# The Sub-Supersolution Method and Extremal Solutions of Quasilinear Elliptic Equations in Orlicz-Sobolev Spaces

Authors: Ge Dong; Xiaochun Fang
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We prove the existence of extremal solutions of the following quasilinear elliptic problem-∑i=1N∂/∂xiai(x,u(x),Du(x))+g(x,u(x),Du(x))=0 under Dirichlet boundary condition in Orlicz-Sobolev spaces W01LM(Ω) and give the enclosure of solutions. The differential part is driven by a Leray-Lions operator in Orlicz-Sobolev spaces, while the nonlinear term g:Ω×R×RN→R is a Carathéodory function satisfying a growth condition. Our approach relies on the method of linear functional analysis theory and the sub-supersolution method.


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## Body

## 1. Introduction

The aim of this paper is to study some qualitative properties of solutions of the following quasilinear elliptic problem:(1)-∑i=1N∂∂xiaix,ux,Dux+gx,ux,Dux=0inΩu=0on∂Ω,on a bounded domain Ω⊂RN with a Lipschitz boundary ∂Ω in Orlicz-Sobolev spaces. The differential part is driven by a Leray-Lions operator, while the nonlinear term g:Ω×R×RN→R is a Carathéodory function satisfying a growth condition.In [1, Chapter 3], the differential part of (1) is a Leray-Lions operator in Sobolev spaces and the nonlinearity g(x,s,ξ) satisfies the growth condition:(2)gx,s,ξ≤k1x+cξp-1,with the constant c>0 and k1(x)∈L+p′(Ω), for a.e. x∈Ω, all s∈R and all ξ∈RN, where p′ is the conjugate Hölder exponent to p, i.e., 1/p+1/p′=1. In [2], the nonlinearity g(x,s,ξ) satisfies the growth condition:(3)gx,s,ξ≤k1x+cξr,with the constant c≥0 and k1(x)∈Lq(Ω), k1≥0, q>p∗′, 0≤r≤p/p∗′, for a.e. x∈Ω, all s∈R and all ξ∈RN, where p∗ is the Sobolev conjugate of p. Faria [2] pointed that the condition (3) is more general than (2) because p-1<p/p∗′. However, p/p∗′=p-1-p/N<p-1 if 0<p<N. Hence, the growth condition (3) is not more general than (2).When trying to weaken the restriction on the Leray-Lions operator and the growth condition (2), one is led to replace W01,p(Ω) with W01LM(Ω) built from an Orlicz space LM(Ω) instead of Lp(Ω), where M is an N-function. The choice M(t)=|t|p, p>1 leads to [1, Theorem 3.17]. A nonstandard example is M(t)=∫0|u|log(1+|t|q)|t|p-2tdt (see, e.g., [2, 3]).Many papers used the surjectivity result for pseudomonotone operators (see, e.g., [1, Theorem 2.99]) defined on reflexive spaces to prove the existence of the solution (see, e.g., [1, 2, 4, 5]). Our method does not need the reflexivity of the spaces. It is well known that the Orlicz space is reflex if and only if both M and its complementary function M¯ satisfy Δ2-condition. However, there exist many spaces without reflexivity. For example, let M(u)=1+uln⁡1+u-|u|; then M satisfies Δ2-condition, but its complementary function M¯(v)=expv-|v|-1 does not satisfy Δ2-condition; i.e., LM(Ω) is not reflexive.In this paper, we get rid of the restriction of the reflexivity of the spaces and get a weak solution for (1) in Orlicz-Sobolev spaces by using a linear functional analysis method. We also give the enclosure of solutions and prove the existence of extremal solutions.This paper is organized as follows. Section2 contains some preliminaries and some technical lemmas which will be needed. In Section 3, we use the linear functional analysis method to prove the existence of solutions for (1) in separable Orlicz-Sobolev spaces and the sub-supersolutions method to give the enclosure of solutions and the existence of extremal solutions between a subsolution and a supersolution. We also get the compactness and directness of the solutions set.For some results, we also refer to [6–13].


## 2. Preliminaries

For quick reference, we recall some basic results of Orlicz spaces. Good references are Adams [14, Chapter 8], Krasnosel’skil [15], Chen [16], and Gossez [17].


### 2.1. N-Function

LetM:R+→R+ be an N-function; i.e., M is continuous, convex, with M(u)>0 for u>0, M(u)/u→0 as u→0, and M(u)/u→+∞ as u→+∞. Equivalently, M admits the representation M(u)=∫0uϕ(t)dt, where ϕ:R+→R+ is a nondecreasing, right continuous function, with ϕ(0)=0, ϕ(t)>0 for t>0, and ϕ(t)→+∞ as t→+∞.TheN-function M¯ conjugated to M is defined by M¯(v)=∫0vψ(s)ds, where ψ:R+→R+ is given by ψ(s)=supt:ϕt≤s.ϕ, ψ are called the right-hand derivatives of M, M¯, respectively.TheN-function M is said to satisfy the Δ2 condition near infinity (M∈Δ2, for short), if, for some k>1 and u~>0, M(2u)≤kM(u), ∀u≥u~.Moreover, one has the following Young inequality:uv≤M(u)+M¯(v), ∀u,v≥0.For theN-function M one defines the Sobolev conjugate M∗ by M∗-1(t)=∫0tM-1(τ)/τN+1/Ndτ,t≥0.LetP,Q be two N-functions, we say that P grows essentially less rapidly than Q near infinity, denoted as P≪Q, if for every ε>0, P(t)/Q(εt)→0 as t→+∞. This is the case if and only if limt→+∞Q-1(t)/P-1(t)=0.We will extend theseN-functions into even functions on all R.For a measurable functionu on Ω, its modular is defined by ρM(u)=∫ΩM(|u(x)|)dx.


### 2.2. Orlicz Spaces

LetΩ be an open and bounded subset of RN and M be an N-function. The Orlicz class KM(Ω) (resp., the Orlicz space LM(Ω)) is defined as the set of (equivalence classes of) real valued measurable functions u on Ω such that ρM(u)<+∞resp.ρM(u/λ)<+∞forsomeλ>0. LM(Ω) is a Banach space under the (Luxemburg) norm:(4)uM=inf⁡λ>0:ρMuλ≤1,and KM(Ω) is a convex subset of LM(Ω) but not necessarily a linear space. The closure in LM(Ω) of the set of bounded measurable functions with compact support in Ω¯ is denoted by EM(Ω).The equalityEM(Ω)=LM(Ω) holds if and only if M∈Δ2; moreover, LM(Ω) is separable.L M ( Ω ) is reflexive if and only if M∈Δ2 and M¯∈Δ2.Convergences in norm and in modular are equivalent if and only ifM∈Δ2.The dual space ofEM(Ω) can be identified with LM¯(Ω) by means of the pairing ∫Ωu(x)v(x)dx, and the dual norm of LM¯(Ω) is equivalent to ·(M¯).


### 2.3. Orlicz-Sobolev Spaces

We now turn to the Orlicz-Sobolev space:W1LM(Ω) (resp., W1EM(Ω)) is the space of all functions u such that u and its distributional partial derivatives lie in LM(Ω) (resp., EM(Ω)). It is a Banach spaces under the norm(5)uΩ,M=∑α≤1DαuM.Denote DuM=DuM and u1,M=uM+DuM. Clearly, u1,M is equivalent to uΩ,M.ThusW1LM(Ω) and W1EM(Ω) can be identified with subspaces of the product of N+1 copies of LM(Ω). Denoting this product by ΠLM, we will use the weak topologies σ(ΠLM,ΠEM¯) and σ(ΠLM,ΠLM¯).IfM∈Δ2, then W1LM(Ω)=W1EM(Ω). If M∈Δ2 and M¯∈Δ2, then W1LM(Ω)=W1EM(Ω) are reflexive; thus the weak topologies σ(ΠLM,ΠEM¯) and σ(ΠLM,ΠLM¯) are equivalent.Lemma 1 (See [18, Lemma 2.2]).
For allu∈W01LM(Ω), one has(6)∫ΩMuxdiam⁡Ωdx≤∫ΩMDuxdx,where diam⁡Ω is the diameter of Ω.Lemma 2 (See [19, Lemma 1]).
Letmeas⁡Ω be bounded and φ:R+→R+, φ(0)=0, φ(r)→+∞ for r→+∞. Then(7)∫ΩφDuxDuxdx∫ΩDuxdx→+∞if∫ΩDuxdx→+∞.Lemma 3 (See [20, Lemma 2.1]).
Ifu∈W1LM(Ω), then u+, u-∈W1LM(Ω) and (8)Du+=Du,ifu>0,0,ifu≤0,Du-=0,ifu≥0,-Du,ifu<0.Hereu+=maxu,0, u-=-minu,0.


## 2.1. N-Function

LetM:R+→R+ be an N-function; i.e., M is continuous, convex, with M(u)>0 for u>0, M(u)/u→0 as u→0, and M(u)/u→+∞ as u→+∞. Equivalently, M admits the representation M(u)=∫0uϕ(t)dt, where ϕ:R+→R+ is a nondecreasing, right continuous function, with ϕ(0)=0, ϕ(t)>0 for t>0, and ϕ(t)→+∞ as t→+∞.TheN-function M¯ conjugated to M is defined by M¯(v)=∫0vψ(s)ds, where ψ:R+→R+ is given by ψ(s)=supt:ϕt≤s.ϕ, ψ are called the right-hand derivatives of M, M¯, respectively.TheN-function M is said to satisfy the Δ2 condition near infinity (M∈Δ2, for short), if, for some k>1 and u~>0, M(2u)≤kM(u), ∀u≥u~.Moreover, one has the following Young inequality:uv≤M(u)+M¯(v), ∀u,v≥0.For theN-function M one defines the Sobolev conjugate M∗ by M∗-1(t)=∫0tM-1(τ)/τN+1/Ndτ,t≥0.LetP,Q be two N-functions, we say that P grows essentially less rapidly than Q near infinity, denoted as P≪Q, if for every ε>0, P(t)/Q(εt)→0 as t→+∞. This is the case if and only if limt→+∞Q-1(t)/P-1(t)=0.We will extend theseN-functions into even functions on all R.For a measurable functionu on Ω, its modular is defined by ρM(u)=∫ΩM(|u(x)|)dx.


## 2.2. Orlicz Spaces

LetΩ be an open and bounded subset of RN and M be an N-function. The Orlicz class KM(Ω) (resp., the Orlicz space LM(Ω)) is defined as the set of (equivalence classes of) real valued measurable functions u on Ω such that ρM(u)<+∞resp.ρM(u/λ)<+∞forsomeλ>0. LM(Ω) is a Banach space under the (Luxemburg) norm:(4)uM=inf⁡λ>0:ρMuλ≤1,and KM(Ω) is a convex subset of LM(Ω) but not necessarily a linear space. The closure in LM(Ω) of the set of bounded measurable functions with compact support in Ω¯ is denoted by EM(Ω).The equalityEM(Ω)=LM(Ω) holds if and only if M∈Δ2; moreover, LM(Ω) is separable.L M ( Ω ) is reflexive if and only if M∈Δ2 and M¯∈Δ2.Convergences in norm and in modular are equivalent if and only ifM∈Δ2.The dual space ofEM(Ω) can be identified with LM¯(Ω) by means of the pairing ∫Ωu(x)v(x)dx, and the dual norm of LM¯(Ω) is equivalent to ·(M¯).


## 2.3. Orlicz-Sobolev Spaces

We now turn to the Orlicz-Sobolev space:W1LM(Ω) (resp., W1EM(Ω)) is the space of all functions u such that u and its distributional partial derivatives lie in LM(Ω) (resp., EM(Ω)). It is a Banach spaces under the norm(5)uΩ,M=∑α≤1DαuM.Denote DuM=DuM and u1,M=uM+DuM. Clearly, u1,M is equivalent to uΩ,M.ThusW1LM(Ω) and W1EM(Ω) can be identified with subspaces of the product of N+1 copies of LM(Ω). Denoting this product by ΠLM, we will use the weak topologies σ(ΠLM,ΠEM¯) and σ(ΠLM,ΠLM¯).IfM∈Δ2, then W1LM(Ω)=W1EM(Ω). If M∈Δ2 and M¯∈Δ2, then W1LM(Ω)=W1EM(Ω) are reflexive; thus the weak topologies σ(ΠLM,ΠEM¯) and σ(ΠLM,ΠLM¯) are equivalent.Lemma 1 (See [18, Lemma 2.2]).
For allu∈W01LM(Ω), one has(6)∫ΩMuxdiam⁡Ωdx≤∫ΩMDuxdx,where diam⁡Ω is the diameter of Ω.Lemma 2 (See [19, Lemma 1]).
Letmeas⁡Ω be bounded and φ:R+→R+, φ(0)=0, φ(r)→+∞ for r→+∞. Then(7)∫ΩφDuxDuxdx∫ΩDuxdx→+∞if∫ΩDuxdx→+∞.Lemma 3 (See [20, Lemma 2.1]).
Ifu∈W1LM(Ω), then u+, u-∈W1LM(Ω) and (8)Du+=Du,ifu>0,0,ifu≤0,Du-=0,ifu≥0,-Du,ifu<0.Hereu+=maxu,0, u-=-minu,0.


## 3. Main Results

LetΩ be a bounded domain in RN(N≥1) with Lipschitz boundary, M,P be two N-functions, and M¯,P¯ be the complementary functions of M,P, respectively. Assume that M satisfies the Δ2 condition near infinity and P≪M. By Theorem 2.2 and Proposition 2.1 in [21] the embeddings W01LM(Ω)↪LP(Ω) and W01LM(Ω)↪LM(Ω) are compact.LetA be the following quasilinear elliptic differential operator in divergence form:(9)Aux=-∑i=1N∂∂xiaix,ux,Dux,where the coefficients ai:Ω×R×RN→R, i=1,…,N, are assumed to satisfy the following: (H1)
Each functionai(x,s,ξ) is a Carathéodory function. Also there exists a positive constant β and a nonnegative function k0∈EM¯(Ω) such that(10)aix,s,ξ≤βk0x+P¯-1Ms+M¯-1Mξfor a.e.x∈Ω and for all s∈R, ξ∈RN.(H2)
∑i=1N(ai(x,s,ξ)-ai(x,s,ξ′))(ξi-ξi′)>0 for a.e. x∈Ω, all s∈R, and all ξ,ξ′∈RN with ξ≠ξ′.(H3)
∑i=1Nai(x,s,ξ)ξi≥νM(|ξ|)-k(x) for a.e. x∈Ω, all s∈R, and all ξ∈RN, with some constant ν>0 and a function k∈L1(Ω).The differential operatorA can be seen as a mapping from W01LM(Ω) into its dual space (W01LM(Ω))∗ given by(11)Au,v=∑i=1N∫Ωaix,ux,Dux∂vx∂xidxfor  allu,v∈W01LMΩ.Example 4.
(1) The p-Laplacian operatorΔp=div(|Du|p-2Du) is form A with the coefficients ai, i=1,…,N, given by ai(x,s,ξ)=|ξ|p-2ξi (see, e.g., [1, Example 2.110]).
(2) Letp(t) be a given positive and continuous function which increases from 0 to +∞ and ai(x,s,ξ)=p(|ξ|)/|ξ|ξi. Then ai, i=1,…,N, satisfy the conditions (H1)-(H3).Consider the following nonlinear elliptic equation:(12)Au+gx,u,Du=0inΩu=0on∂Ω.Here, g:Ω×R×RN→R is assumed to be a Carathéodory function.LetG denote the Nemytskij operator related to g by(13)Gux=gx,ux,Dux,x∈Ω.Foru, v∈L0(Ω), we use the standard notations: u∧v=minu,v,u∨v=maxu,v, u+≔u∨0,u-≔-u∧0, u≤v⇔u(x)≤v(x) for a.e. x∈Ω. A weak solution of (12) is called a solution for short.By Lemma3, W1LM(Ω) is closed under ∨ and ∧. In fact, since u∨v=v+(u-v)+ and u∧v=v-(u-v)-, u∨v, u∧v∈W1LM(Ω), for any u, v∈W1LM(Ω).The following lemma can be found in [5, Remark 3.1] as the setting of Musielak-Orlicz spaces. However, we give another proof.Lemma 5.
(a)W1LM(Ω) (resp., W01LM(Ω)) is closed under “∨” and “∧”, i.e., if u,v∈W1LM(Ω) (resp., W01LM(Ω)), then u∨v,u∧v∈W1LM(Ω) (resp., W01LM(Ω)).
(b) The mappings∨ and ∧: W1LM(Ω)×W1LM(Ω)→W1LM(Ω) (resp., W01LM(Ω)×W01LM(Ω)→W01LM(Ω)) are continuous, i.e., for any sequences {un},{vn} in W1LM(Ω) (resp., W01LM(Ω)), if un→u,vn→v in W1LM(Ω) (resp., W01LM(Ω)), then un∨vn→u∨v, un∧vn→u∧v in W1LM(Ω) (resp., W01LM(Ω)), as n→∞.Proof.
(a) By Lemma3, W1LM(Ω) (resp., W01LM(Ω)) is closed under ∨ and ∧. In fact, since u∨v=v+(u-v)+ and u∧v=v-(u-v)-, u∨v,u∧v∈W1LM(Ω) (resp., W01LM(Ω)), for any u,v∈W1LM(Ω) (resp., W01LM(Ω)).
(b) Letun→u,vn→v in W1LM(Ω) (resp., W01LM(Ω)), as n→∞. Suppose that there exists ε1>0 such that (14)un∨vn-u∨vM>ε1>0for any n∈N, then ρMun∨vn-u∨v/ε1>1.
Therefore, we have∫ΩM8/ε1unx-uxdx→0, and ∫ΩM(8/ε1|vn(x)-v(x)|)dx→0, as n→∞. By passing to a subsequence if necessary, un→u, vn→v, a.e. in Ω, as n→∞, and there exist f1,f2∈L1(Ω) such that M(8/ε1|un(x)-u(x)|)≤f1(x), and M(8/ε1|vn(x)-v(x)|)≤f2(x), which yields that M(4/ε1|un(x)|)≤1/2f1+1/2M(8/ε1|u(x)|), M(4/ε1|vn(x)|)≤1/2f2+1/2M(8/ε1|v(x)|), for a.e. x∈Ω.
Hence,un∨vn→u∨v a.e. in Ω, as n→∞, and (15)M1ε1un∨vnx-u∨vx≤14M4ε1unx+M4ε1vnx+M4ε1ux+M4ε1vx≤1412f1+12M8ε1ux+12f2+12M8ε1vx+M4ε1ux+M4ε1vx,for a.e. x∈Ω.
By Lebesgue’s theorem, we get∫ΩM(1/ε1|(un∨vn)(x)-(u∨v)(x)|)dx→0, as n→∞; this is a contradiction. Consequently, un∨vn-u∨v(M)→0, as n→∞. Similarly, we can deduce that D(un∨vn)-D(u∨v)(M)→0, un∧vn-u∧v(M)→0 and D(un∧vn)-D(u∧v)(M)→0, as n→∞; that is, the mappings ∨ and ∧ are continuous.A functionu is called a (weak) solution of (12) if u∈W01LM(Ω), G(u)∈LP¯(Ω) and u satisfies the following:(16)Au,v+∫ΩGuxvxdx=0,for  allv∈W01LMΩ.A functionu is called a subsolution (resp. supersolution) of (12) if u∈W01LM(Ω), G(u)∈LP¯(Ω), and (16) holds with “=” replaced with “≤” (resp. “≥”) for every nonnegative functions v in W01LM(Ω).By Young inequality andM∈Δ2, there exist K1>1 and u~1>0, such that M¯(ϕ(u))+M(u)=uϕ(u)≤M(2u)≤K1M(u)+M(2u~1) for all u>0. Hence,(17)M¯ϕu≤K1-1Mu+M2u~1.Theorem 6.
Letu_ and u¯ be a subsolution and a supersolution of problem (12), respectively, such that u_≤u¯. Assume (H1)-(H3) and the following local growth condition for the nonlinearity g:(18)gx,s,ξ≤k1x+cP¯-1Mξfor a.e. x∈Ω, all ξ∈RN, and all s∈[u_(x),u¯(x)], with k1∈EP¯(Ω), k1≥0, c≥0. Then there exists at least one solution u∈W01LM(Ω) of problem (12) with u∈[u_,u¯]≔v∈W01LMΩ:u_≤v≤u¯.Proof.
DenoteW01LM(Ω)=V. For x∈Ω, u∈V, we put(19)Tux=u¯x,ifux>u¯xux,ifu_x≤ux≤u¯xu_x,ifux<u_xforu∈V.Then Tu=u∨u_+u∧u¯-u. By Lemma 5, T:V→V is continuous. It is easy to see that T is bounded.
We define the cutoff functionf:Ω×R→R given by(20)fx,s=ϕs-u¯x,ifs>u¯x0,ifu_x≤s≤u¯x-ϕu_x-s,ifs<u_x,for x∈Ω, s∈R. Then f satisfies the following condition:(21)fx,s≤ϕu_+u¯+s,for x∈Ω and all s∈R.
SinceM is convex and M∈Δ2, there exist K2>1 and u~2>0 such that M(|u|)≤K2/2[M(u-u¯(x))+M(|u¯(x)|)]+M(2u~2) whenever u>u¯(x), and M(|u|)≤K2/2[M(u_(x)-u)+M(|u_(x)|)]+M(2u~2) whenever u<u_(x) for x∈Ω, u∈R. For all u∈V, we have (22)∫Ωfx,uxuxdx=∫u>u¯ϕux-u¯xux-u¯xdx+∫u>u¯ϕux-u¯xu¯xdx+∫u<u_ϕu_x-uxu_x-uxdx-∫u<u_ϕu_x-uxu_xdx≥∫u>u¯Mux-u¯xdx-∫u>u¯M¯ε1ϕux-u¯xdx-∫u>u¯M1ε1u¯xdx+∫u<u_Mu_x-uxdx-∫u<u_M¯ε1ϕu_x-uxdx-∫u<u_M1ε1u_xdx≥1-ε1K1-1∫u>u¯Mux-u¯xdx+1-ε1K1-1∫u<u_Mu_x-uxdx-C1≥1-ε1K1-12K2∫ΩMuxdx-C2=1K2∫ΩMuxdx-C2,where ε1=1/2(K1-1) and the constants C1,C2>0.
DefineΓT:V→V∗,(23)ΓTu,w≔∫Ω∑i=1Naix,Tux,Dux∂wx∂xidx+λ∫Ωfx,uxwxdx+∫ΩGTuxwxdx,∀w∈V, where λ>0 is a parameter to be specified later. Then ΓT is well defined.
SinceM∈Δ2, there exists a sequence {wj}j=1∞⊂D(Ω) such that {wj}j=1∞ dense in V. Let Vm=spanw1,…,wm and consider ΓT|Vm. ∫Ω|Du|dx and DuM are two norms of Vm equivalent to the usual norm of finite dimensional vector spaces.
Similar to the proof of Proposition 3.1 in [22], we can deduce that the mapping u→ΓTVmu:Vm→Vm∗ is continuous.
By (H3), (18), and (22), (24)ΓTu,u≥ν∫ΩMDuxdx-∫Ωkxdx+λK2∫ΩMuxdx-λC2-cε2∫ΩMDTuxdx+c∫ΩP1ε2uxdx-C0u1,M≥ν2∫ΩMDuxdx+λK2-1-cK3∫ΩMuxdx-C3-C0u1,M,for every u∈V, where ε2=ν/2c, K3>0 such that P(1/ε2|u(x)|)≤K3M(|u(x)|) and the constants C3,C0>0. Let λ>K2(1+cK3). Then we can deduce that(25)ΓTu,u≥ν2∫ΩMDuxdx-C3-C0u1,M.By Lemma 1, we get(26)∫ΩMDuxdxu1,M≥11+diam⁡Ω∫ΩMDuxdxDuM≥C21+diam⁡Ω∫ΩϕDux/2Dux/2dx∫ΩDux/2dx,where the constant C>0. By Lemma 2, we immediately have(27)∫ΩMDuxdxu1,M→+∞asu1,M→+∞.Combining (25) and (27), we obtain(28)ΓTu,uu1,M→+∞asu1,M→+∞.By Remark 2.1 in [22], for every m, there is a Galerkin solution um∈Vm such that(29)ΓTum,v=0,∀v∈Vm.By the density of {wm}, we get (30)ΓTum,v=0,∀v∈V.
As the same proof in [22], we can deduce that the sequence {um} is bounded in V and there exists u0∈V and a subsequence {uk} of {um}, such that(31)uk⇀u0weaklyinVforσ∏LM,∏EM¯,(32)uk→u0stronglyinLMΩ,(33)uk→u0stronglyinLPΩ,(34)uk→u0a.e.inΩ,(35)ΓTuk⇀0weaklyinV∗forσ∏LM¯,∏EM,as k→+∞.
From (21), {f(x,uk(x))} is bounded in LM¯(Ω). By Lemma 4.4 of [17],(36)fx,ukx⇀fx,u0xweakly  inLM¯ΩforσLM¯Ω,EMΩ,as k→+∞.
On the other hand, thanks to (32) and (33), we have(37)
∫
Ω
f
x
,
u
k
x
u
k
x
-
u
0
x
d
x
→
0
,
∫
Ω
g
x
,
T
u
k
x
,
D
T
u
k
x
u
k
x
-
u
0
x
d
x
→
0
,as k→+∞. Thus we obtain that(38)∫Ω∑i=1Naix,Tukx,Dukx∂ukx∂xi-∂u0x∂xidx→0,ask→+∞.Similar to the proof of Proposition 3.1 in [22], we can construct a subsequence still denoted by {uk} such that(39)Duk→Du0a.e.  inΩ,ask→+∞.Hence,(40)∑i=1Naix,Tuk,Duk→∑i=1Naix,Tu0,Du0a.e.  inΩ,as k→+∞.
Following the lines of Theorem 1 in [2], we can deduce that u_≤uk≤u¯ for every k∈N. By (34), u_≤u0≤u¯.
DenoteΩ0=x∈Ω:u_x≤u0x≤u¯x and Ωk=x∈Ω:u_x≤ukx≤u¯x. Then meas⁡Ω∖Ωk=meas⁡Ω∖Ω0=0, for every k∈N. It follows from (39) that, passing to a subsequence if necessary, (41)DTuk→DTu0a.e.  inΩ,ask→+∞.Therefore,(42)gx,Tuk,DTuk→gx,Tu0,DTu0a.e.  inΩ,ask→+∞.Since {uk} and {Tuk} are bounded in V, {∑i=1Nai(x,Tuk,Duk)} is bounded in LM¯(Ω) and {g(x,Tuk,DTuk)} is bounded in LP¯(Ω). By Lemma 4.4 of [17], ΓTuk⇀ΓTu0 weakly in V∗ for σ(∏LM¯,∏EM). Thanks to (35), one has (ΓTu0,v)=0, for any v∈V. Therefore, we obtain that u0 is a solution of (12).Under the assumptions of Theorem6, we define(43)S=u∈W01LMΩ:uis  a  solution  of12andu_≤u≤u¯.Theorem 7.
Under the assumptions of Theorem6, the set S is compact in W01LM(Ω).Proof.
Let{un} be a sequence in S. It follows from the coerciveness of ΓT that {un} is bounded in W01LM(Ω). As the same proof of Theorem 6, there exists that u0 is a solution of (12) and u_≤u0≤u¯, i.e., u0∈S.To show that the setS is directed with respect to the usual pointwise order, the following additional assumption on the coefficients ai:Ω×R×RN→R is required.(H4)
Let a nonnegative functionk∈LM¯(Ω) and a continuous function ω:R+→R+ exist such that (44)aix,s,ξ-aix,s′,ξ≤kx+P¯-1Ms+P¯-1Ms′+M¯-1Mξωs-s′holds for a.e.x∈Ω, for all s,s′∈R and for all ξ∈RN, where ω satisfies ∫0+dr/ω(r)=+∞, that is, for every ε>0, ∫0εdr/ω(r)=+∞.Similar to the proof of [1, Theorem 3.20], we can deduce the following result.Theorem 8.
Assume hypotheses (H1)-(H4), and letu1 and u2 be subsolutions of (12) such that the Nemytskii operator(45)G:u1∧u2,u1∨u2→LM¯Ωis well defined. Then u1∨u2 is a subsolution of (12). Analogously, if u1 and u2 are supersolutions of (12) with the same assumption on the Nemytskii operator G, then u1∧u2 is a supersolution.Theorem 9.
Let the assumptions of Theorem6 and (H4) hold. Then the following assertions about S are true. (a)
S is a direct set in both directions; that is, if u1,u2∈S then there exist u,v∈S such that u≥u1∨u2 and v≤u1∧u2.(b)
S possesses extremal elements; i.e., there are u∗,u∗∈S such that u∗≤u≤u∗, for all u∈S.Proof.
(a) Letu1,u2∈S. Then u1 and u2 are both subsolutions and supersolutions of (12). It follows, from Theorem 8, u1∨u2 is a subsolution and u1∧u2 is a supersolution of (12). The claim in (a) is now a straightforward consequence of Theorem 6.
(b) SinceW01LM(Ω) is separable, there exists a countable, dense subset {wn:n∈N} of S. Let u1=w1. By (a), we can select un+1∈S such that un∨wn≤un+1≤u¯. Thus, we get a bounded increasing sequence {un}⊂S. Consequently, limn→∞un(x)=supn∈Nun(x)≔u∗(x), for a.e. x∈Ω, and there exists a subsequence {uk}⊂{un} such that uk⇀u∗ weakly in W01LM(Ω) for σ(∏LM,∏EM¯) as k→+∞. Similar to the proof of Theorem 6, we can deduce that u∗∈S. From the density of {wn:n∈N}, we can get that u∗ is the greatest element of S. The existence of the smallest element of S can be deduced in the same way.Remark 10.
A special case in Theorem6 is that P=M. In this case, choice M(t)=|t|p leads to Theorem 3.17 in [1].Remark 11.
The above results can be extended to the more general situation of Musielak-Orlicz-Sobolev spaces following our method developed in this paper.



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