URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES GE DONG Abstract

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Electronic Journal of Differential Equations, Vol. 2008(2008), No. 76, pp. 1–10.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

ELLIPTIC EQUATIONS WITH MEASURE DATA IN ORLICZ SPACES

GE DONG

Abstract. This article shows the existence of solutions to the nonlinear elliptic problemA(u) =f in Orlicz-Sobolev spaces with a measure valued right- hand side, whereA(u) =−diva(x, u,∇u) is a Leray-Lions operator defined on a subset ofW₀¹LM(Ω), with generalM.

1. Introduction

LetM :R→Rbe an N-function; i.e. M is continuous, convex, withM(u)>0 for u > 0,M(t)/t → 0 ast →0, and M(t)/t → ∞as t → ∞. Equivalently, M admits the representationM(u) =Ru

0 φ(t)dt, whereφis the derivative ofM, with φnon-decreasing, right continuous,φ(0) = 0, φ(t)>0 fort >0, andφ(t)→ ∞ as t→ ∞.

TheN-function ¯M conjugate to M is defined by ¯M(v) =Rt

0ψ(s)ds, whereψis given byψ(s) = sup{t:φ(t)≤s}.

The N-functionM is said to satisfy the ∆2 condition, if for some k > 0 and u0>0,

M(2u)≤kM(u), ∀u≥u0.

LetP, Q be two N-functions,P Q means thatP grows essentially less rapidly thanQ; i.e. for eachε >0,P(t)/Q(εt)→0 ast→ ∞. This is the case if and only if limt→∞Q⁻¹(t)/P⁻¹(t) = 0.

Let Ω ⊂ R^N be a bounded domain with the segment property. The class W¹LM(Ω) (resp.,W¹EM(Ω)) consists of all functionsusuch thatuand its distributional derivatives up to order 1 lie inL_M(Ω) (resp.,E_M(Ω)).

Orlicz spacesL_M(Ω) are endowed with the Luxemburg norm kuk_(M)= inf

λ >0 : Z

Ω

M |u(x)|

λ

dx≤1 .

The classesW¹LM(Ω) andW¹EM(Ω) of such functions may be given the norm kukΩ,M = X

|α|≤1

kD^αuk_(M₎.

2000Mathematics Subject Classification. 35J15, 35J20, 35J60.

Key words and phrases. Orlicz-Sobolev spaces; elliptic equation; nonlinear; measure data.

c

2008 Texas State University - San Marcos.

Submitted October 8, 2006. Published May 27, 2008.

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These classes will be Banach spaces under this norm. I refer to spaces of the forms W¹LM(Ω) and W¹EM(Ω) as Orlicz-Sobolev spaces. Thus W¹LM(Ω) and W¹EM(Ω) can be identified with subspaces of the product ofN+1 copies ofLM(Ω).

Denoting this product by ΠL_M, we will use the weak topologies σ(ΠL_M,ΠEM¯) and σ(ΠL_M,ΠLM¯). If M satisfies ∆₂ condition, then L_M(Ω) = E_M(Ω) and W¹L_M(Ω) =W¹E_M(Ω).

The space W₀¹E_M(Ω) is defined as the (norm) closure of C₀^∞(Ω) in W¹E_M(Ω) and the spaceW₀¹L_M(Ω) as the σ(ΠL_M,ΠEM¯) closure ofC₀^∞(Ω) inW¹L_M(Ω).

LetW⁻¹LM¯(Ω) (resp. W⁻¹EM¯(Ω)) denote the space of distributions on which can be written as sums of derivatives of order ≤ 1 of functions in LM¯(Ω) (resp.

EM¯(Ω)). It is a Banach space under the usual quotient norm (see [12]).

If the open set Ω has the segment property, then the space C₀^∞(Ω) is dense in W₀¹LM(Ω) for the modular convergence and thus for the topologyσ(ΠLM,ΠLM¯) (cf. [12, 13]).

Let A(u) = −diva(x, u,∇u) be a Leray-Lions operator defined on W^1,p(Ω), 1< p <∞. Boccardo-Gallouet [7] proved the existence of solutions for the Dirichlet problem for equations of the form

A(u) =f in Ω, (1.1)

u= 0 on∂Ω, (1.2)

where the right handf is a bounded Radon measure on Ω (i.e. f ∈ Mb(Ω)). The functionais supposed to satisfy a polynomial growth condition with respect tou and∇u.

Benkirane [4, 5] proved the existence of solutions to

A(u) +g(x, u,∇u) =f, (1.3)

in Orlicz-Sobolev spaces where

A(u) =−div(a(x, u,∇u)) (1.4) is a Leray-Lions operator defined onD(A)⊂W₀¹LM(Ω),g is supposed to satisfy anatural growth condition withf ∈W⁻¹EM¯(Ω) andf ∈L¹(Ω), respectively, but the result is restricted toN-functions M satisfying a ∆2 condition. Elmahi extend the results of [4, 5] to general N-functions (i.e. without assuming a ∆2-condition onM) in [8, 9], respectively.

The purpose of this paper is to solve (1.1) whenf is a bounded Radon measure, and the Leray-Lions operator A in (1.4) is defined on D(A) ⊂ W₀¹L_M(Ω), with generalM. We show that the solutions to (1.1) belong to the Orlicz-Sobolev space W₀¹L_B(Ω) for any B∈ P_M, whereP_M is a special class ofN-function (see below).

Specific examples to which our results apply include the following:

−div(|∇u|^p−2∇u) =µ in Ω,

−div(|∇u|^p−2∇ulog^β(1 +|∇u|)) =µ in Ω

−divM(|∇u|)∇u

|∇u|² =µ inΩ wherep >1 andµis a given Radon measure on Ω.

For some classical and recent results on elliptic and parabolic problems in Orlicz spaces, I refer the reader to [2, 3, 6, 10, 11, 12, 14, 16, 18].

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2. Preliminaries We define a subset ofN-functions as follows.

PM =n

B:R⁺→R⁺ is anN-function,B⁰⁰/B⁰≤M⁰⁰/M⁰ and

Z 1

0

B◦H⁻¹(1/t^1−1/N)dt <∞o whereH(r) =M(r)/r. Assume that

PM 6=∅ (2.1)

Let Ω ⊂ R^N be a bounded domain with the segment property, M, P be two N-functions such that P M, ¯M ,P¯ be the complementary functions of M, P, respectively,A:D(A)⊂W₀¹L_M(Ω)→W⁻¹LM¯(Ω) be a mapping given byA(u) =

−diva(x, u,∇u) wherea: Ω×R×R^N →R^N be a Caratheodory function satisfying for a.e. x∈Ω and alls∈R, ξ, η∈R^N withξ6=η:

|a(x, s, ξ)| ≤βM(|ξ|)/|ξ| (2.2) [a(x, s, ξ)−a(x, s, η)][ξ−η]>0 (2.3)

a(x, s, ξ)ξ≥αM(|ξ|) (2.4)

whereα, β, γ >0.

Furthermore, assume that there existsD∈ PM such that

D◦H⁻¹ is anN-function. (2.5)

Set Tk(s) = max(−k,min(k, s)), ∀s ∈ R, for all k ≥0. Define by Mb(Ω) as the set of all bounded Radon measure defined on Ω and by T₀^1,M(Ω) as the set of measurable functions Ω→Rsuch thatT_k(u)∈W₀¹L_M(Ω)∩D(A).

Assume that f ∈ M_b(Ω), and consider the following nonlinear elliptic problem with Dirichlet boundary

A(u) =f in Ω. (2.6)

The following lemmas can be found in [4].

Lemma 2.1. Let F :R→Rbe uniformly Lipschitzian, withF(0) = 0. Let M be an N-function, u∈W¹LM(Ω) (resp. W¹EM(Ω)). ThenF(u)∈W¹LM(Ω) (resp.

W¹EM(Ω)). Moreover, if the setD of discontinuity points ofF⁰ is finite, then

∂(F◦u)

∂x_i =

(F⁰(u)_∂x^∂u

i a.e. in{x∈Ω :u(x)6∈D}

0 a.e. in{x∈Ω :u(x)∈D}.

Lemma 2.2. Let F :R→R be uniformly Lipschitzian, withF(0) = 0. I suppose that the set of discontinuity points of F⁰ is finite. Let M be an N-function, then the mappingF :W¹LM(Ω)→W¹LM(Ω)is sequentially continuous with respect to the weak∗ topologyσ(ΠLM,ΠEM¯).

3. Existence theorem

Theorem 3.1. Assume that (2.1)-(2.5)hold andf ∈ Mb(Ω). Then there exists at least one weak solution of the problem

u∈T₀^1,M(Ω)∩W₀¹L_B(Ω), ∀B∈ PM

Z

Ω

a(x, u,∇u)∇φdx=hf, φi, ∀φ∈ D(Ω)

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Proof. DenoteV =W₀¹LM(Ω). (1) Consider the approximate equations u_n ∈V

−diva(x, un,∇un) =fn

(3.1) wherefnis a smooth function which converges tof in the distributional sense that such thatkfnk_L1(Ω)≤ kfk_M_b_(Ω). By [4, Theorem 3.1] or [8], there exists at least one solution{un} to (3.1).

Fork >0, by takingTk(un) as test function in (3.1), one has Z

Ω

a(x, T_k(u_n),∇Tk(u_n))∇Tk(u_n)dx≤Ck.

In view of (2.4), we get Z

Ω

M(|∇Tk(un)|)dx≤Ck. (3.2)

Hence ∇Tk(un) is bounded in (LM(Ω))^N. By [9] there exists usuch that un →u almost everywhere in Ω and

Tk(un)* Tk(u) weakly inW₀¹LM(Ω) forσ(ΠLM,ΠEM¯). (3.3) Fort >0, by taking Th(un−Tt(un)) as test function, we deduce that

Z

t<|un|≤t+h

a(x, un,∇un)∇undx≤hkfkM_b(Ω)

which gives

1 h

Z

t<|un|≤t+h

M(|∇un|)dx≤ kfkM_b(Ω)

and by lettingh→0,

−d dt

Z

|un|>t

M(|∇un|)dx≤ kfkM_b(Ω).

Let nowB∈ PM. Following the lines of [17], it is easy to deduce that Z

Ω

B(|∇un|)dx≤C, ∀n. (3.4)

We shall show that a(x, T_k(u_n),∇Tk(u_n)) is bounded in (LM¯(Ω))^N. Let ϕ ∈ (E_M(Ω))^N withkϕk(M)= 1. By (2.2) and Young inequality, one has

Z

Ω

a(x, Tk(un),∇Tk(un))ϕdx≤β Z

Ω

M¯M(|∇Tk(un)|)

|∇Tk(un)|

dx+β

Z

Ω

M(|ϕ|)dx

≤β Z

Ω

M(|∇Tk(un)|)dx+β

This last inequality is deduced from ¯M(M(u)/u) ≤ M(u), for all u > 0, and R

ΩM(|ϕ|)dx≤1. In view of (3.2), Z

Ω

a(x, Tk(un),∇Tk(un))ϕdx≤Ck+β,

which implies{a(x, Tk(un),∇Tk(un))}n being a bounded sequence in (LM¯(Ω))^N. (2) For the rest of this article, χr, χs and χj,s will denoted respectively the characteristic functions of the sets Ω_r = {x ∈ Ω;|∇Tk(u(x))| ≤ r}, Ωs = {x ∈ Ω;|∇Tk(u(x))| ≤ s} and Ω_j,s = {x ∈ Ω;|∇Tk(v_j(x))| ≤ s}. For the sake of

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simplicity, I will write onlyε(n, j, s) to mean all quantities (possibly different) such that

s→∞lim lim

j→∞ lim

n→∞ε(n, j, s) = 0.

Take a sequence (v_j)⊂ D(Ω) which converges toufor the modular convergence in V (cf. [13]). TakingT_η(u_n−T_k(v_j)) as test function in (3.1), we obtain

Z

Ω

a(x, u_n,∇u_n)∇T_η(u_n−T_k(v_j))dx≤Cη (3.5) On the other hand,

Z

Ω

a(x, u_n,∇u_n)∇T_η(u_n−T_k(v_j))dx

= Z

{|un−T_k(vj)|≤η}∩{|un|≤k}

a(x, Tk(un),∇Tk(un))(∇Tk(un)− ∇Tk(vj))dx +

Z

{|un−Tk(vj)|≤η}∩{|un|>k}

a(x, un,∇un)(∇un− ∇Tk(vj))dx

= Z

{|Tkun−Tk(vj)|≤η}

a(x, Tk(un),∇Tk(un))(∇Tk(un)− ∇Tk(vj))dx +

Z

{|un−Tk(v_j)|≤η}∩{|un|>k}

a(x, un,∇un)∇undx

− Z

a(x, un,∇un)∇Tk(vj)dx By (2.4) the second term of the right-hand side satisfies

Z

{|un−T_k(vj)|≤η}∩{|un|>k}

a(x, u_n,∇u_n)∇u_ndx≥0.

Sincea(x, Tk+η(un),∇Tk+η(un)) is bounded in (LM¯(Ω))^N, there exists somehk+η∈ (LM¯(Ω))^N such that

a(x, Tk+η(un),∇Tk+η(un))* hk+η

weakly in (LM¯(Ω))^N forσ(ΠLM¯,ΠEM). Consequently the third term of the right- hand side satisfies

Z

a(x, u_n,∇un)∇Tk(v_j)dx

= Z

a(x, T_k+η(u_n),∇Tk+η(u_n))∇Tk(v_j)dx

= Z

{|u−T_k(v_j)|≤η}∩{|u|>k}

h_k+η∇T_k(v_j)dx+ε(n) since

∇Tk(vj)χ_{|u_n_−T_k_(v_j_{)|≤η}∩{|u}_n_|>k}→ ∇Tk(vj)χ_{|u−T_k_(v_j)|≤η}∩{|u|>k}

strongly in (E_M(Ω))^N asn→ ∞. Hence Z

{|Tkun−Tk(vj)|≤η}

a(x, Tk(un),∇Tk(un))[∇Tk(un)− ∇Tk(vj)]dx

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≤Cη+ε(n) + Z

{|u−Tk(vj)|≤η}∩{|u|>k}

hk+η∇Tk(vj)dx Let 0< θ <1. Define

Φn,k = [a(x, Tk(un),∇Tk(un))−a(x, Tk(un),∇Tk(u))][∇Tk(un)− ∇Tk(u)].

Forr >0, I have 0≤

Z

Ωr

{[a(x, Tk(un),∇Tk(un))−a(x, Tk(un),∇Tk(u))][∇Tk(un)− ∇Tk(u)]}^θdx

= Z

Ω_r

Φ^θ_n,kχ_{|T_k_(u_n_)−T_k_(v_j_)|>η}dx+ Z

Ω_r

Φ^θ_n,kχ_{|T_k_(u_n_)−T_k_(v_j_)|≤η}dx

Using the H¨older Inequality (with exponents 1/θ and 1/(1−θ)), the first term of the right-side hand is less than

Z

Ω_r

Φn,kdx^θZ

Ω_r

χ_{|T_k(u_n)−Tk(v_j)|>η}dx1−θ

. Noting that

Z

Ω_r

Φn,kdx

= Z

Ωr

a(x, Tk(un),∇Tk(un))∇Tk(un)dx− Z

Ωr

a(x, Tk(un),∇Tk(u))∇Tk(un)dx

− Z

Ω_r

a(x, Tk(un),∇Tk(un))∇Tk(u)dx+ Z

Ω_r

a(x, Tk(un),∇Tk(u))∇Tk(u)dx

≤Ck+β Z

Ωr

M¯M(|∇Tk(u)|)

|∇T_k(u)|

dx+β

Z

Ωr

M(|∇Tk(un)|)dx +β

Z

Ω_r

M¯M(|∇T_k(u_n)|)

|∇Tk(un)|

dx+β Z

Ω_r

M(|∇Tk(u)|)dx +β

Z

Ωr

M(|∇Tk(u)|)dx

≤Ck+β Z

Ω_r

M(|∇T_k(u)|)dx+β Z

Ω

M(|∇T_k(u_n)|)dx +β

Z

Ω

M(|∇Tk(un)|)dx+β Z

Ω_r

M(|∇Tk(u)|)dx+β Z

Ω_r

M(|∇Tk(u)|)dx

≤(2β+ 1)Ck+ 3M(r) meas Ω it follows that

Z

Ωr

Φ^θ_n,kχ_{|T_k_(u_n_)−T_k_(v_j_)|>η}dx≤C(meas{|T˜ _k(u_n)−T_k(v_j)|> η})^1−θ, where ˜C= [(2β+ 1)Ck+ 3M(r) meas Ω]^θ.

Using the H¨older Inequality (with exponents 1/θ and 1/(1−θ)), Z

Ω_r

Φ^θ_n,kχ_{|T_k_(u_n_)−T_k_(v_j_)|≤η}dx

≤Z

Ω_r

Φ_n,kχ_{|T_k_(u_n_)−T_k_(v_j_)|≤η}dxθZ

Ω_r

dx1−θ

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≤Z

Ω_r

Φ_n,kχ_{|T_k_(u_n_)−T_k_(v_j_)|≤η}dx^θ

meas Ω1−θ

Hence 0≤

Z

Ω_r

{[a(x, Tk(u_n),∇Tk(u_n))−a(x, T_k(u_n),∇Tk(u))][∇Tk(u_n)− ∇Tk(u)]}^θdx

≤C˜ meas{|Tk(un)−Tk(vj)|> η}^1−θ +Z

Ω_r

Φn,kχ_{|T_k_(u_n_)−T_k_(v_j_)|≤η}dx^θ

meas Ω1−θ

= ˜C meas{|Tk(un)−Tk(vj)|> η}1−θ

+Z

Ω_r∩{|Tk(u_n)−Tk(v_j)|≤η}

a(x, Tk(un),∇Tk(un))−a(x, Tk(un),∇Tk(u))

×

∇Tk(un)− ∇Tk(u) dx^θ

meas Ω1−θ

For eachs≥r one has 0≤

Z

×

∇Tk(un)− ∇Tk(u) dx

≤ Z

Ω_s∩{|T_k(u_n)−T_k(v_j)|≤η}

a(x, T_k(u_n),∇Tk(u_n))−a(x, T_k(u_n),∇Tk(u))

×

∇Tk(un)− ∇Tk(u)]dx

= Z

Ωs∩{|Tk(un)−Tk(vj)|≤η}

a(x, Tk(un),∇Tk(un))−a(x, Tk(un),∇Tk(u)χs)

×

∇T_k(u_n)− ∇T_k(u)χ_s dx

≤ Z

Ω∩{|Tk(u_n)−Tk(v_j)|≤η}

a(x, Tk(un),∇Tk(un))−a(x, Tk(un),∇Tk(u)χs)

×

∇Tk(un)− ∇Tk(u)χs]dx

= Z

{|Tk(u_n)−Tk(v_j)|≤η}

a(x, T_k(u_n),∇Tk(u_n))−a(x, T_k(u_n),∇Tk(v_j)χ_j,s)

×

∇Tk(un)− ∇Tk(vj)χj,s

dx +

Z

{|Tk(un)−Tk(vj)|≤η}

a(x, Tk(un),∇Tk(un))

∇Tk(vj)χj,s− ∇Tk(u)χs dx

+ Z

a(x, Tk(un),∇Tk(vj)χj,s)

−a(x, T_k(u_n),∇T_k(u)χ_s)

∇T_k(u_n)dx

− Z

a(x, Tk(un),∇Tk(vj)χj,s)∇Tk(vj)χj,sdx +

Z

a(x, Tk(un),∇Tk(u)χs)∇Tk(u)χsdx

=I₁(n, j, s) +I₂(n, j, s) +I₃(n, j, s) +I₄(n, j, s) +I₅(n, j, s)

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On the other hand, Z

a(x, Tk(un),∇Tk(un))[∇Tk(un)− ∇Tk(vj)]dx

= Z

a(x, Tk(un),∇Tk(un))−a(x, Tk(un),∇Tk(vj)χj,s)

×

∇Tk(un)− ∇Tk(vj)χj,s dx +

Z

a(x, T_k(u_n),∇Tk(v_j)χ_j,s)

∇Tk(u_n)− ∇Tk(v_j)χ_j,s dx

− Z

{|T_k(u_n)−T_k(v_j)|≤η}

a(x, T_k(u_n),∇T_k(u_n))∇T_k(v_j)χ_{|∇T_k_(v_j_)|>s}dx The second term of the right-hand side tends to

Z

{|Tk(u)−Tk(v_j)|≤η}

a(x, T_k(u),∇Tk(u)χ_s)[∇Tk(u)− ∇Tk(v_j)χ_s]dx sincea(x, Tk(un),∇Tk(u)χs)χ_{|T_k_(u_n_)−T_k_(v_j_)|≤η} tends to

a(x, Tk(u),∇Tk(u)χs)χ_{|T_k_(u)−T_k_(v_j_)|≤η}

in (EM¯(Ω))^N while∇Tk(un)− ∇Tk(vj)χs tends weakly to∇Tk(u)− ∇Tk(vj)χs in (L_M(Ω))^N forσ(ΠL_M,ΠEM¯).

Since a(x, T_k(u_n),∇Tk(u_n)) is bounded in (LM¯(Ω))^N there exists some h_k ∈ (LM¯(Ω))^N such that (for a subsequence still denoted byu_n)

a(x, T_k(u_n),∇T_k(u_n))* h_k weakly in (LM¯(Ω))^N forσ(ΠLM¯,ΠE_M).

In view of the fact that∇Tk(vj)χ_{|T_k_(u_n_)−T_k_(v_j_)|≤η}→ ∇Tk(vj)χ_{|T_k_(u)−T_k_(v_j_)|≤η}

strongly in (E_M(Ω))^N asn→ ∞the third term of the right-hand side tends to

− Z

h_k∇Tk(v_j)χ_{|∇T_k_(v_j_)|>s}dx.

Hence in view of the modular convergence of (v_j) in V, one has I1(n, j, s)≤Cη+ε(n) +

Z

{|u−Tk(v_j)|≤η}∩{|u|>k}

hk+η∇Tk(vj)dx +

Z

hk∇Tk(vj)χ_{|∇T_k(v_j)|>s}dx

− Z

a(x, Tk(u),∇Tk(u)χs)[∇Tk(u)− ∇Tk(vj)χs]dx

=Cη+ε(n) +ε(j) + Z

Ω

h_k∇Tk(u)χ_{|∇T_k_(u)|>s}dx

− Z

Ω

a(x, T_k(u),0)χ_{|∇T_k_(u)|>s}dx Therefore,

I₁(n, j, s) =Cη+ε(n, j, s) (3.6) For what concernsI2, by lettingn→ ∞, one has

I2(n, j, s) = Z

{|Tk(u)−Tk(vj)|≤η}

hk[∇Tk(vj)χj,s− ∇Tk(u)χs]dx+ε(n)

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since

a(x, Tk(un),∇Tk(un))* hk weakly in (LM¯)^N forσ(ΠLM¯,ΠEM) whileχ_{|T_k_(u_n_)−T_k_(v_j_)|≤η}[∇Tk(vj)χj,s− ∇Tk(u)χs] approaches

χ_{|T_k_(u)−T_k_(v_j_)|≤η}[∇Tk(v_j)χ_j,s− ∇Tk(u)χ_s]

strongly in (E_M)^N. By lettingj→ ∞, and using Lebesgue theorem, then

I₂(n, j, s) =ε(n, j). (3.7)

Similar tools as above, give I₃(n, j, s) =−

Z

Ω

a(x, T_k(u),∇Tk(u)χ_s)∇Tk(u)χ_sdx+ε(n, j) (3.8) Combining (3.6), (3.7), and (3.8), we have

Z

×

∇Tk(un)− ∇Tk(u) dx

≤ε(n, j, s).

Therefore, 0≤

Z

Ω_r

{[a(x, Tk(u_n),∇Tk(u_n))−a(x, T_k(u_n),∇Tk(u))][∇Tk(u_n)− ∇Tk(u)]}^θdx

≤C(meas{|T˜ _k(u_n)−T_k(v_j)|> η})^1−θ+ (meas Ω)^1−θ(ε(n, j, s))^θ Which yields, by passing to the limit superior overn, j, s andη,

n→∞lim Z

Ω_r

a(x, T_k(u_n),∇Tk(u_n))−a(x, T_k(u_n),∇Tk(u))

×

∇Tk(un)− ∇Tk(u) ^θdx= 0.

Thus, passing to a subsequence if necessary, ∇un → ∇ua.e. in Ωr, and sincer is arbitrary,

∇un→ ∇u a.e. in Ω.

By (2.2) and (2.5), Z

Ω

D◦H⁻¹|a(x, u_n,∇u_n)|

β

dx≤ Z

Ω

D(|∇un|)dx≤C

Hence

a(x, un,∇un)* a(x, u,∇u) weakly for σ(ΠL_D◦H−1ΠE_D◦H−1).

Going back to approximate equations (3.1), and usingφ∈ D(Ω) as the test function, one has

Z

Ω

a(x, un,∇un)∇φdx=hfn, φi

in which I can pass to the limit. This completes the proof.

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Ge Dong

1. Department of Mathematics, Shanghai University, No. 99, Shangda Rd., Shanghai, China

2. Department of Basic, Jianqiao College, No. 1500, Kangqiao Rd., Shanghai, China E-mail address:[email protected]