NONLINEAR PROBLEMS IN ORLICZ SPACES

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NONLINEAR PROBLEMS IN ORLICZ SPACES

A. BENKIRANE AND J. BENNOUNA Received 20 October 2001

We study in the framework of Orlicz Sobolev spacesW₀¹LM(Ω), the existence of entropic solutions to the nonlinear elliptic problems:−diva(x,u,∇u) + divφ(u)

=f inΩ, for the case where the second member of the equation f ∈L¹(Ω), and φ∈(C⁰(R))^N.

1. Introduction

Let Ωbe a bounded open subset ofR^N and let A(u)=−diva(x,u,∇u) be a Leray-Lions operator defined onW₀^1,p(Ω), 1< p <∞.

We consider the nonlinear elliptic problem

−diva(x,u,∇u)=f−divφ(u) inΩ,

u=0 on∂Ω, (1.1)

where

f ∈L¹(Ω), φ∈

C⁰(R)^N. (1.2)

Note that no growth hypothesis is assumed on the function φ, which implies that the term divφ(u) may be meaningless, even as a distribution. The notion of entropy solution, used in [8], allows us to give a meaning to a possible solution of (1.1).

In fact Boccardo proved in [8], forpsuch that 2−1/N < p < N, the existence and regularity of an entropy solutionuof problem (1.1), that is,

u∈W₀¹^,q(Ω), q <p˜=(p−1)N N−1 , T_k(u)∈W₀¹^,p(Ω), ∀k >0,

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Ωa(x,u,∇u)∇T_k[u−ϕ]dx≤

Ωf T_k[u−ϕ]dx+

Ωφ(u)∇T_k[u−ϕ]dx

∀ϕ∈W₀¹^,p(Ω)∩L^∞(Ω), (1.3) where

Tk(s)=s if|s| ≤k Tk(s)=k s

|s| if|s|> k. (1.4) For the caseφ=0 and f is a bounded measure, B´enilan et al. proved in [3] the existence and uniqueness of entropy solutions.

We mention as a parallel development, the work of Lions and Murat [14]

who consider similar problems in the context of the renormalized solutions in- troduced by Diperna and Lions [10] for the study of the Boltzmann equations.

They can prove existence and uniqueness of renormalized solution.

The functional setting in these works is that of the usual Sobolev spaceW¹^,p. Accordingly, the functionais supposed to satisfy polynomial growth conditions with respect touand its derivatives∇u. When trying to generalize the growth condition ona, one is led to replaceW^1,pby a Sobolev spaceW¹L_M built from an Orlicz spaceLM instead ofL^p. Here theN-functionM which definesLM is related to the actual growth of the functiona.

It is our purpose, in this paper, to prove the existence of entropy solution for problem (1.1) in the setting of the Orlicz Sobolev spaceW₀¹LM(Ω). Our result, Theorem 3.5, generalizes [8, Theorem 2.1] and gives in particular a refinement of his result (seeRemark 3.6).

For some existence results for strongly nonlinear elliptic equations in Orlicz spaces [4,5,6].

2. Preliminaries

2.1. LetM:R⁺→R⁺be anN-function, that is,Mis continuous, convex, with M(t)>0 fort >0,M(t)/t→0 ast→0 andM(t)/t→ ∞ast→ ∞.

Equivalently, M admits the representation M(t) = _t

0a(τ)dτ, where a : R⁺→R⁺is nondecreasing, right continuous, witha(0)=0,a(t)>0 fort >0 anda(t)→ ∞ast→ ∞.

In the following, we assume, for convenience, that allN-functions are twice continuously diﬀerentiable, see Benkirane and Gossez [7].

TheN-function ¯M conjugate to M is defined by ¯M(t)=_t

0a(τ)dτ, where¯ a¯:R⁺→R⁺is given by ¯a(t)=sup{s:a(s)≤t}, see [1,13].

TheN-functionMis said to satisfy the∆2-condition (resp., near infinity) if for somekand for everyt≥0,

M(2t)≤kM(t) resp., fort≥somet0

. (2.1)

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Let M and P be twoN-functions. The notationP M means thatP grows essentially less rapidly thanM, that is, for each>0,P(t)/M(t)→0 ast→ ∞. This is the case if and only if limt→∞M⁻¹(t)/P⁻¹(t)=0. We will extend allN- functions into even functions on allR.

2.2. LetΩbe an open subset ofR^N. The Orlicz classK_M(Ω) (resp., the Orlicz spaceL_M(Ω)) is defined as the set of (equivalence classes of) real-valued measurable functionsuonΩsuch that

ΩMu(x)dx <∞ (2.2)

(resp.,_ΩM(u(x)/λ)dx <∞for someλ >0). The spaceLM(Ω) is a Banach space under the norm

uM=inf

λ >0 :

ΩM u(x)

λ

dx≤1

(2.3) and K_M(Ω) is a convex subset ofL_M(Ω). The closure in L_M(Ω) of the set of bounded measurable functions with compact support in ¯Ωis denoted byEM(Ω).

The equalityEM(Ω)=LM(Ω) holds if and only ifMsatisfies the∆2condition, for alltor fortlarge according to whetherΩhas infinity measure or not.

The dual ofEM(Ω) can be identified withL_M^¯(Ω) by means of the pairing

Ωu(x)v(x)dx, and the dual norm onL_M^¯(Ω) is equivalent to · _M^¯. We say that u_nconverges toufor the modular convergence inL_M(Ω) if for someλ >0

ΩM un−u λ

dx−→0 asn−→ ∞. (2.4)

IfMsatisfies the∆2-condition, then the modular convergence coincide with the norm convergence.

2.3. The Orlicz Sobolev spaceW¹LM(Ω) (resp.,W¹EM(Ω)) is the space of all functionsusuch thatuand its distributional derivatives up to order one lie in LM(Ω) (resp.,EM(Ω)). It is a Banach space under the norm

u1,M=

|α|≤1

D^αu_M. (2.5)

Thus,W¹L_M(Ω) andW¹E_M(Ω) can be identified with subspaces of the product ofN+ 1 copies ofL_M(Ω). Denoting this product byL_M, we will use the weak topologiesσ(LM,E_M^¯) andσ(LM,L_M^¯).

The spaceW₀¹E_M(Ω) is defined as the norm closure ofᏰ(Ω) inW¹E_M(Ω) and the spaceW₀¹L_M(Ω) as theσ(L_M,EM¯) closure ofᏰ(Ω) inW¹L_M(Ω).

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We say thatu_nconverges toufor the modular convergence inW¹L_M(Ω) if for someλ >0

ΩMD^αu_n−D^αu λ

dx−→0 ∀|α| ≤1. (2.6) This implies the convergenceσ(L_M,L_M^¯).

2.4. LetW⁻¹L_M^¯(Ω) (resp.,W⁻¹E_M^¯(Ω)) denote the space of distributions onΩ which can be written as sums of derivatives of order≤1 of functions inL_M^¯(Ω) (resp.,EM¯(Ω)). It is a Banach space under the usual quotient norm.

If the open setΩhas the segment property, then the spaceᏰ(Ω) is dense inW₀¹LM(Ω) for the modular convergence and thus for the topologyσ(LM, LM¯). Consequently, the action of a distribution inW⁻¹LM¯(Ω) on an element ofW₀¹LM(Ω) is well defined.

2.5. We recall the following lemmas.

Lemma2.1 (see [5]). LetΩbe an open subset ofR^Nwith finite measure. LetM,P, andQbeN-functions such thatQ P, and letf :Ω×R→R^Nbe a Carath´eodory function such that

f(x,s)≤c(x) +k1P⁻¹Mk2|s|

a.e.x∈Ω,∀s∈R, (2.7) wherek1,k2∈R+,c(x)∈EQ(Ω). LetNf be the Nemytskii operator defined from P(EM(Ω),1/k2)={u∈LM(Ω) :d(u,EM(Ω))<1/k2}to(EQ(Ω))^NbyNf(u)(x)=

f(x,u(x)). ThenN_f is strongly continuous.

Lemma2.2 (see [5]). LetF :R→Rbe uniformly Lipschitzian, with F(0)=0.

LetM be anN-function and letu∈W₀¹LM(Ω)(resp.,W₀¹EM(Ω)). ThenF(u)∈ W₀¹LM(Ω)(resp.,W₀¹EM(Ω)). Moreover, if the setDof discontinuity points ofFis finite, then

∂

∂x_iF(u)=







F(u)∂u

∂xi a.e. inx∈Ω:u(x)∈D, 0 a.e. inx∈Ω:u(x)∈D.

(2.8)

ThenF :W₀¹L_M(Ω)→W₀¹L_M(Ω)is sequentially continuous with respect to the weak^∗topologyσ(L_M,E_M^¯).

Lemma 2.3 (see [11]). Let Ω have the segment property. Then for each v ∈ W₀¹LM(Ω), there exists a sequencevn∈Ᏸ(Ω)such thatvn converges tovfor the modular convergence inW₀¹L_M(Ω). Furthermore, ifv∈W₀¹L_M(Ω)∩L^∞(Ω)then

v_n_L∞(Ω)≤(N+ 1)vL^∞(Ω). (2.9)

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2.6. We introduce the following notation, see [2,15].

Definition 2.4. LetMbe anN-function, and define the following set:

ᏭM=

Q:Qis anN-function such thatQ Q ^≤

M M, 1

0

Q◦H⁻¹ 1

r¹⁻¹^/N

dr <∞whereH(r)=M(r) r

.

(2.10)

Remark 2.5. LetM(t)=t^pandQ(t)=t^q, then the conditionQ∈ᏭM is equivalent to the following conditions:

(i) 2−1/N < p < N

(ii)q <p˜=(p−1)N/(N−1), see (1).

Remark 2.6. We can give some examples ofN-functionsMfor which the setᏭM

is not empty. Here, theN-functionsMare defined only at infinity.

(1) ForM(t)=t²logtandQ(t)=tlogt, we haveH(t)=tlogtandH⁻¹(t)= t(logt)⁻¹at infinity, see [13]. Then theN-functionQbelongs toᏭM.

(2) ForM(t)=t²log²tat infinity andQ(t)=tlog²t, we haveH(t)=tlog²t andH⁻¹(t)=t(logt)⁻² at infinity, see [13]. Then theN-functionQbelongs to ᏭM.

3. Definition and existence of entropy solutions

LetΩbe a bounded open subset ofR^N with the segment property. LetM,Pbe twoN-functions such thatP M.

Let A :D(A) ⊂ W₀¹LM(Ω) →W⁻¹L_M^¯(Ω) be a mapping (not defined ev- erywhere) given by A(u)=−diva(x,u,∇u) where a:Ω×R×R^N → R^N is a Carath´eodory function satisfying for a.e.x∈Ωand allt∈R,ξ,ξ¯withξ=ξ¯,

a(x,t,ξ)≤d(x) +k1P¯⁻¹Mk2|t|

+k3M¯⁻¹Mk4|ξ|

, (3.1)

a(x,t,ξ)−ax,t,ξ¯ξ−ξ¯>0, (3.2) a(x,t,ξ)ξ≥αM

|ξ| λ

, (3.3)

whered(x)∈EM¯(Ω),d≥0,α,λ∈R^∗+,k1,k2,k3,k4∈R+. Consider the nonlinear elliptic problem (1.1) where

f ∈L¹(Ω) (3.4)

andφ=(φ1,...,φN) satisfies

φ∈

C⁰(R)^N. (3.5)

As in [8], we define the following notion of an entropy solution, which gives a meaning to a possible solution of (1.1).

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Definition 3.1. Assume that (3.1), (3.2), (3.3), (3.4), and (3.5) hold true and ᏭM=∅. A functionuis an entropy solution of problem (1.1) if

u∈W0¹LQ(Ω) ∀Q∈ᏭM, T_k(u)∈W₀¹L_M(Ω) _∀k >0,

Ωa(x,u,∇u)∇Tk[u−ϕ]dx≤

Ωf Tk[u−ϕ]dx+

Ωφ(u)∇Tk[u−ϕ]dx

∀ϕ∈W₀¹LM(Ω)∩L^∞(Ω).

(3.6)

We cannot use the solutionuas a test function in (1.1), becauseudoes not belong toW₀¹L_M(Ω). An important role is played byT_k(u) and the test functions T_k[u−ϕ], ϕ∈W₀¹L_M(Ω)∩L^∞(Ω) (3.7) because both belong toW₀¹LM(Ω).

InTheorem 3.5, we prove the existence of solution of problem (1.1), in the framework of entropy solutions.

Lemma3.2. LetΩbe a bounded open subset ofR^Nwith the segment property. If u∈(W₀¹LM(Ω))^Nthen_Ωdivudx=0.

Proof ofLemma 3.2. It is suﬃcient to use an approximation ofu.

We recall the following lemma (see [15, Lemma 2]).

Lemma3.3. LetMbe anN-function,u∈W¹LM(Ω)such that_ΩM(|∇u|)dx <∞, then

−µ(t)≥NC¹_N^/Nµ¹⁻¹^/N(t)

×C

−1 NC¹_N^/Nµ¹⁻¹^/N(t)

d dt

{|u|>t}M|∇u| dx

∀t >0, (3.8) whereCis the function defined as

C(t)= 1

supr≥0,H(r)≤t, H(r)=M(r)

r . (3.9)

The functionC_Nis the measure of the unit ball ofR^N, andµ(t)=meas{|u|> t}. Lemma3.4. Let(X,τ,µ)be a measurable set such thatµ(X)<∞. Letγbe a mea- surable functionγ:X→[0,∞)such that

µx∈X:γ(x)=0=0, (3.10)

then for each>0, there existsδ >0such that_Aγ(x)dx < δimplies

µ(A)≤. (3.11)

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Theorem3.5. Under assumptions (3.1), (3.2), (3.3), (3.4), and (3.5), withᏭM=

∅, there exists an entropy solutionuof problem (1.1) (in the sense ofDefinition 3.1).

Remark 3.6. In the caseM(t)=t^p,Theorem 3.5gives a refinement of the regularity result (1) (i.e.,u∈W₀¹^,q(Ω), q <p˜=((p−1)N/N−1)). In fact, byTheorem 3.5, we have u∈ W₀¹LQ(Ω) for eachQ∈ ᏭM (for example for Q(t) =t^p^˜/log^α(e+ t),α >1).

Proof ofTheorem 3.5

Step 1. Define, for eachn >0, the approximations

φ_n(s)=φT_n(s), f_n(s)=T_nf(s). (3.12) Consider the nonlinear elliptic problem

u_n∈W₀¹L_M(Ω), −divax,u_n,∇u_n=f_n−divφ_nu_n inΩ. (3.13) From Gossez and Mustonen [12, Proposition 1, Remark 2], problem (3.13) has at least one solution.

Step 2. We will prove that (u_n) is bounded inW₀¹L_Q(Ω) for eachQ∈ᏭM. Letϕ be the truncation defined, for eacht,h >0, by

ϕ(ξ)=











0 if 0≤ξ≤t, 1

h(ξ−t) ift < ξ < t+h, 1 ifξ≥t+h,

−ϕ(−ξ) ifξ <0.

(3.14)

Using the test functionv=ϕ(un) in (3.13) (v∈W₀¹LM(Ω) byLemma 2.2), we have

Ωax,u_n,∇u_nϕu_n∇u_ndx=

Ωf_nϕu_ndx+

Ωφ_nu_n∇ϕu_ndx. (3.15) We claim now that

Ωφ_nu_n∇ϕu_ndx=0. (3.16) Indeed,

∇ϕun

=ϕun

∇un, (3.17)

where

ϕ(ξ)=



 1

h ift <|ξ|< t+h,

0 otherwise, (3.18)

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defineθ(s)=φ_n(s)(1/h)χ{t<|s|<t+h}, and ˜θ(s)=_s

0θ(τ)dτ, we have byLemma 2.2, θ(u˜ _n)∈(W₀¹L_M(Ω))^N, which implies

Ωφ_nu_n∇ϕu_ndx=

Ωφ_nu_n1

hχ{t<|un|<t+h}∇u_ndx=

Ωθu_n∇u_ndx

=

Ωdivθ˜u_ndx=0 (seeLemma 3.2).

(3.19) This proves (3.16). By (3.3) and (3.15), we have (where we can suppose without loss of generality thatλ=1, since one can takeu_n=un/λ)

α h

t<|un|<t+h

M∇undx≤ f1,Ω. (3.20) Leth→0, then

−d dt

{|un|>t}M∇undx≤C withC=f1,Ω

α . (3.21)

We prove the following inequality, which allows us to obtain the boundedness of (un) inW₀¹LQ(Ω),

− d dt

|un|>tQ∇u_ndx

≤ −µ_n(t)Q◦H⁻¹ − 1 NC¹_N^/Nµ_n(t)^1−1/N

d dt

{|un|>t}M∇undx

.

(3.22)

Indeed, let C(s) =1/H⁻¹(s), where H(r) =M(r)/r andH⁻¹(s)=sup{r ≥ 0, H(r)≤s}. Then

C(s)= s

M◦H⁻¹(s). (3.23)

ByLemma 3.3we have, withµn(t)=meas{|un|> t},

−µ_n(t)≥NC_N¹^/Nµn(t)¹⁻¹^/N

×C − 1

NC_N¹^/Nµn(t)¹⁻¹^/N d dt

|un|>tM∇undx

, (3.24)

then

−µ_n(t)·M◦H⁻¹ − 1 NC¹_N^/Nµ_n(t)¹⁻¹^/N

d dt

|un|>tM∇u_ndx

≥NC_N¹^/Nµn(t)¹⁻¹^/N − 1 NC^1/N_N µ_n(t)¹⁻¹^/N

d dt

|un|>tM∇undx

, (3.25)

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and also 1 µ_n(t)

d dt

{|un|>t}M∇u_ndx

≤M◦H⁻¹ − 1 NC_N¹^/Nµ_n(t)¹⁻¹^/N

d dt

{|un|>t}M∇u_ndx

(3.26)

which gives M⁻¹

1 µ_n(t)

d dt

{|un|>t}M∇undx

≤H⁻¹

− 1

NC¹_N^/Nµ_n(t)^1−1/N d dt

{|un|>t}M∇undx

.

(3.27)

LetQ ∈ᏭM and let D(s)=M(Q⁻¹(s)),D is then convex, and the Jensen’s inequality gives

D ^{t<|uⁿ^|<t⁺^h}Q∇undx

−µ_n(t+h) +µ_n(t)

≤

{t<|un|<t+h}M∇undx

−µ_n(t+h) +µ_n(t) , (3.28) then

Q⁻¹ 1 µ_n(t)

d dt

{|un|>t}Q∇u_ndx

≤M⁻¹ 1 µn(t)

d dt

{|un|>t}M∇u_ndx

≤H⁻¹ − 1 NC_N¹^/Nµn(t)¹⁻¹^/N

d dt

{|un|>t}M∇undx

(3.29)

which gives (3.22). By (3.21) and (3.22) and since the function t−→

{|un|>t}Q∇u_ndx (3.30) is absolutely continuous (see [15]), we have

ΩQ∇u_ndx= _∞

0 −d

dt

{|un|>t}Q∇u_n

dt

≤ _∞

0 −µ_n(t)Q◦H⁻¹ C NC_N¹^/Nµn(t)¹⁻¹^/N

dt

≤ 1 C

_C_·meas(Ω)

0

Q◦H⁻¹ 1 r¹⁻¹^/N

dr <∞

(3.31)

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which implies that (_∇u_n) is bounded in L_Q(Ω) for eachQ∈ ᏭM. Then u_nis bounded inW₀¹LQ(Ω) for eachQ∈ᏭM. Passing to a subsequence if necessary, we can assume that

unu weakly inW0¹LQ(Ω) forσLQ,E_Q^¯, a.e. inΩ. (3.32) Step 3. We prove thatTk(un)Tk(u) weakly inW₀¹LM(Ω) for allk >0. Using the test functionT_k(u_n) in (3.13), we obtain

Ωax,un,∇un

∇Tk un

dx=

ΩfnTk un

dx+

Ωφn un

∇Tk un

dx. (3.33) We claim that

Ωφ_nu_n∇T_ku_ndx=0. (3.34) Indeed,∇Tk(un)=∇unχ{|un|≤k}, defineθ(t)=φn(t)χ{|t|≤k}, and ˜θ(t)=_t

0θ(τ)dτ, we have byLemma 2.2, ˜θ(u_n)_∈(W₀¹L_M(Ω))^N,

Ωφ_nu_n∇T_ku_ndx=

Ωφ_nu_nχ{|un|≤k}∇u_ndx

=

Ωθun

∇undx

=

Ωdivθ˜u_ndx=0 (byLemma 3.2)

(3.35)

which proves the claim.

On the other hand, (3.33) can be written as

Ωax,un,∇un

∇Tk un

dx=

Ωax,un,∇Tk un

∇Tk un

dx

=

Ωf_nT_ku_ndx,

(3.36)

which implies, with (3.3), that∇Tk(un) is bounded in (LM(Ω))^N, andTk(un) is bounded in (W₀¹L_M(Ω))^N. Sinceu_n→ua.e. inΩthenT_k(u_n)→T_k(u) a.e. inΩ.

Then

T_ku_nT_k(u) weakly inW₀¹L_M(Ω) forσL_M,EM¯

. (3.37) Step 4. We will prove that∇un→ ∇ua.e. inΩ. Letλ >0,>0. ForB >1,k >0, we consider as in [9] forn,m∈N,

E1=∇un> B∪∇um> B∪un> B∪um> B, E2=u_n−u_m> k,

E3=un−um≤k,un≤B,um≤B,∇un≤B, ∇um≤B,∇un−∇um≥λ,

(3.38)

we have_{|∇u_n−∇u_m| ≥λ} ⊂E1∪E2∪E3.

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Since (u_n) and (_∇u_n) are bounded in L¹(Ω) (since u_n is bounded in W₀¹LQ(Ω)), we have

2BµE1

<

E1

∇u_n+u_ndx <

Ω

∇u_n+u_ndx < C. (3.39) Then measE1 ≤forBsuﬃciently large enough, independently ofn,m. Thus we fixBin order to have

measE1≤. (3.40)

Now we claim that measE3≤fornandmlarge. LetC1be such thatu_n1≤ C1 and ∇un1 ≤C1. As in [9], the assumption (3.2) gives the existence of a measurable functionγ(x) such that

measx∈Ω:γ(x)=0=0,

a(x,t,ξ)−ax,t,ξ¯ξ−ξ¯≥γ(x)>0, (3.41) for allt∈R,ξ,ξ¯∈R^N,|t|,|ξ|,|ξ¯| ≤B,|ξ−ξ¯| ≥λa.e. inΩ. We have

E3

γ(x)dx≤

E3

ax,u_n,∇u_n−ax,u_n,∇u_m∇u_n−∇u_mdx

≤

E3

ax,um,∇um

−ax,un,∇um

∇un−∇um dx +

E3

ax,un,∇un

−ax,um,∇um

∇un−∇um dx.

(3.42)

Using the test functionT_k(u_n−u_m) in (3.13) and integrating onE3, we obtain

E3

ax,u_n,∇u_n−ax,u_m,∇u_m∇T_ku_n−u_mdx

=

E3

f_n−f_mT_ku_n−u_mdx

+

E3

φn un

−φm um

∇Tk

un−um dx,

(3.43)

with

E3

φn un

−φm um

∇Tk

un−um dx

≤2B

E3

φn un

−φm umdx

≤2B

E³

φTn un

−φun+φun

−φum +φum

−φTm

umdx.

(3.44)

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Letn0≥B, then forn,m ≥n0 we haveT_n(u_n)₌u_n andT_m(u_m)₌u_m on E3, which implies that the first and the third integral of the last inequality vanish.

The second integral of (3.42) is bounded forn,m≥n0by 2kf1,Ω+ 2B

E3

φu_n−φu_mdx. (3.45) For a.e.x∈Ωand1>0 there existη(x)≥0 (meas{x∈Ω:η(x)=0}=0) such that|s−s| ≤η(x),|s|,|s|,|ξ| ≤Bimplies

a(x,s,ξ)−ax,s,ξ≤1. (3.46) We use now the continuity ofφ, to obtain for a.e.x∈ Ωand 2 >0, η1(x)≥ 0 (meas{x ∈ Ω : η1(x) = 0} = 0) such that |s−s| ≤ η1(x), |s|,|s| ≤ B implies

φ(s)−φs≤2. (3.47)

Then

E3

γ(x)dx≤

E3∩{x∈Ω:η(x)<k}

ax,u_m,∇u_m−ax,u_n,∇u_m

×

∇un−∇um dx +

E3∩{x∈Ω:η(x)≥k}

ax,u_m,∇u_m−ax,u_n,∇u_m

×

∇u_n−∇u_mdx + 2kf1,Ω+ 2B

E3∩{x∈Ω:η1(x)<k}

φun

−φumdx +

E3∩{x∈Ω:η1(x)≥k}

φu_n−φu_mdx.

(3.48)

By using for the first integral the definition ofE3 and condition (3.1), for the second integral the definition ofE3and (3.46), for the fourth integral the definition ofE3and|φ(u_n)| ≤C(B) (since|u_n| ≤Bandφcontinuous), and for the last integral the definition ofE3and (3.47), we obtain

E3

γ(x)dx≤C(B)

E3∩{x∈Ω:η(x)<k}

1 +d(x)dx+ 2C1(B)1

+ 2kf1,Ω+ 2C(B) measx∈Ω:η1(x)< k+C22.

(3.49)

We have meas{x∈Ω:η(x)< k} →0 whenk→0, and meas{x∈Ω:η1(x)<

k} →0 whenk→0. Let>0 and letδbe the real, inLemma 3.4, corresponding to, we choose1,2such that

2C1(B)1≤δ

5, C22≤δ

5, (3.50)

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andksuch that C(B)

E3∩{x∈Ω:η(x)<k}

1 +d(x)dx < δ

5, 2kf1,Ω≤δ 5, 2C(B) measx∈Ω:η1(x)< k<δ

5.

(3.51)

Then_E

3γ(x)dx < δandLemma 3.4implies that

measE3< ∀n,m≥n0. (3.52) This completes the proof of the claim.

Let the lastkbe fixed,u_na Cauchy sequence in measure, we choosen1such that

measE2≤ ∀n,m≥n1. (3.53)

Then

measx∈Ω:_∇u_n−∇u_m≥λ≤ ∀n,m≥maxn1,n0

(3.54) and∇un→ ∇uin measure, consequently

∇u_n−→ ∇u a.e. inΩ. (3.55)

Step 5. Letϕ∈W₀¹LM(Ω)∩L^∞(Ω). From Lemma 2.3, there exists a sequence (ϕ_j) _∈ Ᏸ(Ω) such that ϕ_j converges to ϕ for the modular convergence in W₀¹LM(Ω) with

ϕ_j_L∞(Ω)≤(N+ 1)ϕL^∞(Ω). (3.56) UsingT_k[u_n−ϕ_j] as a test function in (3.13) we obtain

Ωax,un,∇un

∇Tk un−ϕj

dx

=

Ωf_nT_ku_n−ϕ_jdx+

Ωφ_nu_n∇T_ku_n−ϕ_jdx

(3.57)

which gives, ifn→ ∞, lim inf

n→∞

Ωax,un,∇un

∇Tk un−ϕj

dx

≥lim inf

n→∞

Ω

ax,u_n,∇u_n−ax,u_n,∇ϕ_j∇T_ku_n−ϕ_jdx

+ lim

n→∞

Ωax,Tk+ϕjL∞(Ω)

un ,∇ϕj

∇Tk un−ϕj

dx

≥

Ω

ax,u,∇u−ax,u,∇ϕ_j∇T_ku−ϕ_jdx

+

Ωax,u,∇ϕj

∇Tk u−ϕj

dx,

(3.58)

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where we have used Fatou lemma for the first integral, and for the second the convergences∇Tk[un−ϕj]→ ∇Tk[u−ϕj] by (3.37) in (LM(Ω))^N forσ(LM, EM¯) and a(x,T_k+ϕjL∞(Ω)(u_n),∇ϕ_j) → a(x,T_k+ϕjL∞(Ω)(u),∇ϕ_j) strongly in (E_M^¯(Ω))^Nby (3.1), which implies that

lim inf

n→∞

Ωax,un,∇un

∇Tk un−ϕj

dx≥

Ωax,u,∇u∇Tk u−ϕj

dx. (3.59) Forn≥k+ (N+ 1)ϕL^∞(Ω),

Ωφn un

∇Tk un−ϕj

dx=

ΩφTk+(N+1)ϕL∞(Ω)

un

∇Tk un−ϕj

dx

n→∞−→

ΩφT_k+(N+1)ϕL∞(Ω)(u)∇T_ku−ϕ_jdx, (3.60) we have used the convergences∇T_k[u_n−ϕ_j]∇T_k[u−ϕ_j] by (3.37) in (L_M(Ω))^N andφ(T_k+(N+1)ϕL∞(Ω)(u_n))_→φ(T_k+(N+1)ϕL∞(Ω)(u)) strongly in (EM¯(Ω))^Nsince φis continuous. On the other hand, since fn→f strongly inL¹(Ω) andTk[un− ϕj]Tk[u−ϕj] weakly^∗inL^∞(Ω), we have

Ωf_nT_ku_n−ϕ_jdx−→

Ωf T_ku−ϕ_jdx. (3.61) Then

Ωa(x,u,∇u)∇T_ku−ϕ_jdx≥

ΩφT_k+(N+1)ϕL∞(Ω)(u)∇T_ku−ϕ_jdx +

Ωf Tk u−ϕj

dx.

(3.62)

Now, ifj→ ∞in (3.62), we get lim inf

j→∞

Ωa(x,u,∇u)∇Tk u−ϕj

dx

≥lim inf

j→∞

Ω

a(x,u,∇u)−ax,u,∇ϕ_j∇T_ku−ϕ_jdx

+ lim

j→∞

Ωax,u,∇ϕ_j∇T_ku−ϕ_jdx

≥

Ω

a(x,u,∇u)−a(x,u,∇ϕ)∇T_k[u−ϕ]dx

+

Ωa(x,u,∇ϕ)∇Tk[u−ϕ]dx,

(3.63)

where we have used Fatou lemma for the first integral, and for the second the convergences∇T_k[u−ϕ_j]∇T_k[u−ϕ] in (L_M(Ω))^N for the modular convergence anda(x,u,∇ϕ_j)_→a(x,u,∇ϕ) in (LM¯(Ω))^Nfor the modular convergence,

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which implies that lim inf

j→∞

Ωa(x,u,∇u)∇T_ku−ϕj dx≥

Ωa(x,u,∇u)∇T_k[u−ϕ]dx. (3.64) On the other hand, since∇T_k[u−ϕ_j]→ ∇T_k[u−ϕ] in (L_M(Ω))^Nfor the modular convergence, then weakly forσ(L_M,L_M^¯) andφ(T_k+(N+1)ϕL∞(Ω)(u))∈ (L_M^¯(Ω))^Nwe have

ΩφTk+(N+1)ϕL∞(Ω)(u)∇Tk u−ϕj

dx

j→∞−→

ΩφT_k+(N+1)ϕL∞(Ω)(u)∇T_k[u−ϕ]dx

=

Ωφ(u)∇T_k[u−ϕ]dx.

(3.65)

Since f ∈L¹(Ω) andT_k[u−ϕ_j]T_k[u−ϕ] weakly^∗inL^∞(Ω), we have

Ωf T_ku−ϕ_jdx−→

Ωf T_k[u−ϕ]dx. (3.66) Then

Ωa(x,u,∇u)∇T_k[u−ϕ]dx≥

Ωφ(u)∇T_k[u−ϕ]dx+

Ωf T_k[u−ϕ]dx (3.67)

anduis an entropy solution of problem (1.1).

Theorem3.7. Suppose, inTheorem 3.5, that theN-functionMsatisfies, further- more, the∆2-condition and f ≥0, then the entropy solutionuof problem (1.1) satisfiesu≥0.

Proof ofTheorem 3.7. Usingϕ=T_l(u⁺) as test function in the definition of entropy solution, we obtain

Ωa(x,u,∇u)∇T_ku−T_lu⁺dx

≤

Ωf Tk u−Tl

u⁺dx+

Ωφ(u)∇Tk u−Tl

u⁺dx.

(3.68)

We have

Ωf Tk u−Tl

u⁺dx≤

{u≥l}f Tk

u−Tl(u)dx. (3.69)

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Indeed,

Ωf T_ku−T_lu⁺dx=

u≥lf T_ku−T_lu⁺dx +

0<u<l

f Tk u−Tl

u⁺dx +

u≤0f Tk u−Tl

u⁺dx.

(3.70)

If 0< u < lthenu−T_l(u⁺)=0 and0<u<lf T_k[u−T_l(u⁺)]dx=0. Ifu≤0 then u−Tl(u⁺)=uand_u≤0f Tk[u−Tl(u⁺)]dx ≤0 since f is positive. Ifu≥lthen u⁺=uand_u≥lf T_k[u−T_l(u⁺)]dx≤

u≥lf T_k[u−T_l(u)]dx.

On the other hand, we claim that

Ωφ(u)∇T_ku−T_lu⁺dx=0. (3.71) Indeed, if 0< u < l, thenu−T_l(u⁺)=0, ₀_<u<lφ(u)∇T_k[u−T_l(u⁺)]dx =0. If u≤0, thenu−T_l(u⁺)=u,

u≤0φ(u)∇Tk u−Tl

u⁺dx=

−k≤u≤0φ(u)∇udx

=

Ωφ(u)∇uχ{−k≤u≤0}dx.

(3.72)

We verify that the third integral of the last inequality vanishes. For this, de- fineθ(t)=φ(t)χ{−k≤t≤0}, and ˜θ(t)=_t

0θ(τ)dτ we have, byLemma 2.2, ˜θ(u)∈ (W₀¹L_M(Ω))^Nwhich implies

Ωφ(u)∇uχ{−k≤u≤0}dx=

Ωθ(u)∇udx

=

Ωdivθ(u)˜ dx=0 (byLemma 3.2).

(3.73)

Ifu≥lthenu⁺=uand

{u≥l}φ(u)∇Tk u−Tl

u⁺dx=

l≤u≤l+k

φ(u)∇udx

=

Ωφ(u)∇uχ{l≤u≤l+k}dx.

(3.74)

Similarly, we verify that

Ωφ(u)∇uχ{l≤u≤l+k}dx=0. (3.75)