19)–(22), the problem (P) has at least one entropy solution.

Proof of Theorem 2.

We divide our proof in six steps:

Step 1: A priori estimate of { u^m }. We consider the following test function:

v=um+ηexp(G(|um|))T1(um−Tj(um)),

with η small enough, we get:

∑i=1N∫Ωσim(x,um,∇um).∇exp(G(|um|))T1(um−Tj(um))dx+∑i=1N∫Ωbim(x,um,∇um).exp(G(|um|))T1(um−Tj(um))dx+∫Ωm.Tm(um−ζ)−.sg1m(um).exp(G(|um|))T1(um−Tj(um))≤∫Ωfm(x).exp(G(|um|))T1(um−Tj(um))dx,

according to (20) and (22) we obtain:

∑i=1N∫Ωσim(x,um,∇um).exp(G(|um|))∇T1(um−Tj(um))dx+∫Ωm.Tm(um−ζ)−.sg1m(um).exp(G(|um|))T1(um−Tj(um))≤∫Ωfm(x)+h(x)+φ(x).l(um)a¯.exp(G(|um|))T1(um−Tj(um))dx,

since f, h, φ ∈ L¹(Ω), l∈L1(R)∩L∞(R), exp(G(±∞))≤exp‖l‖L1(R)a¯ and the fact T₁(u_m − T_j(u_m)) → 0 is weakly in W∘M1(Ω) as j → ∞ (proposition 4). We have:

∫Ωfm(x)+h(x)+φ(x).l(um)a¯.exp(G(|um|))T1(um−Tj(um))dx→0asm→∞,

then,

∑i=1N∫Ωσim(x,um,∇um).exp(G(|um|)).∇T1(um)−Tj(um)dx+∫Ωm.Tm(um−ζ)−.sg1m(um).exp(G(|um|)).T1(um)−Tj(um)dx≤0.

Hence,

limm→∞limj→∞∑i=1N∫{Ω:K<|um|<j+1}σim(x,um,∇um).∇umdx=0,

and

(33) limm→∞∫Ωm.Tm(um−ζ)−.sg1m(um)dx=0.

• Step 2. Convergence of the gradient:

In this step we consider again the following test function:

v=um+ηexp(G(|um|))(TK(um)−TK(u))hj(um),

with, hj(um)=1−|T1(um−Tj(um))|=1 if{|um|≥j},0 if{|um|≥j+1},j+1−|um|if{j<|um|<j+1}, and |T_K(u_m) − T_K(u)| at the same sign when u_m ∈{ |u_m| > K} where j ≥ K > 0 and η are small enough, we obtain

∑i=1N∫Ωσim(x,um,∇um).∇(exp(G(|um|))(TK(um)−TK(u))hj(um))dx+∫Ωm.TK(um−ζ)−.sg1m(um).exp(G(|um|))(TK(um)−TK(u))hj(um)dx+∑i=1N∫Ωbim(x,um,∇um).exp(G(|um|))(TK(um)−TK(u))hj(um)dx≤∫Ωfm(x).exp(G(|um|))(TK(um)−TK(u))hj(um)dx,

by (20), (22) and the fact j ≥ K > 0 we have this:

(34) ∑i=1N∫{|um|≤K}σim(x,TK(um),∇TK(um)).exp(G(|um|)).∇(TK(um)−TK(u))dx+∑i=1N∫Ωσim(x,um,∇um).exp(G(|um|)).(TK(um)−TK(u)).∇hj(um)dx+∫Ωm.Tm(um−ζ)−.sg1m(um).exp(G(|um|)).(TK(um)−TK(u)).hj(um)dx≤∫Ωfm(x)+h(x)+φ(x).l(|um|)a¯.exp(G(|um|)).(TK(um)−TK(u)).hj(um)dx,

then, by the condition (c) in proposition 4 we have T_K(u_m) → T_K(u) weakly in W∘M1(Ω), and since f^m, h(x), φ ∈ L¹(Ω) we get

(35) ∫Ωfm(x)+h(x)+φ(x).l(|um|)a¯.exp(G(|um|)).(TK(um)−TK(u)).hj(um)dx→0,

and

∫Ωm.Tm(um−ζ)−.sg1m(um).exp(G(|um|)).(TK(um)−TK(u)).hj(um)dx=0,

and

(36) ∑i=1N∫Ωσim(x,um,∇um).exp(G(|um|)).(TK(um)−TK(u)).∇hj(um)dx=∑i=1N∫{Ω:j<|um|<j+1}σim(x,um,∇um).∇um.exp(G(|um|)).(TK(um)−TK(u))dx≤2K.exp‖l‖L1(Ω)a¯.∑i=1N∫{Ω:j<|um|<j+1}σim(x,um,∇um).∇umdx→0asj→∞,

combining (34) − (36)

∑i=1N∫{|um|≤K}σim(x,TK(um),∇TK(um)).exp(G(|um|)).∇(TK(um)−TK(u))dx≤ϵ(i,j,m),

thus,

∑i=1N∫Ωσim(x,TK(um),∇TK(um))−σim(x,TK(um),∇TK(u)).∇(TK(um)−TK(u)).exp(G(|um|))dx≤−∑i=1N∫Ωσim(x,TK(um),∇TK(u)).∇(TK(um)−TK(u)).exp(G(|um|))dx−∑i=1N∫{|um|>K}σim(x,TK(um),∇TK(um)).∇(TK(um)−TK(u)).exp(G(|um|))dx+ϵ(i,j,m),

letting i, j, m tend to infinity, we have:

(37) ∑i=1N∫Ωσim(x,TK(um),∇TK(um))−σim(x,TK(um),∇TK(u)).∇(TK(um)−TK(u)).exp(G(|um|))dx→0asm→∞,

which is implied by Proposition 2

(38) M(|∇um|)→M(|∇u|)inL1(Ω).

Hence, we obtain for a subsequence:

(39) ∇um→∇ua.e inΩ.

• Step 3. The equi-integrability of bim(x,um,∇um):

In this section we will show that:

(40) bim(x,um,∇um)→bi(x,u,∇u).

Therefore, it is enough to show that bim(x,um,∇um) is uniformly equi-integrable. We take the following test function:

v=um+ηexp(G(|um|))∫|um|0∫{|s|>j}l(s)dsdx,

we obtain:

∑i=1N∫Ωσim(x,um,∇um).∇exp(G(|um|))∫|um|0∫{|s|>j}l(s)dsdx+∑i=1N∫Ωbim(x,um,∇um).exp(G(|um|))∫|um|0∫{|s|>j}l(s)dsdx+∫Ωm.Tm(um−ζ)−.sg1m(um).exp(G(|um|))∫|um|0∫{|s|>j}l(s)dsdx=∫Ωfm(x).exp(G(|um|))∫|um|0∫{|s|>j}l(s)dsdx,

by (20) and (22) we get:

a¯∑i=1N∫ΩMi(|∇um|).exp(G(|um|)).∫{|um|>j}l(um)dx+∫Ωm.Tm(um−ζ)−.sg1m(um).exp(G(|um|))∫|um|0∫{|s|>j}l(s)dsdx≤∫Ωfm+h(x)+φ(x).l(|um|)a¯.exp(G(|um|))∫|um|0∫{|s|>j}l(s)dsdx+∫Ωφ(x).exp(G(|um|)).∫{|um|>j}l(um)dx,

which implies:

a¯∑i=1N∫ΩMi(|∇um|).exp(G(|um|)).l(um).χ{|um|>j}dx≤c1∫|um|0l(um).χ{|um|>j}dx.

Therefore,

∑i=1N∫{|um|>j}l(um).Mi(|∇um|)dx≤c2∫|um|0l(|um|).χ{|um|>j}dx.

and

(41) 0≤∫Ωm.Tm(um−ζ)−.sg1m(um).exp(G(|um|))∫|um|0∫{|s|>j}l(s)dsdx≤c3,

and since l∈L1(R)∩L∞(R) we deduce that:

limj→∞supm∈{1,…,N}∑i=1N∫{|um|}l(|um|).Mi(|∇um|)dx=0,

by (37) and (31) we conclude (30)

• Step 4. Passing to the limit

Let ϕ∈W∘M1(Ω)∩L∞(Ω) we take the following test function:

v=um−ηTj(um−ϕ),

and |u_m|−‖ϕ‖_∞ < |u_m − ϕ| ≤ j. Then, {|u_m − ϕ| ≤ j}⊂{|u_m| ≤ j + ‖ϕ‖_∞} we obtain:

∑i=1N∫Ωσim(x,um,∇um).∇Tj(um−ϕ)dx+∑i=1N∫Ωbim(x,um,∇um).Tj(um−ϕ)dx+∫Ωm.Tm(um−ζ)−.sg1m(um).Tj(um−ϕ)dx≤∫Ωfm(x).Tj(um−ϕ)dx,

which implies that:

∑i=1N∫Ωσim(x,um,∇um).∇Tj(um−ϕ)dx=∑i=1N∫Ωσim(x,Tj+‖ϕ‖∞(um),∇Tj+‖ϕ‖∞(um))−σim(x,Tj+‖ϕ‖∞(um),∇ϕ)×∇Tj+‖ϕ‖∞(um−ϕ).χ{|um−ϕ|<j}dx+∑i=1N∫Ωσim(x,Tj+‖ϕ‖∞(um),∇ϕ)∇Tj+‖ϕ‖∞(um−ϕ).χ{|um−ϕ|<j}dx,

by Fatou’s Lemma we get:

limm→∞inf∑i=1N∫Ωσim(x,Tj+‖ϕ‖∞(um),∇ϕ)∇Tj+‖ϕ‖∞(um−ϕ).χ{|um−ϕ|<j}dx=∑i=1N∫Ωσim(x,Tj+‖ϕ‖∞(u),∇ϕ)∇Tj+‖ϕ‖∞(u−ϕ).χ{|u−ϕ|<j}dx,

and the fact that

(42) ∑i=1Nσim(x,Tj+‖ϕ‖∞(um),∇Tj+‖ϕ‖∞(um))⇀∑i=1Nσim(x,Tj+‖ϕ‖∞(u),∇Tj+‖ϕ‖∞(u))

weakly in W∘M1(Ω). And since T_j(u_m − ϕ) ⇀ T_j(u − ϕ) weakly in W∘M1(Ω), and by (39) we obtain:

∑i=1N∫Ωbim(x,um,∇um)Tj(um−ϕ)dx→∑i=1N∫Ωbi(x,u,∇u)Tj(u−ϕ)dx,

and

∫Ωfm(x)Tj(um−ϕ)dx→∫Ωf(x)Tj(u−ϕ)dx,

and

∫Ωm.Tm(um−ζ)−.sg1m(um).Tj(um−ϕ)dx→∫Ωm.Tm(u−ζ)−.sg1m(u).Tj(u−ϕ)dx,

which completes the proof of Theorem 2.

Remark 3.

For the demonstration of the uniqueness solution to this problem (P) in unbounded domain is obtained in [31] with the operator bi(x,u,∇u):Ω×R×RN→R are strictly monotonic, at least for a broad class of lower order term, and in [27] with the operator bi(x,u,∇u):Ω×R×RN→R for i = 1, …, N are contraction Lipschitz continuous functions which do not satisfy any sign condition, and

∑i=1Nσi(x,ξ,∇ξ)−σi(x,ξ′,∇ξ′).(∇ξ−∇ξ′)dx>0.

Appendix

Let

A:W∘M1(Ω)→(W∘M1(Ω))′v→<A(u),v>=∫Ω∑i=1Nσi(x,u,∇u).∂v∂xi+bi(x,u,∇u).vdx−∫Ωf(x).vdx,

and let denote LM¯(Ω)=∏k=1NLM¯i(Ω) with the norm:

‖v‖LM¯(Ω)=∑i=1N‖vi‖M¯i,Ωv=(v1,…,vN)∈LM¯(Ω).

where Mi¯(t) are N-functions satisfying the Δ₂−conditions.

Sobolev-space W∘M1(Ω) is the completion of the space C0∞(Ω).

σ(x,s,ξ)=σ1(x,s,ξ),…,σN(x,s,ξ)

and

b(x,s,ξ)=b1(x,s,ξ),…,bN(x,s,ξ).

Let’s show that operator A is bounded. So, for u∈W∘M1(Ω), according to (9) and (23) we get:

(43)‖σ(x,u,∇u)‖LM¯(Ω)=∑i=1N‖σi(x,u,∇u)‖LM¯i(Ω)≤∑i=1N∫ΩM¯i(σi(x,u,∇u))dx+N≤ã(Ω).‖M(u)‖1,Ω+‖ϕ‖1,Ω+N.

Further, for σ(x,u,∇u)∈LM¯i(Ω),v∈W∘M1(Ω) using Hölder’s inequality we have:

(44)|<A(u),v>Ω|≤2‖σ(x,u,∇u)‖LM¯(Ω).‖v‖W∘M1(Ω)+2‖b(x,u,∇u)‖LM(Ω).‖v‖W∘M1(Ω)+c0.‖v‖W∘M1(Ω).

Thus, A is bounded.

And that A is coercive. So, for u∈W∘M1(Ω)

<A(u),u>Ω=∑i=1N∫Ωσi(x,u,∇u).∂u∂xidx+∑i=1N∫Ωbi(x,u,∇u).udx−∫Ωf(x).udx.

Then,

<A(u),u>Ω‖u‖W∘M1(Ω)≥1‖u‖W∘M1(Ω).a¯∑i=1N∫ΩMi∂u∂xidx−c1−c0−l(u).∑i=1N∫ΩMi∂u∂xidx−∫Ωh(x)dx≥1‖u‖W∘M1(Ω).(a¯(Ω)−c2).∑i=1N∫ΩMi∂u∂xidx−c0−c1−c3.

According to (18) we have for all k > 0, ∃ α₀ > 0 such that:

bi(|uxi|)>kbi|uxi|‖uxi‖Mi,Ω,i=1,…,N.

We take ‖uxi‖Mi,Ω>α0i=1,…,N.

Suppose that ‖uxi‖W∘M1(Ω)→0 as j → ∞, we can assume that:

‖ux1j‖M1,Ω+⋯+‖uxNj‖MN,Ω≥Nα0.

According to (9) for c > 1, we have:

|uj|b(|uj|)<cM(uj),

then, by (2.8) we obtain:

<A(uj),uj>Ω‖uj‖W∘M1(Ω)≥a¯(Ω)−c2Nα0.∑i=1N∫ΩMi∂u∂xidx−c4Nα0≥a¯(Ω)−c2Nα0.∑i=1N∫Ω|uxij|b(|uxij|)dx−c4Nα0≥(a¯(Ω)−c2).kcN‖uxij‖Mi.∑i=1N∫Ω|uxij|bi|uxij|‖uxij‖Mi,Ωdx−c4Nα0≥(a¯(Ω)−c2).kcN.∑i=1N∫ΩMi|uxij|‖uxij‖Mi,Ωdx−c4Nα0≥(a¯(Ω)−c2).kcN−c4Nα0,

which shows that A is coercive because k is arbitrary.

And finally that A is pseudo-monotonic. Following up this assumption and since the space W∘M1(Ω) is separable, then ∃(uj)∈C0∞(Ω) such as:

(45)uj⇀uinW∘M1(Ω),

and

(46)A(uj)⇀yin(W∘M1(Ω))′;

according to (45), we have for all subsequences denoted again by u^j,

‖uj‖W∘M1(Ω)≤c2,j∈N

(uj)j∈N is bounded in W∘M1(Ω), and since W∘M1(Ω) is continuously and compactly injected into L_M(Ω)

uj⇀uweakly inLM(Ω),

uj→ua.e. inΩ,j∈N,

and according to (39), we have:

σim(x,uj,∇uj)→σim(x,u,∇u)a.e. inΩ,j∈N

and

bim(x,uj,∇uj)→bim(x,u,∇u)a.e. inΩ,j∈N

and

m.Tm(uj−ζ)−.sg1m(uj)→m.Tm(u−ζ)−.sg1m(u)a.e. inΩ,j∈N

from (45) and (46), there exist σ̃m in LM¯(Ω) such as:

(47)σim(x,uj,∇uj)⇀σ̃m,j∈N

and there exist b̃m in L_M(Ω) such as:

(48)bim(x,uj,∇uj)⇀b̃m,j∈N.

By (33) it is clear that for any v∈W∘M1(Ω), we get:

(49)<y,v>=limj→∞∑i=1N∫Ωσim(x,uj,∇uj).∇vdx+limj→∞∑i=1N∫Ωbim(x,uj,∇uj).vdx=∫Ωσ̃m.∇vdx+∫Ωb̃m.vdx,

whereof:

(50)limj→∞sup<A(uj),uj>=limj→∞sup∑i=1N∫Ωσim(x,uj,∇uj)∇ujdx+limj→∞∑i=1N∫Ωbim(x,uj,∇uj)ujdx≤∫Ωσ̃m∇ujdx+∫Ωb̃mujdx.

By (48), we have:

(51)∫Ωbm(x,uj,∇uj)ujdx→∫Ωb̃mudx.

Consequently,

(52)limj→∞sup∑i=1N∫Ωσim(x,uj,∇uj)∇ujdx≤∫Ωσ̃m∇ujdx.

On the other hand, we have by the condition of monotony:

∑i=1N(σim(x,uj,∇uj)−σim(x,uj,∇u)).∇(uj−u)≥0,

which implies

(53)∑i=1N(σi(x,Tm(uj),∇uj)−σi(x,Tj(uj),∇u)).∇(uj−u)≥0,

then,

∑i=1Nσi(x,Tm(uj),∇uj).∇uj≥∑i=1Nσi(x,Tm(uj),∇u).∇(uj−u)+∑i=1Nσi(x,Tm(uj),∇uj).∇u,

and by Step 2, we get:

∑i=1Nσi(x,Tm(uj),∇u)→∑i=1Nσi(x,Tm(u),∇u)inLM¯(Ω),

according to (47), we obtain:

(54)limj→∞inf∑i=1N∫Ωσim(x,uj,∇uj).∇ujdx≥∫Ωσ̃m.∇ujdx.

Therefore, from (52), we have:

(55)limj→∞∑i=1N∫Ωσim(x,uj,∇uj).∇ujdx=∫Ωσ̃m.∇ujdx,

According to (49), (51) and (54) we get:

<A(uj),uj>→<y,u> as j→∞.

Hence, from (55), and (39) we obtain:

limj→∞∑i=1N∫Ω(σim(x,uj,∇uj)−σim(x,uj,∇u)).∇(uj−u)dx=0.

By (49) we can conclude that

<y,u>=<A(u),u>∀u∈W∘M1(Ω).