# Existence of Solution for Nonlinear Elliptic Equations with Unbounded Coefficients andL1 Data

Authors: Hicham Redwane
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

An existence result of a renormalized solution for a class of nonlinear elliptic equations is established. The diffusion functionsa(x,u,∇u) may not be in (Lloc1(Ω))N for a finite value of the unknown and the data belong to L1(Ω).


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## Body

## 1. Introduction

In this paper we investigate the problem of existence of a renormalized solutions for elliptic equations of the type(1.1)-div(a(x,u,∇u))=finΩ,u=0on∂Ω,
where Ω is an open bounded subset of ℝN,N≥1, with the data f in L1(Ω). The operator -div(a(x,u,∇u)) is a Leray-Lions operator defined on the weighted sobolev spaces W01,p(Ω,w), but which is not restricted by any growth condition with respect to u (see assumptions (2.2), (2.4), and (2.5) of Section 3). The function a(x,s,ξ) is controlled by a real function b:]-∞,m[→ℝ which blows up for a finite value m>0 (see (2.2), (2.3)).There are mainly two types of difficulties that are studying Problem (1.1). One consists to give a sense to the flux a(x,u,∇u) on the set {x∈Ω;u(x)=m}. The second one is that the data f only belong to L1, so that proving existence of a weak solution (i.e., in the distribution meaning) seems to  be an arduous task. To overcome this difficulty we use in this paper the framework of renormalized solutions. This notion was introduced by DiPerna and Lions [1] for the study of Boltzmann equation (see also Lions [2] for a few applications to fluid mechanics models). This notion was then adapted to elliptic vesion of (1.1) in Boccardo et al. [3], and Murat [4, 5] (see also [6, 7] for nonlinear parabolic problems). At the same time the equivalent notion of entropy solutions has been developed independently by Bénilan et al. [8] for the study of nonlinear elliptic problems.In the case wherea(x,u,∇u) is replaced by (d(u)+A(u))∇u (problems with diffusion matrices that have at least one diagonal coefficient that blows up for a finite value of the unknown) and f∈L2(Ω), existence and uniqueness has been established in Blanchard and Redwane [9, 10].As far as we have the stationary and evolution equations case (1.1), the existence and a partial uniqueness of renormalized solutions have been proved in Blanchard et al. [11] in the case where a(x,u,∇u) is replaced by A(x,u)∇u (where A(x,s) is a Carathéodory symmetric matrices, such that A(x,s) blows up as s→m- uniformly with respect to x). It has also been applied to the study of linear and nonlinear elliptic and parabolic equations when the diffusion coefficient has a singularity for a finite value of the unknown (see García Vázquez and Ortegón Gallego [12, 13] and Orsina [14]).The paper is organized as follows. In Section2 we will precise some basic properties of weighted Sobolev spaces. Section 3 is devoted to specify the assumptions on a(x,s,ξ),b(s), and f needed in the present study and gives the definition of a renormalized solution of (1.1). In Section 4 (Theorem 4.1) we establish the existence of such a solution.


## 2. Preliminaries

Throughout the paper, we assume that the following assumptions hold true.Ω is a bounded open subset on ℝN,N≥1. Let us suppose that 1<p<∞ is a real number, and ω(x)={ωi(x)}{0≤i≤N} is a vector of weight functions. Furthermore we suppose that every component ωi(x) is a measurable function which is strictly positive and satisfies(2.1)ωi∈Lloc1(Ω),ωi-1/(p-1)∈Lloc1(Ω).
We define the weighted Lebesgue space Lp(Ω,ω0) with weight ω0, as the space of all real-valued measurable functions u for which(2.2)∥u∥p,ω0=(∫Ω|u(x)|pω0(x)dx)1/p<+∞.
In order to define the weighted Sobolev space of W1,p(Ω,ω), as the space of all real-valued functions u∈Lp(Ω,ω0) such that the derivaties in the sense of distributions satisfy ∂u/∂xi∈Lp(Ω,ωi) for all i=1,…,N. This set of functions forms a Banach space under the norm
(2.3)∥u∥1,p,ω=(∫Ω|u(x)|pω0(x)dx+∑i=1N∫Ω|∂u∂xi|pωi(x)dx)1/p.
To deal with the Dirichlet problem, we use the space X=W01,p(Ω,ω) defined as the closure of C0∞(Ω) with respect to the norm ∥·∥1,p,ω. Note that, C0∞(Ω) is dense in W01,p(Ω,ω) and (W01,p(Ω,ω),∥·∥1,p,ω) is a reflexive Banach space. Note that the expression(2.4)∥u∥X=(∑i=1N∫Ω|∂u∂xi|pωi(x)dx)1/p
is a norm defined on X and is equivalent to the norm (2.3). Moreover (X,∥·∥X) is a reflexive Banach space, and there exist a weight function σ on Ω and a parameter 1<q<∞ such that the Hardy inequality(2.5)(∫Ω|u|qσ(x)dx)1/q≤C(∑i=1N∫Ω|∂u∂xi|pωi(x)dx)1/p
holds for every u∈X with a constant C>0 independent of u. Moreover, the imbedding X↪Lq(Ω,σ) is compact.We recall that the dual of the weighted Sobolev spacesW01,p(Ω,ω) is equivalent to W-1,p′(Ω,ω*), where ω*={ωi*=ωi1-p′;i=1,…,N} and p′=p/(p-1) is the conjugate of p. For more details we refer the reader to [15] (see also [16]).


## 3. Assumptions on the Data and Definition of a Renormalized Solution

Throughout the paper, we assume that the following assumptions hold true.Ω is a bounded open set on ℝN,N≥1. Let 1<p<∞, and let ω(x)={ωi(x)}{0≤i≤N} be a vector of weight functions.Let now-div(a(x,u,∇u)) be a Leray-Lions operator defined on W01,p(Ω,ω) into W-1,p′(Ω,ω*) and where
(3.1)a:Ω×ℝ×ℝN→ℝNisaCarathéodoryfunction,suchthat
there exists a positive function b∈C0((-∞,m)) which satisfies(3.2)limr→m-b(r)=+∞;∫0mb(s)ds<+∞,b(r)≥α>0∀r∈]-∞,m[,a(x,s,ξ)·ξ≥b(s)p-1∑i=1Nωi(x)|ξi|p,a(x,s,0)=0,
for almost every x∈Ω, for every s∈ℝ and ξ∈ℝN.For anyi=1,…,N,
(3.3)|ai(x,s,ξ)|≤ωi(x)1/p[L(x)+σ(x)1/p′|s|q/p′+b(s)p-1∑j=1Nωj1/p′(x)|ξj|p-1],
for almost every x∈Ω, for every s and ξ, and where L(x) is a positive function in Lp′(Ω)(3.4)[a(x,s,ξ)-a(x,s,ξ′)][ξ-ξ′]≥0,
for any ξ,ξ′∈ℝN, for any s∈ℝ and for almost every x∈Ω(3.5)fisanelementofL1(Ω).Remark 3.1.
As already mentioned in the introduction, problem (1.1) does not admit a weak solution under assumptions (3.1)–(3.5) since the growth of a(x,u,∇u) is not controlled with respect to u, the field a(x,u,∇u) is not, in general, defined as a distribution because the difficulty is defining the field a(x,u,∇u) on the subset {x∈Ω;u(x)=m} of Ω, (since on this set, b(u)=+∞).
The following notations will be used throughout the paper. For anyK≥0, the truncation at height K is defined by TK(r)=max(-K,min(r,K)), for any positive numbers l and K, the functions TlK are defined by
(3.6)TlK(r)={-K,ifr≤-K,r,if-K≤r≤l,l,ifr≥l.
We define for n≥1 fixed
(3.7)θn(r)=T1(r-Tn(r))={0,if|r|≤n,r-nsg(s),ifn≤|r|≤n+1,sg(s),if|r|≥n+1,
and Sn(r)=1-|θn(r)|, for all r∈ℝ.
The definition of a renormalized solution for Problem (1.1) can be stated as follows.Definition 3.2.
A measurable functionu defined on Ω is a renormalized solution of Problem (1.1) if
(3.8)TK(u)∈W01,p(Ω,ω)∀K≥0,(3.9)u(x)≤mforalmosteveryx∈Ω,(3.10)a(x,TmK(u),∇TmK(u))χ{u<m}∈∏i=1NLp′(Ω,wi1-p′),(3.11)∫{-n-1≤u(x)≤-n}a(x,u,∇u)∇udx→0asn→+∞,(3.12)1δ∫{m-2δ≤u(x)≤m-δ}a(x,u,∇u)∇udx→∫{u=m}fdxasδ→0,
and if, for every function S in W1,∞(ℝ) such that supp(S) is compact and S(m)=0,u satisfies
(3.13)∫Ωa(x,u,∇u)∇(S(u)φ)dx=∫ΩfS(u)φdx,∀φ∈W01,p(Ω,ω)∩L∞(Ω).The following remarks are concerned with Definition3.2.Remark 3.3.
Notice that, thanks to our regularity assumptions (3.8), (3.9), (3.10) and the choice of S, all terms in (3.13) are well defined.
The following two identifications are made in (3.13):
(i)
a(x,u,∇u)∇(S(u)φ) identifies with a(x,TmK(u),∇TmK(u))∇(S(u)φ) for almost every x∈Ω, where K>0 and supp(S)⊂[-K,K]. As a consequence of (3.8), (3.9), and (3.10), and of S∈W1,∞(ℝ),φ∈W01,p(Ω,ω)∩L∞(Ω), it follows that
(3.14)a(x,TmK(u),∇TmK(u))∇(S′(u)φ)∈L1(Ω),(ii)
fS(u)φ∈L1(Ω), because f∈L1(Ω) and S(u)φ∈L∞(Ω).


## 4. Existence Result

This section is devoted to establish the following existence theorem.Theorem 4.1.
Under assumptions (3.1)–(3.5) there exists a renormalized solution u of Problem (1.1).Proof.
The proof is divided into 7 steps. In Step1, we introduce an approximate problem. Step 2 is devoted to establish a few a priori estimates, the limit u of the approximate solutions uε is introduced and it is shown that u satisfies (3.8) and (3.9). Step 3 is devoted to prove an energy estimate (Lemma 4.2) which is a key point for the monotonicity arguments that are developed in Step 4. Step 5 is devoted to prove that u satisfies (3.11). In Step 6 we prove that u satisfies (3.12). Finally, Step 7 is devoted to prove that u satisfies (3.13) of Definition 3.2.Step 1.
Let us introduce the following regularization of the data:(4.1)bε(r)=b(Tm-ε1/ε(r))∀r∈ℝforε>0,(4.2)aε(x,s,ξ)=a(x,Tm-ε1/ε(s),ξ)a.e.inΩ,∀s∈ℝ,∀ξ∈ℝN,(4.3)fε∈Lp′(Ω);∥fε∥L1(Ω)≤∥f∥L1(Ω):fε→fstronglyinL1(Ω)asεtendsto0.
Let us now consider the following regularized problem:
(4.4)-div(aε(x,uε,∇uε))=fεinΩ,(4.5)uε=0on∂Ω.
In view of (3.3), (4.1), and (4.2), aε satisfy. For i=1,…,N(4.6)|aiε(x,s,ξ)|≤ωi(x)1/p[L(x)+σ(x)1/p′|Tm-ε1/ε(s)|q/p′+bε(s)p-1∑j=1Nωj1/p′(x)|ξj|p-1]
a.e. x∈Ω, for all s∈ℝ,ξ∈ℝN. And
(4.7)α≤bε(r)≤max{-1/ε≤r≤m-ε}b(r)=Cε∀r∈ℝ.As a consequence, proving existence of a weak solutionuε∈W01,p(Ω,ω) of (4.4) and (4.5) is an easy task (see, e.g., Theorem 2.1 and Remark 2.1 in Chapter 2 of [17] and see also [18]).Step 2.
A priori estimates and pointwise convergence ofuε.
UsingTK(uε) as a test function in (4.4) leads to
(4.8)∫Ωaε(x,uε,∇uε)∇TK(uε)dx=∫ΩfεTK(uε)dx≤K∥f∥L1(Ω).
Since aε satisfies (3.2), (4.2), and owing to (4.8) we have
(4.9)∫Ωbε(uε)p-1∑i=1N|∂TK(uε)∂xi|pωi(x)dx≤K∥f∥L1(Ω),(4.10)αp-1∫Ω∑i=1N|∂TK(uε)∂xi|pωi(x)dx≤K∥f∥L1(Ω).
From (4.10) we deduce with a classical argument (see, e.g., [18]) that, for a subsequence still indexed by ε,
(4.11)uε→ua.e.inΩ,(4.12)TK(uε)→TK(u)weaklyinW01,p(Ω,ω)andstronglyinLq(Ω,σ),
as ε tends to 0, where u is a measurable function defined on Ω which is finite a.e. in Ω.
Taking nowZε=∫0TmK(uε)bε(s)ds as a test function in (4.4) gives
(4.13)∫Ωaε(x,uε,∇uε)∇Zεdx=∫ΩfεZεdx.
Since aε satisfies (3.2) and b satisfies (3.1), permit to deduce from (4.13) that
(4.14)∫Ω∑i=1N|∂Zε∂xi|pωi(x)dx≤CK∥f∥L1(Ω),
where |Zε|≤∫-Kmb(s)ds=CK is a constant independent of ε.
Now for a fixedK>0, assumption (3.3) gives for i=1,…,N,
(4.15)|aiε(x,TmK(uε),∇TmK(uε))|≤ωi(x)1/p[L(x)+σ(x)1/p′max(K,m)q/p′+∑j=1Nωj1/p′(x)|∂Zε∂xj|p-1].
In view of (4.14) and (4.15), we deduce that
(4.16)aε(x,TmK(uε),∇TmK(uε))isboundedin∏i=1NLp′(Ω,wi1-p′),
then there exists a function XK∈∏i=1NLp′(Ω,wi1-p′) such that
(4.17)aε(x,TmK(uε),∇TmK(uε))⇀XKweaklyin∏i=1NLp′(Ω,wi1-p′)asε→0.
To prove that u is less or equal to m is an easy task which is performed exactly as in [10, 11]. Using T2m+(uε)-Tm+(uε) as a test function in (4.4) leads to
(4.18)∫Ωaε(x,uε,∇uε)∇(T2m+(uε)-Tm+(uε))dx=∫Ωfε(T2m+(uε)-Tm+(uε))dx,
which implies easily that
(4.19)∫Ωaε(x,uε,∇(T2m+(uε)-Tm+(uε)))∇(T2m+(uε)-Tm+(uε))dx≤m∥f∥L1(Ω).
Then (3.2), (4.1), and (4.2) yield
(4.20)b(m-ε)p-1∫Ω∑i=1N|∂(T2m+(uε)-Tm+(uε))∂xi|pωi(x)dx≤m∥f∥L1(Ω).
With the help of Poincaré's inequality, we have
(4.21)∫Ω|T2m+(uε)-Tm+(uε)|pω0(x)dx≤Cmb(m-ε)p-1∥f∥L1(Ω),
where C does not depend on ε. Then in view of (3.1), (4.11), and ω0>0, we can pass to the limit in (4.21) as ε tends to 0, to deduce that
(4.22)T2m+(u)-Tm+(u)=0a.e.inΩ,u≤ma.e.inΩ.
Let us now takeTK(vε) as a test function in (4.4), where vε=∫0uεbε(s)ds. We obtain
(4.23)∫Ωaε(x,uε,∇uε)∇TK(vε)dx=∫ΩfεTK(vε)dx≤K∥f∥L1(Ω).
Then (3.2) yields
(4.24)∫Ω∑i=1N|∂TK(vε)∂xi|pωi(x)dx≤K∥f∥L1(Ω).
We deduce with a classical argument that, for a subsequence still indexed by ε,
(4.25)vε→va.e.inΩ,(4.26)TK(vε)→TK(v)weaklyinW01,p(Ω,ω),
as ε tends to 0, where v is a measurable function defined on Ω which is finite a.e. in Ω.
Using the admissible test functionθn(vε) in (4.4) leads to
(4.27)∫Ωaε(x,uε,∇uε)∇θn(vε)dx=∫Ωfεθn(vε)dx.
As a consequence of the previous convergence results, we are in a position to pass to the limit as ε tends to 0 in (4.27)
(4.28)limε→0∫Ωaε(x,uε,∇uε)∇θn(vε)dx=∫Ωfθn(v)dx.
Using the pointwise convergence of θn(u) to 0 as n tends to +∞ and |θn(u)|≤1 a.e. in Ω independently of n, since f∈L1(Ω), Lebesgue's convergence theorem shows that ∫Ωfθn(v)dx→0, as n tends to +∞. Passing to the limit in (4.28) we obtain
(4.29)limn→+∞limε→0∫{n≤|vε|≤n+1}aε(x,uε,∇uε)∇vεdx=0.Step 3.
In this step we prove the following monotonicity estimate.Lemma 4.2.
The subsequence ofuε defined in Step 1 satisfies for any K≥0(4.30)limε→0∫Ω[aε(TmK(uε),∇TmK(uε))bε(uε)p-1-aε(TmK(uε),∇TmK(u))bε(uε)p-1][∇TmK(uε)-∇TmK(u)]dx=0.Proof.
LetK≥0 be fixed. Equality (4.30) is split into
(4.31)∫Ω[aε(TmK(uε),∇TmK(uε))bε(uε)p-1-aε(TmK(uε),∇TmK(u))bε(uε)p-1][∇TmK(uε)-∇TmK(u)]dx=A1ε+A2ε+A3ε,
where
(4.32)A1ε=∫Ωaε(TmK(uε),∇TmK(uε))bε(uε)p-1∇TmK(uε)dxdsdt,A2ε=-∫Ωaε(TmK(uε),∇TmK(uε))bε(uε)p-1∇TmK(u)dxdsdt,A3ε=-∫Ωaε(TmK(uε),∇TmK(u))bε(uε)p-1(∇TmK(uε)-∇TmK(u))dxdsdt.
In the sequel we pass to the limit in (4.31) when ε tends to 0.Limit ofA1ε
Using the admissible test functionSn(vε)∫0TmK(u)(1/b(s)p-1)ds in (4.4) leads to
(4.33)∫ΩSn(vε)aε(x,uε,∇uε)∇TmK(u)b(u)p-1dx+∫Ωaε(uε,∇uε)∇Sn(vε)·(∫0TmK(u)1b(s)p-1ds)dx=∫ΩfεSn(vε)∫0TmK(u)1b(s)p-1dsdx,
where vε=∫0uεbε(s)ds, pass to the limit as ε tends to 0 in (4.33).
Since supp(Sn)⊂[-(n+1),n+1] and {x∈Ω;|vε|≤n+1}⊂{x∈Ω;|uε|≤(n+1)/α}, we have for i=1,…,N and ε≤α/(n+1)
(4.34)|aiε(x,uε,∇uε)Sn(vε)|=|aiε(x,T(n+1)/α(uε),∇T(n+1)/α(uε))Sn(vε)|≤∥Sn∥L∞(ℝ)ωi(x)1/p[∑j=1NL(x)+σ(x)1/p′|T(n+1)/α(uε)|q/p′+∑j=1Nωj1/p′(x)|∂Tn+1(vε)∂xj|p-1].
In view of (4.24), (4.34) we deduce that for fixed n≥1:
(4.35)aε(x,T(n+1)/α(uε),∇T(n+1)/α(uε))Sn(vε)isboundedin∏i=1NLp′(Ω,wi1-p′),
independently of ε≤α/(n+1). Then there exists a function Yn∈∏i=1NLp′(Ω,wi1-p′) such that for fixed n≥1:
(4.36)Sn(vε)aε(x,T(n+1)/α(uε),∇T(n+1)/α(uε))⇀Ynweaklyin∏i=1NLp′(Ω,wi1-p′)asε→0.
Now for max(K,m)≤n/α, we have
(4.37)Sn(vε)aε(x,T(n+1)/α(uε),∇T(n+1)/α(uε))χ{-K<uε<m}=Sn(vε)aε(x,TKm(uε),∇TKm(uε))χ{-K<uε<m}
a.e. in Ω, which implies that, through the use of (4.17), (4.25), and (4.36) and passing to the limit as ε tends to 0,
(4.38)Ynχ{-K<u<m}=Sn(v)XKχ{-K<u<m}
a.e. in Ω-{{u=-K}∪{u=m}}formax(K,m)≤n/α. As a consequence of (4.38) we have for max(K,m)≤n/α(4.39)Yn∇TmK(u)=Sn(v)XK∇mK(u)a.e.inΩ.
We are now in a position to exploit (4.33), which gives together with (4.36) and (4.39)
(4.40)limε→0∫ΩSn(vε)aε(x,uε,∇uε)∇TmK(u)b(u)p-1dx=limε→0∫ΩSn(vε)aε(x,T(n+1)/α(uε),∇T(n+1)/α(uε))∇TmK(u)b(u)p-1dx=∫ΩYn∇TmK(u)b(u)p-1dx=∫ΩSn(v)XK∇TmK(u)b(u)p-1dx.
Passing to the limit as n tends to +∞ in (4.40) leads to
(4.41)limn→+∞limε→0∫ΩSn(vε)aε(x,uε,∇uε)∇TmK(u)b(u)p-1dx=∫ΩXK∇TmK(u)b(u)p-1dx.
The second term of (4.33)
(4.42)|∫Ωaε(x,uε,∇uε)∇Sn(vε).(∫0TmK(u)1b(s)p-1ds)dx|≤max(m,K)αp-1∫{n≤|vε|≤n+1}aε(x,uε,∇uε)∇vεdx.
Then (4.29) implies that
(4.43)limn→+∞limε→0∫Ωaε(x,uε,∇uε)∇Sn(vε).(∫0TmK(u)1b(s)p-1ds)dx=0.
In view (4.41) and (4.43), passing to the limit as ε tends to 0 and as n tends to +∞ in (4.33) is an easy task and leads to
(4.44)∫ΩXK∇TmK(u)b(u)p-1dx=∫Ωf∫0TmK(u)1b(s)p-1dsdx.
We are now in a position to exploit (4.44).
The use of the test function∫0TmK(uε)(1/bε(s)p-1)ds in (4.4), yields
(4.45)∫Ωaε(x,uε,∇uε)∇TmK(uε)bε(uε)p-1dx=∫Ωfε∫0TmK(uε)1bε(s)p-1dsdx.
Passing to the limit as ε tends to 0 in (4.45), in view (4.44), we have
(4.46)limε→0A1ε=limε→0∫Ωaε(x,uε,∇uε)∇TmK(uε)bε(uε)p-1dx=∫ΩXK∇TmK(u)b(u)p-1dx.Limit ofA2ε
In view of (4.12), (4.17) and since 1/bε(uε)p-1 converges to 1/b(u)p-1 a.e. in Ω and due to the bound 1/bε(uε)p-1≤1/αp-1 a.e. in Ω, we have
(4.47)limε→0A2ε=-∫ΩXK∇TmK(u)b(u)p-1dx.Limit ofA3ε
Let us remark that (3.1), (4.1), and (4.11) imply that
(4.48)aε(x,TmK(uε),∇TmK(u))bε(uε)p-1→a(x,TmK(u),DTmK(u))b(u)p-1a.e.inΩ,
as ε tends to 0, and that for i=1,…,N(4.49)|aiε(TmK(uε),∇TmK(u))bε(uε)p-1|≤wi1/p(x)[1αp-1L(x)+max(K,m)q/p′αp-1σ1/p′(x)+∑j=1Nwj1/p′|∂TmK(u)∂xj|p-1]
a.e. in Ω, uniformly with respect to ε.
It follows that whenε tends to 0
(4.50)a(x,TmK(uε),∇TmK(u))bε(uε)p-1→a(x,TmK(u),∇TK(u))b(u)p-1stronglyin∏i=1NLp′(Ω,wi1-p′).
In view of (4.12), we conclude that
(4.51)(∇TmK(uε)-∇TmK(u))⇀0weaklyin∏i=1NLp(Ω,wi),asεgoesto0.
As a consequence of (4.50) and (4.51) we have for all K>0(4.52)limε→0A3ε=0.
Equations (4.46), (4.46), (4.47), and (4.46) allow to pass to the limit as ε tends to zero in (4.31) and to obtain (4.30) of Lemma 4.2.Step 4.
In this step we identify the weak limitXK and we prove the weak L1 convergence of the “truncated” energy (aε(x,TmK(uε),∇TmK(uε))/bε(uε)p-1)∇TmK(uε) as ε tends to 0.Lemma 4.3.
For fixedK≥0, one has
(4.53)XK=a(x,TmK(u),∇TmK(u))a.e.in{x∈Ω;u(x)<m}.
And as ε tends to 0
(4.54)aε(x,TmK(uε),∇TmK(uε))bε(uε)p-1∇TmK(uε)⇀a(x,TmK(u),∇TmK(u))b(u)p-1∇TmK(u)weaklyinL1(Ω).Proof.
LetK≥0 be fixed. From (4.11) and (4.50) together with (4.30) of Lemma 4.2, we obtain
(4.55)limε→0∫Ωaε(x,TmK(uε),∇TmK(uε))b(TmK(uε))p-1∇TmK(uε)dx=∫ΩXKb(TmK(u))p-1∇TK(u)dx.
We remark the monotone character a (with respect to ξ) and since 1/bε(uε)p-1 converges to 1/b(u)p-1 a.e. in Ω and due to the bound 1/bε(uε)p-1≤1/αp-1 a.e. in Ω, we conclude that for all ψ∈∏i=1NLp(Ω,wi) we have
(4.56)0≤limε→0∫Ω[aε(x,TmK(uε),∇TmK(uε))bε(uε)p-1-aε(x,TmK(uε),ψ)bε(uε)p-1][∇TmK(uε)-ψ]dx=limε→0∫Ωaε(x,TmK(uε),∇TmK(uε))bε(uε)p-1[∇TmK(uε)-ψ]dx-limε→0∫Ωaε(x,TmK(uε),ψ)bε(uε)p-1[∇TmK(uε)-ψ]dx=∫ΩXKb(u)p-1[∇TmK(u)-ψ]dx-∫Ωa(x,TmK(u),ψ)b(u)p-1[∇TmK(u)-ψ]dx×∫Ω[XKb(u)p-1-a(x,TmK(u),ψ)b(u)p-1][∇TmK(u)-ψ]dx.
The usual Minty's argument applies in view of (4.56). It follows that
(4.57)XKb(u)p-1=a(x,TmK(u),∇TmK(u))b(u)p-1a.e.inΩ
which together with (4.20) yields (4.53) of Lemma 4.3.
In order to prove (4.54), we observe that the monotone character of a (with respect to ξ) and (4.30) give
(4.58)[aε(x,TmK(uε),∇TmK(uε))bε(uε)p-1-aε(x,TmK(uε),∇TmK(u))bε(uε)p-1][∇TmK(uε)-∇TmK(u)]→0
strongly in L1(Ω) as ε tends to 0. Moreover (4.12), (4.17), (4.50), and (4.53) imply that
(4.59)aε(x,TmK(uε),∇TmK(uε))bε(uε)p-1∇TmK(u)⇀a(x,TmK(u),∇TmK(u))b(u)p-1∇TmK(u)
weakly in L1(Ω) as ε tends to 0
(4.60)aε(x,TmK(uε),∇TmK(u))bε(uε)p-1∇TmK(uε)⇀a(x,TmK(u),∇TmK(u))b(u)p-1∇TmK(u)
weakly in L1(Ω) as ε tends to 0, and
(4.61)aε(x,TmK(uε),∇TmK(u))bε(uε)p-1∇TmK(u)⇀a(x,TmK(u),∇TmK(u))b(u)p-1∇TmK(u)
strongly in L1(Ω) as ε tends to 0.
Using the above convergence results (4.59), (4.60), and (4.61) in (4.58) we obtain that for any K≥0
(4.62)aε(x,TmK(uε),∇TmK(uε))bε(uε)p-1∇TmK(uε)⇀a(x,TmK(u),∇TmK(u))b(u)p-1∇TmK(u)
weakly in L1(Ω) as ε tends to 0.Step 5.
In this step we prove thatu satisfies (3.11).
Using(Tmn+1(uε)-Tmn(uε)) as a test function in (4.4) leads to
(4.63)∫Ωaε(x,uε,∇uε)∇(Tmn+1(uε)-Tmn(uε))dx=∫Ωfε(Tmn+1(uε)-Tmn(uε))dx.
Since supp(Tmn+1(·)-Tmn(·))⊂[-(n+1),-n], we have
(4.64)∫{-n-1≤uε(x)≤-n}aε(x,uε,∇uε)∇uεdx=∫Ωaε(x,uε,∇uε)∇(Tmn+1(uε)-Tmn(uε))dx=∫Ωaε(x,uε,∇uε)bε(uε)p-1∇(Tmn+1(uε)-Tmn(uε))b(Tm-1n+1(uε))p-1dx=∫Ωaε(x,Tmn+1(uε),∇Tmn+1(uε))bε(uε)p-1∇Tmn+1(uε)b(Tm-1n+1(uε))p-1dx-∫Ωaε(x,Tmn(uε),∇Tmn(uε))bε(uε)p-1∇Tmn(uε)b(Tm-1n+1(uε))p-1dx.
In view of (4.54) of Lemma 4.3 and since b(Tm-1n+1(uε))p-1 converges to b(Tm-1n+1(u))p-1 a.e. in Ω and due to the bound b(Tm-1n+1(uε))p-1≤maxs∈[-n-1,m-1]b(s)p-1 a.e. in Ω, we can pass to the limit as ε tends to 0 for fixed n≥0 to obtain
(4.65)limε→0∫{-n-1≤uε(x)≤-n}aε(x,uε,∇uε)∇uεdx=∫Ωa(x,Tmn+1(u),∇Tmn+1(u))b(u)p-1∇Tmn+1(u)b(Tm-1n+1(u))p-1dx-∫Ωa(x,Tmn(u),∇Tmn(u))b(u)p-1∇Tmn(u)b(Tm-1n+1(u))p-1dx=∫{-n-1≤u(x)≤-n}a(x,u,∇u)∇udx.
Taking the limit as ε tends to 0 and n tends to +∞ in (4.63) and using the estimate (4.64) and (4.65) show that
(4.66)limn→+∞∫{-n-1≤u(x)≤-n}a(x,u,∇u)∇udx≤limn→+∞∫{u≤-n}|f|dx=0.Step 6.
In this step we prove thatu satisfies (3.12).
UsingSn(vε)(1/δ)(Tm-δ+(u)-Tm-2δ+(u)) as a test function in (4.4) leads to
(4.67)1δ∫ΩSn(vε)aε(x,uε,∇uε)∇(Tm-δ+(u)-Tm-2δ+(u))dx=∫ΩSn(vε)fε(Tm-δ+(u)-Tm-2δ+(u))δdx,
where vε=∫0uεbε(s)ds. Since supp(Sn)⊂[-(n+1),n+1] and {x∈Ω;|vε|≤n+1}⊂{x∈Ω;|uε|≤(n+1)/α} we have
(4.68)1δ∫ΩSn(vε)aε(x,uε,∇uε)∇(Tm-δ+(u)-Tm-2δ+(u))dx=1δ∫ΩSn(vε)aε(x,T(n+1)/α(uε),∇T(n+1)/α(uε))∇(Tm-δ+(u)-Tm-2δ+(u))dx.
In view of (4.22), (4.36), (4.39), and (4.53), passing to the limit as ε tends to 0 and n tends to +∞(4.69)limn→+∞limε→01δ∫ΩSn(vε)aε(x,T(n+1)/α(uε),∇T(n+1)/α(uε))∇(Tm-δ+(u)-Tm-2δ+(u))dx=limn→+∞1δ∫ΩSn(v)a(x,T(n+1)/α(u),∇T(n+1)/α(u))∇(Tm-δ+(u)-Tm-2δ+(u))dx=limn→+∞1δ∫ΩSn(v)a(x,Tm-δ(u),∇Tm-δ(u))∇(Tm-δ+(u)-Tm-2δ+(u))dx=1δ∫Ωa(x,Tm-δ(u),∇Tm-δ(u))∇(Tm-δ+(u)-Tm-2δ+(u))dx=1δ∫{m-2δ≤u≤m-δ}a(x,u,∇u)∇udx.
Taking the limit as ε tends to 0, n tends to +∞ and δ tends to 0 in (4.67) and using the estimate (4.68) and (4.69) show that
(4.70)limδ→01δ∫{m-2δ≤u≤m-δ}a(x,u,∇u)∇udx=limδ→0∫Ωf(Tm-δ+(u)-Tm-2δ+(u))δdx=∫{u=m}f(x)dx.Step 7.
In this step,u is shown to satisfy (3.13). Let φ∈W01,p(Ω,ω)∩L∞(Ω) and let S be a function in W1,∞(ℝ) such that S has a compact support and S(m)=0. Let K be a positive real number such that supp(S)⊂[-K,K] and vε=∫0uεbε(s)ds. Using S(u)Sn(vε)φ as a test function in (4.4) leads to
(4.71)∫ΩSn(vε)aε(x,uε,∇uε)∇(S(u)φ)dx+∫ΩS(u)φaε(x,uε,∇uε)∇Sn(vε)dx=∫ΩfεSn(vε)S(u)φdx.
In what follows we pass to the limit asε tends to 0 and n tends to +∞ in each term of (4.71).Limit of First Term in (4.71)
Since suppSn⊂[-(n+1),n+1] and {x∈Ω;|vε|≤n+1}⊂{x∈Ω;|uε|≤(n+1)/α}, we have
(4.72)Sn(vε)aε(x,uε,∇uε)=Sn(vε)aε(x,T(n+1)/α(uε),∇T(n+1)/α(uε))a.e.inΩ.
In view of (4.22), (4.36), (4.39), and (4.53), passing to the limit as ε tends to 0
(4.73)limε→0∫ΩSn(vε)aε(x,uε,∇uε)∇(S(u)φ)dx=limε→0∫ΩSn(vε)aε(x,T(n+1)/α(uε),∇T(n+1)/α(uε))∇(S(u)φ)dx=∫ΩSn(v)a(x,T(n+1)/α(u),∇T(n+1)/α(u))∇(S(u)φ)dx=∫ΩSn(v)a(x,TmK(u),∇TmK(u))∇(S(u)φ)dx,limn→+∞limε→0∫ΩSn(vε)aε(x,uε,∇uε)∇(S(u)φ)dx=limn→+∞∫ΩSn(v)a(x,TmK(u),∇TmK(u))∇(S(u)φ)dx=∫Ωa(x,TmK(u),∇TmK(u))∇(S(u)φ)dx=∫Ωa(x,u,∇u)∇(S(u)φ)dx.Limit of Second Term in (4.71)
Since supp(Sn′)⊂[-(n+1),-n]∪[n+1,n] for any n≥1. As a consequence
(4.74)|∫ΩS(u)φaε(x,uε,∇uε)∇Sn(vε)dx|≤∥S∥L∞(Ω)∥φ∥L∞(Ω)∫{n≤|vε|≤n+1}aε(x,uε,∇uε)∇vεdx.
Taking the limit as ε tends to 0 and n tends to +∞ in (4.74) and using the estimate (4.29) show that
(4.75)limn→+∞limε→0|∫ΩS(u)φaε(x,uε,∇uε)∇Sn(vε)dx|=0.Limit of the Right-Hand Side of (4.71)
Due to (4.3) and (4.25), we have
(4.76)limn→+∞limε→0∫ΩfεSn(vε)S(u)φdx=∫ΩfS(u)φdx.
As a consequence of the previous convergence results, we are in a position to pass to the limit as ε tends to 0 in (4.71) and to conclude that u satisfies (3.13). The proof of Theorem 4.1 is achieved.



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*Source: 219586-2009-12-23.xml*