Existence of Solution for Nonlinear Elliptic Equations with Unbounded Coefficients and L1 Data

International Journal of Mathematics and Mathematical Sciences Volume 2009, Article ID 219586,18pages

doi:10.1155/2009/219586

Research Article

Existence of Solution for

Nonlinear Elliptic Equations with

Unbounded Coefficients and

L

1

Data

Hicham Redwane

Facult´e des Sciences Juridiques, ´Economiques et Sociales, Universit´e Hassan 1, B.P. 784, Settat 26 000, Morocco

Correspondence should be addressed to Hicham Redwane,redwane [email protected]

Received 20 July 2009; Accepted 17 November 2009

Recommended by M ´onica Clapp

An existence result of a renormalized solution for a class of nonlinear elliptic equations is established. The diffusion functionsax, u,∇umay not be inL1

locΩ

N

for a finite value of the unknown and the data belong toL1_Ω.

Copyrightq2009 Hicham Redwane. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In this paper we investigate the problem of existence of a renormalized solutions for elliptic equations of the type

−divax, u,∇u f inΩ,

u0 on∂Ω, 1.1

where Ω is an open bounded subset of RN_{, N ≥ 1, with the data f in L}1_{Ω. The}

operator−divax, u,∇uis a Leray-Lions operator defined on the weighted sobolev spaces

W₀1,pΩ, w, but which is not restricted by any growth condition with respect to u see

assumptions2.2,2.4, and2.5ofSection 3. The functionax, s, ξis controlled by a real functionb:− ∞, m→ Rwhich blows up for a finite valuem >0see2.2,2.3.

i.e., in the distribution meaningseems 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 Lions1for the study of Boltzmann equationsee also Lions2for 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 B´enilan et al.8for the study of nonlinear elliptic problems.

In the case whereax, u,∇uis replaced bydu Au∇uproblems with diffusion

matrices that have at least one diagonal coefficient that blows up for a finite value of the unknownandf ∈L2_{Ω, existence and uniqueness has been established in Blanchard and}

Redwane9,10.

As far as we have the stationary and evolution equations case1.1, the existence and a partial uniqueness of renormalized solutions have been proved in Blanchard et al.11in the case whereax, u,∇uis replaced byAx, u∇uwhereAx, sis a Carath´eodory symmetric matrices, such thatAx, sblows 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 unknownsee Garc´ıa V´azquez and Orteg ´on Gallego12,13and Orsina14.

The paper is organized as follows. InSection 2we will precise some basic properties of weighted Sobolev spaces.Section 3is devoted to specify the assumptions onax, s, ξ, bs,

andfneeded in the present study and gives the definition of a renormalized solution of1.1. InSection 4Theorem 4.1we establish the existence of such a solution.

2. Preliminaries

Throughout the paper, we assume that the following assumptions hold true.Ωis a bounded open subset onRN, N ≥ 1. Let us suppose that 1 < p < ∞ is a real number, andωx {ωix}_{0≤i≤N}is a vector of weight functions. Furthermore we suppose that every component

ωixis a measurable function which is strictly positive and satisfies

ωi∈L1_locΩ, ωi−1/p−1∈L1_locΩ. 2.1

We define the weighted Lebesgue spaceLp_{Ω, ω}

0with weight ω0, as the space of all

real-valued measurable functionsufor which

up,ω0

Ω|ux|

p_ω

0xdx

1/p

<∞. 2.2

In order to define the weighted Sobolev space ofW1,pΩ, ω, as the space of all real-valued

functionsu∈Lp_{Ω, ω}

0such that the derivaties in the sense of distributions satisfy∂u/∂xi∈

Lp_{Ω, ωifor alli1, . . . , N. This set of functions forms a Banach space under the norm}

u1,p,ω

Ω|ux|

p_ω

0xdx

N

i1

Ω

_∂x∂u_ipωixdx 1/p

To deal with the Dirichlet problem, we use the spaceX W₀1,pΩ, ωdefined as the closure of C₀∞Ω with respect to the norm · _1,p,ω. Note that, C∞₀ Ω is dense inW₀1,pΩ, ωand W₀1,pΩ, ω, · _1,p,ωis a reflexive Banach space. Note that the expression

u_X

_N

i1

Ω

_∂x∂u_ipωixdx 1/p

2.4

is a norm defined onXand is equivalent to the norm2.3. MoreoverX, · _Xis a reflexive Banach space, and there exist a weight functionσonΩand a parameter 1< q <∞such that the Hardy inequality

Ω|u|

q_σxdx

1/q

≤C

_N

i1

Ω

_∂x∂u_ipωixdx 1/p

2.5

holds for everyu ∈ X with a constantC > 0 independent of u. Moreover, the imbedding

X →Lq_{Ω, σis compact.}

We recall that the dual of the weighted Sobolev spacesW₀1,pΩ, ω is equivalent to

W−1,p_{Ω, ω}∗_{, whereω}∗ _{ω∗ i ω

1−p

i ;i1, . . . , N}andp p/p−1is the conjugate ofp. For more details we refer the reader to15 see also16.

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

Throughout the paper, we assume that the following assumptions hold true.Ωis a bounded open set onRN_{, N ≥1. Let 1 < p <∞, and letωx {ωix}}

{0≤i≤N}be a vector of weight functions.

Let now −divax, u,∇u be a Leray-Lions operator defined on W₀1,pΩ, ω into

W−1,p_{Ω, ω}∗_{and where}

a:Ω×R×RN−→RN is a Carath´eodory function, such that 3.1

there exists a positive functionb∈C0_{−∞, mwhich satisfies}

lim

r→m−br ∞;

_m

0

bsds <∞, br≥α >0∀r ∈−∞, m,

ax, s, ξ·ξ≥bsp−1

N

i1

ωix|ξi|p, ax, s,0 0,

3.2

for almost everyx∈Ω, for everys∈Randξ∈RN_. For anyi1, . . . , N,

|aix, s, ξ| ≤ωix1/p ⎡

⎣Lx σx1/p|s|q/p bsp−1

N

j1

ω1_j/pxξjp−1

⎤

for almost everyx∈Ω, for everysandξ, and whereLxis a positive function inLp_Ω

ax, s, ξ−ax, s, ξξ−ξ≥0, 3.4

for anyξ, ξ ∈RN_{, for anys∈Rand for almost everyx∈Ω}

f is an element ofL1Ω. 3.5

Remark 3.1. As already mentioned in the introduction, problem1.1does not admit a weak solution under assumptions3.1–3.5since the growth ofax, u,∇uis not controlled with respect to u, the field ax, u,∇u is not, in general, defined as a distribution because the difficulty is defining the field ax, u,∇uon the subset{x ∈ Ω;ux m}ofΩ,since on this set,bu ∞.

The following notations will be used throughout the paper. For any K ≥ 0, the truncation at heightKis defined byTKr max−K, minr, K, for any positive numbersl

andK, the functionsTK

l are defined by

T_lKr ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

−K, ifr≤ −K,

r, if −K≤r≤l,

l, ifr≥l.

3.6

We define forn≥1 fixed

θnr T1r−Tnr

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

0, if |r| ≤n,

r−n sgs, if n≤ |r| ≤n1,

sgs, if |r| ≥n1,

3.7

andSnr 1− |θnr|, for allr∈R.

The definition of a renormalized solution for Problem1.1can be stated as follows.

Definition 3.2. A measurable functionudefined onΩis a renormalized solution of Problem 1.1if

TKu∈W₀1,pΩ, ω ∀K≥0, 3.8

ux≤m for almost everyx∈Ω, 3.9

ax, T_mKu,∇T_mKuχ{u<m}∈ N

i1

{−n−1≤ux≤−n}ax, u,∇u∇udx−→0 asn−→∞, 3.11

1

δ

{m−2δ≤ux≤m−δ}ax, u,∇u∇udx−→

{um}fdx asδ−→0, 3.12

and if, for every functionSinW1,∞_{Rsuch that suppSis compact andSm 0, usatisfies}

Ωax, u,∇u∇

Suϕdx

ΩfSuϕdx, ∀ϕ∈W

1,p

0 Ω, ω∩L∞Ω. 3.13

The following remarks are concerned withDefinition 3.2.

Remark 3.3. Notice that, thanks to our regularity assumptions 3.8, 3.9, 3.10 and the choice ofS, all terms in3.13are well defined.

The following two identifications are made in3.13:

iax, u,∇u∇Suϕidentifies withax, TK

mu,∇TmKu∇Suϕfor almost every

x∈ Ω, whereK > 0 and suppS⊂ −K, K. As a consequence of3.8,3.9, and 3.10, and ofS∈W1,∞_{R, ϕ∈W}1,p

0 Ω, ω∩L∞Ω, it follows that

ax, T_mKu,∇T_mKu∇Suϕ∈L1Ω, 3.14

iifSuϕ∈L1Ω, becausef∈L1ΩandSuϕ∈L∞Ω.

4. Existence Result

This section is devoted to establish the following existence theorem.

Theorem 4.1. Under assumptions3.1–3.5there exists a renormalized solutionuof Problem1.1.

Proof. The proof is divided into 7 steps. In Step 1, we introduce an approximate problem. Step 2is devoted to establish a few a priori estimates, the limituof the approximate solutions

uε_{is introduced and it is shown thatusatisfies3.8and3.9.Step 3is devoted to prove an} energy estimateLemma 4.2which is a key point for the monotonicity arguments that are developed inStep 4.Step 5is devoted to prove thatusatisfies3.11. InStep 6we prove that

usatisfies3.12. Finally,Step 7is devoted to prove thatusatisfies3.13ofDefinition 3.2.

Step 1. Let us introduce the following regularization of the data:

bε_{r bT}1/ε m−εr

∀r∈Rforε >0, 4.1

aεx, s, ξ ax, T_m1/ε_−εs, ξ a.e.inΩ, ∀s∈R, ∀ξ∈RN, 4.2

fε_∈Lp_{Ω; f}ε

Let us now consider the following regularized problem:

−divaεx, uε,∇uε fε inΩ, 4.4

uε_{0 on∂Ω. 4.5}

In view of3.3,4.1, and4.2,aε_{satisfy. Fori1, . . . , N}

_aε

ix, s, ξ≤ωix1/p

⎡

⎣Lx σx1/pTm1/ε−εs q/p

bεsp−1

N

j1

ω1_j/pxξjp−1

⎤

⎦ 4.6

a.e.x∈Ω, for alls∈R, ξ∈RN.And

α≤bεr≤ max

{−1/ε≤r≤m−ε}br Cε ∀r∈R. 4.7

As a consequence, proving existence of a weak solutionuε _{∈ W}1,p

0 Ω, ωof4.4and

4.5is an easy tasksee, e.g., Theorem 2.1 and Remark 2.1 in Chapter 2 of17and see also 18.

Step 2. A priori estimates and pointwise convergence ofuε_. UsingTKuε_{as a test function in4.4leads to}

Ωa

ε_{x, u}ε_,∇uε_∇TKuε_dx

Ωf

ε_TKuε_dx≤Kf

L1_Ω. 4.8

Sinceaε_{satisfies3.2,4.2, and owing to4.8we have}

Ωb

ε_uεp−1N

i1

∂TK_∂xu_i εpωixdx≤Kf_L1_Ω, 4.9

αp−1

Ω

N

i1

∂TK_∂xu_iεpωixdx≤Kf_L1_Ω. 4.10

From4.10we deduce with a classical argumentsee, e.g.,18that, for a subsequence still indexed byε,

uε−→u a.e.inΩ, 4.11

TKuε−→TKuweakly inW₀1,pΩ, ωand strongly inLqΩ, σ, 4.12

Taking nowZεTmKuε

0 bεsdsas a test function in4.4gives

Ωa

ε_{x, u}ε_,∇uε_∇Zε_dx

Ωf

ε_Zε_{dx. 4.13}

Sinceaε_{satisfies3.2andbsatisfies3.1, permit to deduce from4.13that}

Ω

N

i1

∂Z_∂xiεpωixdx≤CKf_L1_Ω, 4.14

where|Zε_{| ≤}m

−KbsdsCKis a constant independent ofε.

Now for a fixedK >0, assumption3.3gives fori1, . . . , N,

aε

i

x, T_mKuε,∇T_mKuε

≤ωix1/p

⎡

⎣Lx σx1/p maxK, mq/p

N

j1

ω1_j/px∂Z

ε

∂xj

p−1⎤

⎦.

4.15

In view of4.14and4.15, we deduce that

aεx, T_mKuε,∇T_mKuεis bounded in N

i1

LpΩ, w1_i−p, 4.16

then there exists a functionXK∈

_N

i1LpΩ, w 1−p

i such that

aεx, T_mKuε,∇T_mKuε XKweakly in N

i1

LpΩ, w_i1−p asε−→0. 4.17

To prove thatuis less or equal tomis an easy task which is performed exactly as in10,11. UsingT_2muε−T_muεas a test function in4.4leads to

Ωa

ε_{x, u}ε_,∇uε_∇T

2muε−Tmuε

dx

Ωf

ε_T

2muε−Tmuε

dx, 4.18

which implies easily that

Ωa

ε_{x, u}ε_,∇T

2muε−Tmuε

∇T_2muε−T_muεdx≤mf_L1_Ω. 4.19

Then3.2,4.1, and4.2yield

bm−εp−1

Ω

N

i1

∂T_2muε_−T

muε

∂xi

p

With the help of Poincar´e’s inequality, we have

Ω

T_2muε−T_muεpω0xdx≤

Cm

bm−εp−1f_L1_Ω

, _4.21

whereCdoes not depend onε. Then in view of3.1,4.11, andω0 >0, we can pass to the

limit in4.21asεtends to 0, to deduce that

T_2mu−T_mu 0 a.e.inΩ,

u≤m a.e.inΩ. 4.22

Let us now takeTKvε_{as a test function in4.4, wherev}εuε

0 bεsds. We obtain

Ωa

ε_{x, u}ε_,∇uε_∇TKvε_dx

Ωf

ε_TKvε_dx≤Kf

L1_Ω. 4.23

Then3.2yields

Ω

N

i1

∂T_∂xKv_iεpωixdx≤Kf_L1_Ω. 4.24

We deduce with a classical argument that, for a subsequence still indexed byε,

vε−→v a.e.inΩ, 4.25

TKvε−→TKvweakly inW₀1,pΩ, ω, 4.26

asεtends to 0, wherevis a measurable function defined onΩwhich is finite a.e. inΩ. Using the admissible test functionθnvε_{in4.4leads to}

Ωa

ε_{x, u}ε_,∇uε_∇θnvε_dx

Ωf

ε_θnvε_{dx. 4.27}

As a consequence of the previous convergence results, we are in a position to pass to the limit asεtends to 0 in4.27

lim ε→0

Ωa

ε_{x, u}ε_,∇uε_∇θnvε_dx

Using the pointwise convergence of θnu to 0 as n tends to ∞ and |θnu| ≤ 1 a.e. in Ω independently of n, since f ∈ L1_{Ω, Lebesgue’s convergence theorem shows that}

Ωfθnvdx → 0,asntends to∞. Passing to the limit in4.28we obtain

lim n→∞εlim→0

{n≤|vε_|≤n1}

aεx, uε,∇uε∇vεdx0. 4.29

Step 3. In this step we prove the following monotonicity estimate.

Lemma 4.2. The subsequence ofuε_{defined in Step 1 satisfies for anyK≥0}

lim ε→0

Ω

aε_TK

muε,∇TmKuε

bε_uεp−1 −

aε_TK

muε,∇TmKu

bε_uεp−1

∇TK

muε− ∇TmKu

dx0. 4.30

Proof. LetK≥0 be fixed. Equality4.30is split into

Ω

aε_TK

muε,∇TmKuε

bε_uεp−1 −

aε_TK

muε,∇TmKu

bε_uεp−1

∇TK

muε− ∇TmKu

dx

Aε₁Aε₂Aε₃,

4.31

where

Aε

1

Ω aε_TK

muε,∇TmKuε

bε_uεp−1 ∇T K

muεdx ds dt,

Aε₂−

Ω aε_TK

muε,∇TmKuε

bε_uεp−1 ∇T K

mudx ds dt,

Aε₃−

Ω aε_TK

muε,∇TmKu

bε_uεp−1

∇TK

muε− ∇TmKu

dx ds dt.

4.32

In the sequel we pass to the limit in4.31whenεtends to 0.

Limit of

A

Using the admissible test functionSnvεTK mu

0 1/bs

p−1_{dsin4.4leads to}

ΩSnv

ε_aε_{x, u}ε_,∇uε∇TmKu

bup−1dx

Ωa

ε_uε_,∇uε_∇Snvε_·

TK

mu

0

1

bsp−1ds

dx

Ωf

ε_Snvε

_TK mu

0

1

bsp−1dsdx,

4.33

wherevεuε

Since suppSn⊂−n1, n1and{x∈Ω;|vε_{| ≤n1} ⊂ {x∈Ω;|u}ε_{| ≤n1/α},} we have fori1, . . . , Nandε≤α/n1

_aε

ix, uε,∇uεSnvεaεi

x, Tn1/α uε,∇Tn1/αuε

Snvε

≤ SnL∞_Rωix1/p

⎡

⎣Lx σx1/pTn1/αuεq/p

N j1

ω_j1/px∂Tn1v

ε

∂xj

p−1⎤

⎦.

4.34

In view of4.24,4.34we deduce that for fixedn≥1:

aεx, Tn1/αuε,∇Tn1/αuεSnvεis bounded in

N

i1

LpΩ, w1_i−p, 4.35

independently ofε≤α/n1. Then there exists a functionYn ∈Ni1LpΩ, w 1−p

i such that for fixedn≥1:

Snvεaεx, Tn1/αuε,∇Tn1/αuε Yn weakly in N

i1

LpΩ, w1_i−p asε−→0.

4.36

Now for maxK, m≤n/α, we have

Snvε_aε_{x, T}

n1/αuε,∇Tn1/αuε

χ{−K<uε_<m}

Snvεaεx, T_Kmuε,∇T_Kmuεχ{−K<uε_<m}

4.37

a.e. inΩ,which implies that, through the use of4.17,4.25, and4.36and passing to the limit asεtends to 0,

Ynχ{−K<u<m}SnvXKχ{−K<u<m} 4.38

a.e. inΩ− {{u−K} ∪ {um}} for maxK, m≤n/α. As a consequence of4.38we have for maxK, m≤n/α

We are now in a position to exploit4.33, which gives together with4.36and4.39

lim ε→0

ΩSnv

ε_aε_{x, u}ε_,∇uε∇TmKu

bup−1dx

lim ε→0

ΩSnv

ε_aε_{x, T}

n1/αuε,∇Tn1/αuε∇T

K mu

bup−1dx

ΩYn

∇TK

mu

bup−1 dx

ΩSnvXK

∇TK

mu

bup−1dx.

4.40

Passing to the limit asntends to∞in4.40leads to

lim n→∞εlim→0

ΩSnv

ε_aε_{x, u}ε_,∇uε∇TmKu

bup−1dx

ΩXK

∇TK

mu

bup−1dx. 4.41

The second term of4.33

Ωa

ε_{x, u}ε_,∇uε_∇Snvε_.

TK

mu

0

1

bsp−1ds

dx

≤ maxm, K

αp−1

{n≤|vε_|≤n1}

aεx, uε,∇uε∇vεdx.

4.42

Then4.29implies that

lim n→∞εlim→0

Ωa

ε_{x, u}ε_,∇uε_∇Snvε_.

TK

mu

0

1

bsp−1ds

dx0. 4.43

In view4.41and4.43, passing to the limit asεtends to 0 and asntends to∞in4.33is an easy task and leads to

ΩXK

∇TK

mu

bup−1 dx

Ωf TK

mu

0

1

bsp−1ds dx. 4.44

We are now in a position to exploit4.44. The use of the test functionTmKuε

0 1/bεs

p−1_{dsin4.4, yields}

Ωa

ε_{x, u}ε_,∇uε∇TmKuε

bε_uεp−1dx

Ωf

ε

TK muε

0

1

Passing to the limit asεtends to 0 in4.45, in view4.44, we have

lim ε→0A

ε

1_εlim_→0

Ωa

ε_{x, u}ε_,∇uε∇TmKuε

bε_uεp−1dx

ΩXK

∇TK

mu

bup−1 dx. 4.46

Limit of

A

In view of4.12,4.17and since 1/bε_uεp−1_{converges to 1/bu}p−1_{a.e. inΩand due to the}

bound 1/bε_uεp−1_≤1/αp−1_{a.e. inΩ, we have}

lim ε→0A

ε

2−

ΩXK

∇TK

mu

bup−1dx. 4.47

Limit of

A

Let us remark that3.1,4.1, and4.11imply that

aε_{x, T}K

muε,∇TmKu

bε_uεp−1 −→

ax, TK

mu, DTmKu

bup−1 a.e.inΩ, 4.48

asεtends to 0, and that fori1, . . . , N

aε_iT_mKuε,∇T_mKu bε_uεp−1

≤w1_i/px

⎡

⎣ 1

αp−1Lx

maxK, mq/p

αp−1 σ

1/p_x N j1

w1_j/p∂T

K mu

∂xj

p−1⎤

⎦

4.49

a.e. inΩ, uniformly with respect toε. It follows that whenεtends to 0

ax, TK

muε,∇TmKu

bε_uεp−1 −→

ax, TK

mu,∇TKu

bup−1 strongly in

N

i1

LpΩ, w1_i−p. 4.50

In view of4.12, we conclude that

∇TK

muε− ∇TmKu

0 weakly in N

i1

LpΩ, wi, asεgoes to 0. 4.51

As a consequence of4.50and4.51we have for allK >0

lim ε→0A

ε

Equations4.46,4.46,4.47, and4.46allow to pass to the limit asεtends to zero in4.31 and to obtain4.30ofLemma 4.2.

Step 4. In this step we identify the weak limitXKand we prove the weakL1convergence of the “truncated” energyaε_{x, T}K

muε,∇TmKuε/bεuεp−1∇TmKuεasεtends to 0. Lemma 4.3. For fixedK≥0, one has

XKa

x, T_mKu,∇T_mKu a.e.in{x∈Ω;ux< m}. 4.53

And asεtends to 0

aε_{x, T}K

muε,∇TmKuε

bε_uεp−1 ∇T

K muε

ax, TK

mu,∇TmKu

bup−1 ∇T

K

muweakly inL1Ω. 4.54

Proof. LetK≥0 be fixed. From4.11and4.50together with4.30ofLemma 4.2, we obtain

lim ε→0

Ω

aε_{x, T}K

muε,∇TmKuε

bTmKuε

p−1 ∇T

K

muεdx

Ω

XK

bTmKu

p−1 ∇TKudx. 4.55

We remark the monotone characterawith respect toξand since 1/bε_uεp−1 _{converges to}

1/bup−1a.e. inΩand due to the bound 1/bεuεp−1≤1/αp−1a.e. inΩ, we conclude that for allψ∈N_i1Lp_{Ω, wiwe have}

0≤lim ε→0

Ω

aε_{x, T}K

muε,∇TmKuε

bε_uεp−1 −

aε_{x, T}K muε, ψ

bε_uεp−1

∇TK

muε−ψ

dx

lim ε→0

Ω

aε_{x, T}K

muε,∇TmKuε

bε_uεp−1

∇TK

muε−ψ

dx

−lim

ε→0

Ω

aε_{x, T}K muε, ψ

bε_uεp−1

∇TK

muε−ψ

dx Ω XK

bup−1

∇TK

mu−ψ

dx−

Ω

ax, TK mu, ψ

bup−1

∇TK

mu−ψ

dx × Ω XK

bup−1 −

ax, TK mu, ψ

bup−1

∇TK

mu−ψ

dx.

4.56

The usual Minty’s argument applies in view of4.56. It follows that

XK

bup−1

ax, TK

mu,∇TmKu

bup−1 a.e.inΩ 4.57

In order to prove4.54, we observe that the monotone character ofawith respect to ξand4.30give

aεx, T_mKuε,∇T_mKuε bε_uεp−1 −

aεx, T_mKuε,∇T_mKu bε_uεp−1

∇TK

muε− ∇TmKu

−→0 4.58

strongly inL1_{Ωasεtends to 0. Moreover4.12,4.17,4.50, and4.53imply that}

aε_{x, T}K

muε,∇TmKuε

bε_uεp−1 ∇T

K mu

ax, TK

mu,∇TmKu

bup−1 ∇T

K

mu 4.59

weakly inL1_{Ωasεtends to 0}

aε_{x, T}K

muε,∇TmKu

bε_uεp−1 ∇T

K muε

ax, TK

mu,∇TmKu

bup−1 ∇T

K

mu 4.60

weakly inL1_{Ωasεtends to 0, and}

aε_{x, T}K

muε,∇TmKu

bε_uεp−1 ∇T K mu

ax, TK

mu,∇TmKu

bup−1 ∇T

K

mu 4.61

strongly inL1_{Ωasεtends to 0.}

Using the above convergence results4.59,4.60, and4.61in4.58we obtain that for anyK≥0

aε_{x, T}K

muε,∇TmKuε

bε_uεp−1 ∇T

K muε

ax, TK

mu,∇TmKu

bup−1 ∇T

K

mu 4.62

weakly inL1_{Ωasεtends to 0.}

Step 5. In this step we prove thatusatisfies3.11. UsingTn1

m uε−Tmnuεas a test function in4.4leads to

Ωa

ε_{x, u}ε_,∇uε_∇Tn1

m uε−Tmnuε

dx

Ωf

ε_Tn1

m uε−Tmnuε

Since suppTn1

m ·−Tmn·⊂−n1,−n, we have

{−n−1≤uε_x≤−n}

aεx, uε,∇uε∇uεdx

Ωa

ε_{x, u}ε_,∇uε_∇Tn1

m uε−Tmnuε

dx

Ω

aε_{x, u}ε_,∇uε

bε_uεp−1 ∇

T_mn1uε−T_mnuεbT_mn₋1₁uεp−1dx

Ω

aε_{x, T}n1

m uε,∇Tmn1uε

bε_uεp−1 ∇T

n1

m uεb

T_mn₋1₁uεp−1dx

−

Ω

aε_{x, T}n

muε,∇Tmnuε

bε_uεp−1 ∇T n muεb

T_mn₋1₁uεp−1dx.

4.64

In view of4.54ofLemma 4.3and sincebT_mn₋1₁uεp−1 converges tobT_mn₋1₁up−1a.e. inΩ and due to the boundbTn1

m−1uε

p−1 _{≤ max}

s∈−n−1,m−1bsp−1 a.e. inΩ, we can pass to the

limit asεtends to 0 for fixedn≥0 to obtain

lim ε→0

{−n−1≤uε_x≤−n}

aεx, uε,∇uε∇uεdx

Ω

ax, Tn1

m u,∇Tmn1u

bup−1 ∇T

n1

m ub

Tn1

m−1u

p−1

dx

−

Ω ax, Tn

mu,∇Tmnu

bup−1 ∇T

n mub

T_mn₋1₁up−1dx

{−n−1≤ux≤−n}ax, u,∇u∇udx.

4.65

Taking the limit asεtends to 0 andntends to∞in4.63and using the estimate4.64and 4.65show that

lim n→∞

{−n−1≤ux≤−n}ax, u,∇u∇udx≤nlim→∞

{u≤−n}

fdx0. 4.66

Step 6. In this step we prove thatusatisfies3.12. UsingSnvε_1/δT

m−δu−Tm−2δuas a test function in4.4leads to

1

δ

ΩSnv

ε_aε_{x, u}ε_,∇uε_∇T

m−δu−Tm−2δu

dx

ΩSnv

ε_fε

T_m−δu−T_m−2δu

δ dx,

wherevε uε

0 bεsds. Since suppSn ⊂ −n1, n1and{x ∈ Ω;|vε| ≤n1} ⊂ {x ∈

Ω;|uε_{| ≤n1/α}we have}

1

δ

ΩSnv

ε_aε_{x, u}ε_,∇uε_∇T

m−δu−Tm−2δu

dx

1

δ

ΩSnv

ε_aε_{x, T}

n1/αuε,∇Tn1/αuε

∇T_m−δu−T_m−2δudx.

4.68

In view of4.22,4.36,4.39, and4.53, passing to the limit asεtends to 0 andntends to ∞

lim n→∞εlim→0

1

δ

ΩSnv

ε_aε_{x, T}

n1/αuε,∇Tn1/αuε∇Tm−δu−Tm−2δu

dx

lim n→∞

1

δ

ΩSnva

x, Tn1/αu,∇Tn1/αu

∇T_m−δu−T_m−2δudx

lim n→∞

1

δ

ΩSnvax, Tm−δu,∇Tm−δu∇

T_m−δu−T_m−2δudx

1

δ

Ωax, Tm−δu,∇Tm−δu∇

T_m−δu−T_m−2δudx

1

δ

{m−2δ≤u≤m−δ}ax, u,∇u∇udx.

4.69

Taking the limit asεtends to 0,ntends to∞andδtends to 0 in4.67and using the estimate 4.68and4.69show that

lim δ→0

1

δ

{m−2δ≤u≤m−δ}

ax, u,∇u∇udx lim δ→0

Ωf

T_m−δu−T_m−2δu

δ dx

{um}fxdx.

4.70

Step 7. In this step,uis shown to satisfy3.13. Letϕ ∈W₀1,pΩ, ω∩L∞Ωand letSbe a function inW1,∞_{Rsuch that Shas a compact support andSm 0. LetKbe a positive} real number such that suppS ⊂ −K, Kandvε uε

0 bεsds. UsingSuSnvεϕas a test

function in4.4leads to

ΩSnv

ε_aε_{x, u}ε_,∇uε_∇Suϕdx

ΩSuϕa

ε_{x, u}ε_,∇uε_∇Snvε_dx

Ωf

ε_Snvε_Suϕdx.

4.71

Limit of First Term in

4.71

Since suppSn⊂−n1, n1and{x∈Ω;|vε| ≤n1} ⊂ {x∈Ω;|uε| ≤n1/α}, we have

Snvεaεx, uε,∇uε Snvεaεx, Tn1/αuε,∇Tn1/αuε

a.e.inΩ. 4.72

In view of4.22,4.36,4.39, and4.53, passing to the limit asεtends to 0

lim ε→0

ΩSnv

ε_aε_{x, u}ε_,∇uε_∇Suϕdx

lim ε→0

ΩSnv

ε_aε_{x, T}

n1/αuε,∇Tn1/αuε∇Suϕdx

ΩSnva

x, Tn1/αu,∇Tn1/αu∇Suϕdx

ΩSnva

x, T_mKu,∇T_mKu∇Suϕdx,

lim n→∞εlim→0

ΩSnv

ε_aε_{x, u}ε_,∇uε_∇Suϕdx

lim n→∞

ΩSnva

x, T_mKu,∇T_mKu∇Suϕdx

Ωa

x, T_mKu,∇T_mKu∇Suϕdx

Ωax, u,∇u∇

Suϕdx.

4.73

Limit of Second Term in

4.71

Since suppS_n⊂−n1,−n∪n1, nfor anyn≥1. As a consequence

ΩSuϕa

ε_{x, u}ε_,∇uε_∇Snvε_{dx≤ S}

L∞_Ωϕ_L∞_Ω

{n≤|vε_|≤n1}

aεx, uε,∇uε∇vεdx.

4.74

Taking the limit asεtends to 0 andntends to∞in4.74and using the estimate4.29show that

lim n→∞εlim→0

_ΩSuϕaεx, uε,∇uε∇Snvεdx0. 4.75

Limit of the Right-Hand Side of

4.71

Due to4.3and4.25, we have

lim n→∞εlim→0

Ωf

ε_Snvε_Suϕdx

As a consequence of the previous convergence results, we are in a position to pass to the limit asεtends to 0 in4.71and to conclude thatusatisfies3.13. The proof of Theorem 4.1is achieved.

Acknowledgment

The author would like to thank the anonymous referees for interesting remarks.

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