Existence of solutions of degenerated unilateral problems with <span class="mathjax-formula">$L^1$</span> data

BLAISE PASCAL

Lahsen Aharouch, Youssef Akdim

Existence of solutions of degenerated unilateral problems with L

data

Volume 11, n^o1 (2004), p. 47-66.

<http://ambp.cedram.org/item?id=AMBP_2004__11_1_47_0>

©Annales mathématiques Blaise Pascal, 2004, tous droits réservés.

L’accès aux articles de la revue « Annales mathématiques Blaise Pascal » (http://ambp.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://ambp.cedram.org/legal/). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copy- right.

Publication éditée par le laboratoire de mathématiques de l’université Blaise-Pascal, UMR 6620 du CNRS

Clermont-Ferrand — France

cedram

Article mis en ligne dans le cadre du

Centre de diffusion des revues académiques de mathématiques

Existence of solutions of degenerated unilateral problems with L

data

Lahsen Aharouch Youssef Akdim

Abstract

In this paper, we shall be concerned with the existence result of the Degenerated unilateral problem associated to the equation of the type

Au+g(x, u,∇u) =f−divF,

where A is a Leray-Lions operator and g is a Carathéodory func- tion having natural growth with respect to |∇u| and satisfying the sign condition. The second term is such that, f ∈ L¹(Ω) and F ∈ Π^N_i=1L^p0(Ω, w^1−p_i ⁰).

1 Introduction

LetΩbe a bounded open set ofR^N,pbe a real number such that1< p <∞ and w = {w_i(x), 0 ≤ i ≤ N} be a vector of weight functions on Ω, i.e.

each w_i(x) is a measurable a.e. strictly positive function on Ω, satisfying some integrability conditions (see section 2). Now we consider the obstacle problem associated to the following differential equations

Au+g(x, u,∇u) =f−divF. (1.1) WhereAis a Leray-Lions operator fromW₀^1,p(Ω, w)into its dualW^−1,p0(Ω, w^∗) defined by Au = −diva(x, u,∇u) and where g is a nonlinear lower order term having natural growth (order p) with respect to |∇u|, with respect to |u|, we do not assume any growth restrictions, but we assume only the

”sign-condition"

g(x, s, ξ)s≥0.

As regards the second member, we suppose thatf ∈L¹(Ω)and that F ∈Π^N_i=1L^p0(Ω, w_i^1−p0).

In the case where F = 0, an existence theorem has been proved in [2] with f ∈W^−1,p0(Ω, w), and in [3] with f ∈ L¹(Ω) and where the nonlinearity g satisfies further the following coercivity condition

|g(x, s, ξ)| ≥β

N

X

i=1

w_i|ξ_i|^p for |s|> γ. (1.2) Our purpose, in this paper, is to prove an existence result for degenerated unilateral problems associated to(1.1)in the case whereF 6= 0and without assuming the coercivity condition(1.2). So that, we generalize The previous results given in [3].

Let us point out that another work in theL^pcase can be found in[6, 12]in the case of equation, and in [11] in the case of obstacle problems.

This paper is organized as follows, sections2containe some preliminaries and some technical lemmas, section 3 is concerned with main results and basic assumptions, in section 4, we prove main results and we study the stability and the positivity of solution.

2 Preliminaries

Let Ω be a bounded open subset of R^N(N ≥ 1). Let 1 < p < ∞, and let w={w_i(x);i= 1, ..., N},

0 ≤i ≤N be a vector of weight functions i.e. every component w_i(x) is a measurable function which is strictly positive a.e. inΩ. Further, we suppose in all our considerations that for0≤i≤N

w_i∈L¹_loc(Ω) and w⁻

1 p−1

i ∈L¹_loc(Ω). (2.1)

We define the weighted space with weightγ in Ω as L^p(Ω, γ) ={u(x) :uγ^1p ∈L^p(Ω)}, which is endowed with, we define the norm

kuk_p,γ = ( Z

Ω

|u(x)|^pγ(x)dx)^1p.

We denote byW^1,p(Ω, w)the space of all real-valued functionsu∈L^p(Ω, w₀) such that the derivatives in the sense of distributions satisfy

∂u

∂x_i ∈L^p(Ω, w_i) for all i= 1, ..., N.

This set of functions forms a Banach space under the norm kuk_1,p,w=

Z

Ω

|u(x)|^pw₀dx+

N

X

i=1

Z

Ω

|∂u

∂x_i|^pw_i(x)dx

!¹_p

. (2.2) To deal with the Dirichlet problem, we use the space

X =W₀^1,p(Ω, w),

defined as the closure ofC₀^∞(Ω)with respect to the norm (2.2). Note that, C₀^∞(Ω)is dense in W₀^1,p(Ω, w) and (X,k.k_1,p,w)is a reflexive Banach space.

We recall that the dual space of the weighted Sobolev spaces W₀^1,p(Ω, w) is equivalent toW^−1,p0(Ω, w^∗), wherew^∗= {w_i^∗= w_i^1−p0}, i = 1, ..., N and p⁰ is the conjugate ofp i.e. p⁰ = _p−1^p . For more details we refer the reader to [10].

We now introduce the functional spaces we will need later.

For p ∈(1,∞), τ₀^1,p(Ω, w) is defined as the set of measurable functions u : Ω→Rsuch that fork >0the truncated functions T_k(u)∈W₀^1,p(Ω, w).

We gives the following lemma this is a generalization of Lemma 2.1 [4] in weighted spaces.

Lemma 2.1: For every u ∈ τ₀^1,p(Ω, w), there exists a unique measurable function v: Ω→Rsuch that

∇T_k(u) =vχ_{|v|<k}.

Lemma 2.2: Letλ∈Rand letuandv be two measurable functions defined on Ω which are finite almost everywhere, and which are such that T_k(u), T_k(v) and T_k(u+λv) belong toW₀^1,p(Ω, w) for every k >0then

∇(u+λv) =∇(u) +λ∇(v)a.e. in Ω

where∇(u),∇(v)and ∇(u+λv)are the gradients ofu, vand u+λv intro- duced in Lemma 2.1.

The proof of this lemma is similar to the proof of Lemma 2.12 [9] for the non weighted case.

Now, we state the following assumptions.

(H₁)-The expression

kuk_X =

N

X

i=1

Z

Ω

|∂u

∂x_i|^pw_i(x)dx

!¹_p

, (2.3)

is a norm defined on X and is equivalent to the norm (2.2). (Note that (X,kuk_X)is a uniformly convex (and reflexive) Banach space.

-There exist a weight functionσ on Ω and a parameterq, 1< q <∞, such that

1< q < p+p⁰ (2.4) and

σ^1−q0 ∈L¹_loc(Ω). (2.5)

withq⁰= _q−1^q and such that the Hardy inequality Z

Ω

|u|^qσ(x)dx ¹_q

≤C

N

X

i=1

Z

Ω

|∂u

∂x_i|^pw_i(x)dx

!¹_p

, (2.6)

holds for every u ∈X with a constant C > 0 independent of u. Moreover, the imbeding

X ,→L^q(Ω, σ) (2.7)

determined by the inequality(2.6) is compact.

Now, we state the following technical lemmas which are needed later.

Lemma 2.3:[1] Letg∈L^r(Ω, γ)and letg_n∈L^r(Ω, γ), with kg_nk_Ω,r ≤c,1<

r <∞. If g_n(x)→g(x) a.e. in Ω, then g_n * g weakly in L^r(Ω, γ).

Lemma 2.4:[1]: Assume that (H₁) holds. Let F : R → R be unifomly Lipschitzian, with F(0) = 0. Let u ∈W₀^1,p(Ω, w). Then F(u)∈ W₀^1,p(Ω, w).

Moreover, if the setD of discontinuity points of F⁰ is finite, then

∂(F ◦u)

∂x_i =

F⁰(u)_∂x^∂u

i a.e. in {x∈Ω :u(x)∈/D}

0 a.e. in {x∈Ω :u(x)∈D}.

The previous lemma, we deduce the following.

Lemma 2.5:[1]: Assume that (H₁) holds. Let u ∈ W₀^1,p(Ω, w), and let T_k(u), k ∈R⁺, be the usual truncation then T_k(u) ∈ W₀^1,p(Ω, w). Moreover, we have

T_k(u)→u strongly in W₀^1,p(Ω, w).

3 Main results

Let A be a nonlinear operator from W₀^1,p(Ω, w) into its dual W^−1,p0(Ω, w^∗) defined as

Au=−div(a(x, u,∇u)),

where a : Ω×R×R^N → R^N is a Carathéodory function satisfying the following assumptions:

(H₂)

|a_i(x, s, ξ)| ≤w

1 p

i (x)[k(x) +σ^p10|s|^pq0 +

N

X

j=1

w

1 p0

j (x)|ξ_j|^p−1] for i= 1, ..., N (3.1) [a(x, s, ξ)−a(x, s, η)](ξ−η)>0 for all ξ6=η∈R^N, (3.2)

a(x, s, ξ)ξ≥α

N

X

i=1

w_i(x)|ξ_i|^p, (3.3) wherek(x) is a positive function inL^p0(Ω)and αis a positive constants.

(H₃) g(x, s, ξ) is a Carathéodory function satisfying

g(x, s, ξ).s≥0 (3.4)

|g(x, s, ξ)| ≤b(|s|)(

N

X

i=1

w_i(x)|ξ_i|^p+c(x)), (3.5) whereb:R⁺→R⁺is a nonnegative increasing function andc(x)is a positive function which inL¹(Ω).

Let

K_ψ ={u∈W₀^1,p(Ω, w)/ u ≥ψ a.e. in Ω},

where ψ: Ω→Ris a measurable function on Ωsuch that

ψ⁺∈W₀^1,p(Ω, w)∩L^∞(Ω). (3.6) Finally, we assume that

f ∈L¹(Ω), (3.7)

and

F ∈Π^N_i=1L^p0(Ω, w_i^1−p0). (3.8) We defined, fors and k inR,k ≥0, T_k(s) =max(−k, min(k, s)).

For the nonlinear Dirichlet boundary value problem(1.1), we state our main result as follows.

Theorem 3.1:: Assume that the assumption (H₁)−(H₃) and (3.6)−(3.8) hold, then, there exists at least one solution of(1.1)in the following sense:

(P)



















u≥ψ, g(x, u,∇u)∈L¹(Ω) T_k(u)∈W₀^1,p(Ω, w),

Z

Ω

a(x, u,∇u)∇T_k(u−v)dx+ Z

Ω

g(x, u,∇u)T_k(u−v)dx

≤ Z

Ω

f T_k(u−v)dx+ Z

Ω

F∇T_k(u−v)dx

∀ v∈K_ψ∩L^∞(Ω) ∀k >0.

Remarks 3.2:

1. We obtain the same results of our theorem if we suppose that the signe condition (3.4) is only near infinity.

2. The statement of Theorem 3.1 generalizes in weighted case the analo- gous one in [12] and [11].

4 Proof of main results

We recall the following lemma wich play an important rôle in the proof of our main result,

Lemma 4.1:[1] Assume that (H₁) and (H₂) are satisfied, and let (u_n) be a sequence inW₀^1,p(Ω, w) such thatu_n* u weakly in W₀^1,p(Ω, w) and

Z

Ω

[a(x, u_n,∇u_n)−a(x, u_n,∇u)]∇(u_n−u)dx→0

then, u_n→u inW₀^1,p(Ω, w).

Proof of Theorem 3.1

For the prove of the existence theorem we proceed by steps.

Step 1. A priori estimates

Let us defined the following sequence of the unilaterals problems















u_n ∈K_ψ, g(x, u_n,∇u_n)∈L¹(Ω), g(x, u_n,∇u_n)u_n∈L¹(Ω) hAu_n, u_n−vi+

Z

Ω

g(x, u_n,∇u_n)(u_n−v)dx

≤ Z

Ω

f_n(u_n−v)dx+ Z

Ω

F_n∇(u_n−v)dx,

∀ v∈K_ψ∩L^∞(Ω).

(4.1)

wheref_nandF_n are a regular functions such thatf_nstrongly converges tof inL¹(Ω) andF_n strongly converges toF in Π^N_i=1L^p0(Ω, w^1−p_i ⁰). By Theorem 3.1of [2], there exists at least one solution of (4.1).

Taking v ∈K_ψ and choosing h≥ kψ⁺k∞ so asw˜ =T_h(u_n−T_k(u_n−v))∈ K_ψ∩L^∞(Ω). The choice ofw˜as a test function in(4.1)and lettingh→+∞, we obtain

(P_n)











hAu_n, T_k(u_n−v)i+ Z

Ω

g(x, u_n,∇u_n)T_k(u_n−v)dx

≤ Z

Ω

f_nT_k(u_n−v)dx+ Z

Ω

F_n∇T_k(u_n−v)dx,

∀v∈K_ψ.

The use ofv=ψ⁺ as test function in(P_n) gives Z

Ω

a(x, u_n,∇u_n)∇T_k(u_n−ψ⁺)dx+ Z

Ω

g(x, u_n,∇u_n)T_k(u_n−ψ⁺)dx

≤ Z

Ω

f_nT_k(u_n−ψ⁺)dx+ Z

Ω

F_n∇T_k(u_n−ψ⁺)dx,

since g(x, u_n,∇u_n)T_k(u_n−ψ⁺)≥0, then Z

{|un−ψ⁺|≤k}

a(x, u_n,∇u_n)∇T_k(u_n−ψ⁺)dx

≤Ck+ Z

{|un−ψ⁺|≤k}

F_n∇T_k(u_n−ψ⁺)dx by Young’s inequality and(3.3), one easily has

α 2 Z

Ω N

X

i=1

|∂T_k(u_n)

∂x_i |^pw_i(x)dx≤c₁k ∀k >1. (4.2) Now, as in [5], we prove that u_n converges to some function u locally in measure (and therefore, we can aloways assume that the convergence is a.e.

after passing to a suitable subsequence). We shall show thatu_n is a Cauchy sequence in measure in any ballB_R.

Letk >0large enough, we have k meas({|u_n|> k} ∩B_R) =

Z

{|un|>k}∩BR

|T_k(u_n)|dx≤ Z

BR

|T_k(u_n)|dx

≤ Z

Ω

|T_k(u_n)|^qσ dx ¹_qZ

BR

σ^1−q0dx _q0¹

≤ c_R Z

Ω N

X

i=1

|∂T_k(u_n)

∂x_i |^pw_i(x)dx

!¹_p

≤ c₁k^1p which implies

meas({|u_n|> k} ∩B_R)≤ c₁ k^1−1p

∀k >1. (4.3) We have, for everyδ >0,

meas({|u_n−u_m|> δ} ∩B_R)≤meas({|u_n|> k} ∩B_R)

+meas({|u_m|> k} ∩B_R) +meas{|T_k(u_n)−T_k(u_m)|> δ}. (4.4) Since T_k(u_n) is bounded inW₀^1,p(Ω, w), there exists some v_k ∈ W₀^1,p(Ω, w), such that

T_k(u_n)* v_k weakly in W₀^1,p(Ω, w)

T_k(u_n)→v_k strongly in L^q(Ω, σ) and a.e. in Ω.

Consequently, we can assume that T_k(u_n) is a Cauchy sequence in measure inΩ.

Let ε > 0, then, by (4.3) and (4.4), there exists some k(ε) > 0 such that meas({|u_n−u_m|> δ} ∩B_R)< ε for all n, m≥n₀(k(ε), δ, R). This proves that (u_n) is a Cauchy sequence in measure in B_R, thus converges almost everywhere to some measurable functionu. Then

T_k(u_n)* T_k(u) weakly in W₀^1,p(Ω, w),

T_k(u_n)→T_k(u) strongly in L^q(Ω, σ)and a.e. inΩ.

Which implies, by using (3.1), for all k > 0 there exists a function h_k ∈ Π^N_i=1L^p0(Ω, w_i), such that

a(x, T_k(u_n),∇T_k(u_n))* h_k weakly in Π^N_i=1L^p0(Ω, w_i). (4.5) Step 2. Strong convergence of truncation

Let k > 0 large enough such that k > kψ⁺k_∞, we consider the function φ(t) = te^γt2, with γ >(^b(k)

2α )² (this function is introduce by [7, 8]). Thanks to Lemma 1 of [8], we have the following inequality

φ⁰(s)−b(k)

α |φ(s)| ≥ 1

2 (4.6)

hold for alls∈R.

Here, we definew_n =T_2k(u_n−T_h(u_n) +T_k(u_n)−T_k(u)) where h >2k >0.

Forη= exp(−4γk²), we define the following function as

v_n,h=u_n−ηφ(w_n). (4.7)

The use ofv_n,h as test function in(P_n), we obtain, for all l >0 hA(u_n), T_l(ηφ(w_n))i+

Z

Ω

g(x, u_n,∇u_n)T_l(ηφ(w_n))dx

≤ Z

Ω

f_nT_l(ηφ(w_n))dx+ Z

Ω

F_n∇T_l(ηφ(w_n))dx, we take alsollarge enough, we have

hA(u_n), φ(w_n)i+ Z

Ω

g(x, u_n,∇u_n)φ(w_n)dx

≤ Z

Ω

f_nφ(w_n)dx+ Z

Ω

F_n∇φ(w_n)dx.

(4.8)

Note that,∇w_n= 0on the set where{|u_n|> h+ 4k}, therefore, settingM = 4k+h, and denoting by ε¹_h(n), ε²_h(n), ... various sequences of real numbers which converge to zero asntends to infinity for any fixed value ofh, we get, by(4.8),

Z

Ω

a(x, T_M(u_n),∇T_M(u_n))∇w_nφ⁰(w_n)dx+ Z

Ω

g(x, u_n,∇u_n)φ(w_n)dx

≤ Z

Ω

f_nφ(w_n)dx+ Z

Ω

F_n∇w_nφ⁰(w_n)dx.

(4.9) sinceφ(w_n)g(x, u_n,∇u_n)≥0on the subset{x∈Ω :|u_n(x)|> k}, we deduce from(4.9)that

Z

Ω

a(x, T_M(u_n),∇T_M(u_n))∇w_nφ⁰(w_n)dx+ Z

{|un|≤k}

g(x, u_n,∇u_n)φ(w_n)dx

≤ Z

Ω

f_nφ(w_n)dx+ Z

Ω

F_n∇w_nφ⁰(w_n)dx.

(4.10) Splitting the first integral on the left hand side of(4.10) where|u_n| ≤k and

|u_n|> k, we can write, by using(3.3):

Z

Ω

a(x, T_M(u_n),∇T_M(u_n))∇w_nφ⁰(w_n)dx

≥ Z

Ω

a(x, T_k(u_n),∇T_k(u_n))[∇T_k(u_n)− ∇T_k(u)]φ⁰(w_n)dx

−C_k Z

{|un|>k}

|a(x, T_M(u_n),∇T_M(u_n))||∇T_k(u)|dx,

(4.11) where C_k=φ⁰(2k). Since, whenn tends to infinity, we have

for all i = 1, ..., N,^∂(Tk(u))

∂xi χ_{|un|>k} tends to 0 strongly in L^p(Ω, w_i) while, (a_i(x, T_M(u_n),∇T_M(u_n)))_n is bounded in L^p0(Ω, w_i^1−p0) hence the last term in the previous inequality tends to zero for everyhfixed asntends to infinity.

Now, observe that Z

Ω

a(x, T_k(u_n),∇T_k(u_n))[∇T_k(u_n)− ∇T_k(u)]φ⁰(w_n)dx

= Z

Ω

[a(x, T_k(u_n),∇T_k(u_n))−a(x, T_k(u_n),∇T_k(u))]

×[∇T_k(u_n)− ∇T_k(u)]φ⁰(w_n)dx +

Z

Ω

a(x, T_k(u_n),∇T_k(u))[∇T_k(u_n)− ∇T_k(u)]φ⁰(w_n)dx.

(4.12) By the continuity of the Nymetskii operator, we have for alli= 1, ..., N

a_i(x, T_k(u_n),∇T_k(u))φ⁰(w_n))→a_i(x, T_k(u),∇T_k(u))φ⁰(T_2k(u−T_h(u))

strongly in L^p0(Ω, w_i^1−p0) and since ^∂(T_∂x^k(un))

i * ^∂(T_∂x^k(u))

i weakly in L^p(Ω, w_i), the second term of the right hand side of(4.12)tends to0 asn→ ∞.

So that(4.11) yields Z

Ω

a(x, T_M(u_n),∇T_M(u_n))[∇T_k(u_n)− ∇T_k(u)]φ⁰(w_n)dx

≥ Z

Ω

[a(x, T_k(u_n),∇T_k(u_n))−a(x, T_k(u_n),∇T_k(u))]

×[∇T_k(u_n)− ∇T_k(u)]φ⁰(w_n)dx+ε²_h(n).

(4.13) For the second term of the left hand side of(4.10), we can estimate as follows

Z

{|un|≤k}

g(x, u_n,∇u_n)φ(w_n)dx

≤ Z

{|un|≤k}

b(k)(c(x) +

N

X

i=1

w_i|∂T_k(u_n)

∂x_i |^p|φ(w_n)|dx

≤b(k) Z

Ω

c(x)|φ(w_n)|dx +^b(k)

α

Z

Ω

a(x, T_k(u_n),∇T_k(u_n))∇T_k(u_n)|φ(w_n)|dx, (4.14)

remark that, we have Z

Ω

a(x, T_k(u_n),∇T_k(u_n))∇T_k(u_n)|φ(w_n)|dx

= Z

Ω

[a(x, T_k(u_n),∇T_k(u_n))−a(x, T_k(u_n),∇T_k(u))]

×[∇T_k(u_n)− ∇T_k(u)]|φ(w_n)|dx +

Z

Ω

a(x, T_k(u_n),∇T_k(u_n))∇T_k(u)|φ(w_n)|dx +

Z

Ω

a(x, T_k(u_n),∇T_k(u))[∇T_k(u_n)− ∇T_k(u)]|φ(w_n)|dx.

(4.15) By the Lebesgue’s Theorem, we have

∇T_k(u)|φ(w_n)| → ∇T_k(u)|φ(T_2k(u−T_h(u)))| strongly in Π^N_i=1L^p(Ω, w_i).

Moreover, in view of (4.5) the second term of the right hand side of (4.15) tends to

Z

Ω

h_k∇T_k(u)|φ(T_2k(u−T_h(u)))|dx.

The third term of the right hand side of (4.15) tends to 0 since for all i= 1, ..., N

a_i(x, T_k(u_n),∇T_k(u))|φ(w_n)| →a_i(x, T_k(u),∇T_k(u))|φ(T_2k(u−T_h(u)))|strongly inL^p0(Ω, w_i^1−p0), while

∂(T_k(u_n))

∂x_i * ∂(T_k(u))

∂x_i weakly in L^p(Ω, w_i).

From(4.14)and (4.15), we obtain Z

{|un|≤k}

g(x, u_n,∇u_n)φ(w_n)dx

≤ Z

Ω

[a(x, T_k(u_n),∇T_k(u_n))−a(x, T_k(u_n),∇T_k(u))]

×[∇T_k(u_n)− ∇T_k(u)]|φ(w_n)|dx +ε³_h(n) +

Z

Ω

h_k∇T_k(u)|φ(T_2k(u−T_h(u)))|dx +b(k)

Z

Ω

c(x)|φ(w_n)|dx.

(4.16)

Now, by the strongly convergence off_n and F_n and in fact that

w_n* T_2k(u−T_h(u)) weakly in W₀^1,p(Ω, w) and weakly-∗ in L^∞(Ω), (4.17) moreover, combining(4.13)and(4.16), we conclude that

Z

Ω

[a(x, T_k(u_n),∇T_k(u_n))−a(x, T_k(u_n),∇T_k(u))]

×[∇T_k(u_n)− ∇T_k(u)](φ⁰(w_n)−^b(k)_α |φ(w_n)|)dx

≤ Z

Ω

h_k∇T_k(u)|φ(T_2k(u−T_h(u)))|dx+ε⁵_h(n) +b(k)

Z

Ω

c(x)φ(T_2k(u−T_h(u)))dx+ Z

Ω

f φ(T_2k(u−T_h(u)))dx +

Z

Ω

F∇T_2k(u−T_h(u))φ⁰(T_2k(u−T_h(u)))dx.

which and using(4.6), implies that Z

Ω

[a(x, T_k(u_n),∇T_k(u_n))−a(x, T_k(u_n),∇T_k(u))][∇T_k(u_n)− ∇T_k(u)]dx

≤2 Z

Ω

h_k∇T_k(u)|φ(T_2k(u−T_h(u)))|dx+ε⁵_h(n) +2b(k)

Z

Ω

c(x)φ(T_2k(u−T_h(u)))dx+ 2 Z

Ω

f φ(T_2k(u−T_h(u)))dx +2

Z

Ω

F∇T_2k(u−T_h(u))φ⁰(T_2k(u−T_h(u)))dx.

Hence, passing to the limit overn, we get lim sup

n→∞

Z

Ω

[a(x, T_k(u_n),∇T_k(u_n))−a(x, T_k(u_n),∇T_k(u))][∇T_k(u_n)−∇T_k(u)]dx

≤2 Z

Ω

h_k∇T_k(u)|φ(T_2k(u−T_h(u)))|dx +2b(k)

Z

Ω

c(x)φ(T_2k(u−T_h(u)))dx+ 2 Z

Ω

f φ(T_2k(u−T_h(u)))dx +2

Z

Ω

F∇T_2k(u−T_h(u))φ⁰(T_2k(u−T_h(u)))dx.

(4.18) It remains to show, for our purposes, that the all term on the right hand side of(4.18) converge to zero ashgoes to infinity. The only difficulty that exists is in the last term. For the other terms it suffices to apply Lebesgue’s

theorem.

We deal now with this term. Let us observe that, if we takeu_n−ηφ(T_2k(u_n− T_h(u_n))) as test function in(P_n), we obtain by using(3.3):

α Z

{h≤|un|≤2k+h}

N

X

i=1

|∂u_n

∂x_i|^pw_iφ⁰(T_2k(u_n−T_h(u_n)))dx +

Z

Ω

g(x, u_n,∇u_n)φ(T_2k(u_n−T_h(u_n)))dx

≤ Z

{h≤|un|≤2k+h}

F_n∇u_nφ⁰(T_2k(u_n−T_h(u_n)))dx +

Z

Ω

f_nφ(T_2k(u_n−T_h(u_n)))dx.

Sinceg(x, u_n,∇u_n)φ(T_2k(u_n−T_h(u_n)))≥0, We have α

Z

{h≤|un|≤2k+h}

N

X

i=1

|∂u_n

∂x_i|^pw_iφ⁰(T_2k(u_n−T_h(u_n)))dx

≤ Z

{h≤|un|≤2k+h}

F_n∇u_nφ⁰(T_2k(u_n−T_h(u_n)))dx +

Z

Ω

f_nφ(T_2k(u_n−T_h(u_n)))dx, which yields, thanks to Young’s inequalities

α 2

Z

{h≤|un|≤2k+h}

N

X

i=1

|∂u_n

∂x_i|^pw_iφ⁰(T_2k(u_n−T_h(u_n)))dx

≤ Z

Ω

f_nφ(T_2k(u_n−T_h(u_n)))dx+c_k Z

{h≤|un|}

|w^−1p F_n|^p0 dx,

consequently, thanks to the strong convergence of|w^−1p F_n|^p0 andf_n, inL¹(Ω) and letting firstlyn→ ∞afterh tend to infinity, we obtain

lim sup

h→∞

Z

{h≤|un|≤2k+h}

N

X

i=1

|∂u

∂x_i|^pw_iφ⁰(T_2k(u−T_h(u)))dx= 0, so that

h→∞lim Z

Ω

F∇T_2k(u−T_h(u))φ⁰(T_2k(u−T_h(u)))dx= 0.

Therefore by(4.18), lettingh go to infinity, we conclude,

n→∞lim Z

Ω

[a(x, T_k(u_n),∇T_k(u_n))−a(x, T_k(u_n),∇T_k(u))]

·[∇T_k(u_n)− ∇T_k(u)]dx= 0, which implies that by using Lemma 4.1

T_k(u_n)→T_k(u) strongly in W₀^1,p(Ω, w) ∀k >0. (4.19) Step 3. Passing to the limit

We takev∈K_ψ∩L^∞(Ω)as test function in(P_n), we can write Z

Ω

a(x, T_k+kvk∞(u_n),∇T_k+kvk∞(u_n))∇T_k(u_n−v)dx +

Z

Ω

g(x, u_n,∇u_n)T_k(u_n−v)dx

≤ Z

Ω

f_nT_k(u_n−v)dx+ Z

Ω

F_n∇T_k(u_n−v)dx.

(4.20)

By Fatou’s lemma and in fact that

a(x, Tk+kvk_∞(u_n),∇Tk+kvk_∞(u_n))* a(x, Tk+kvk_∞(u),∇Tk+kvk_∞(u)) weakly inΠ^N_i=1L^p0(Ω, w_i^1−p0). It is easily see that

Z

Ω

a(x, Tk+kvk_∞(u),∇Tk+kvk_∞(u))∇T_k(u−v)dx

≤lim inf

n→∞

Z

Ω

a(x, T_k+kvk∞(u_n),∇T_k+kvk∞(u_n))∇T_k(u_n−v)dx.

(4.21) On the other hand, by using the strong convergence ofF_n and

∇T_k(u_n−v)*∇T_k(u−v) weakly in Π^N_i=1L^p(Ω, w_i), we deduce that the integral

Z

Ω

F_n∇T_k(u_n−v)dx→ Z

Ω

F∇T_k(u−v)dx as n→ ∞.

Now, we need to prove that

g(x, u_n,∇u_n)→g(x, u,∇u) strongly in L¹(Ω), (4.22)

in particular it is enough to prove the equiintegrable of g(x, u_n,∇u_n). To this purpose, we takeT_l+1(u_n)−T_l(u_n)as test function in (P_n), we obtain

Z

{|un|>l+1}

|g(x, u_n,∇u_n)|dx= Z

{|un|>l}

|f_n|dx.

Letε >0. Then there exists l(ε)≥1such that Z

{|un|>l(ε)}

|g(x, u_n,∇u_n)|dx < ε/2. (4.23) For any measurable subsetE⊂Ω, we have

Z

E

|g(x, u_n,∇u_n)|dx ≤ Z

E

b(l(ε)) c(x) +

N

X

i=1

w_i|∂(T_l(ε)(u_n))

∂x_i |^p

! dx

+ Z

{|un|>l(ε)}

|g(x, u_n,∇u_n)|dx.

In view of(4.19), there existsη(ε)>0such that Z

E

b(l(ε)) c(x) +

N

X

i=1

w_i|^{∂(Tl(ε)(un))}

∂xi |^p

!

dx < ε/2 (4.24) for all E such that |E|< η(ε).

Finally, by combining (4.23)and (4.24)one easily has Z

E

|g(x, u_n,∇u_n)|dx < ε for all E such that |E|< η(ε),

which allows us, by using(4.21)and(4.22), we can pass to the limit in(4.20).

This completes the proof of Theorem 3.1.

Remark 4.2: Note that, we obtain the existence result withowt assuming the coercivity condition. However one can overcome this difficulty by in- troduced the function w_n = T_2k(u_n−T_h(u_n) +T_k(u_n)−T_k(u)) in the test function (4.7).

Corollary 4.3: Let 1 < p < ∞. Assume that the hypothesis (H₁) − (H₃),(3.6) and (3.7) holds, let f_n any sequence of function in L¹(Ω) con- verge tof weakly inL¹(Ω)and letu_n the solution of the following unilateral

problem

(P_n⁰)



















u_n≥ψ a.e. in Ω.

T_k(u_n)∈W₀^1,p(Ω, w), g(x, u_n,∇u_n)∈L¹(Ω) Z

Ω

a(x, u_n,∇u_n)∇T_k(u_n−v)dx+ Z

Ω

g(x, u_n,∇u_n)T_k(u_n−v)dx

≤ Z

Ω

f_nT_k(u_n−v)dx,

∀v∈K_ψ∩L^∞(Ω), ∀ k >0.

Then, there exists a subsequence ofu_nstill denotedu_n such thatu_nconverges to ualmost everywhere and T_k(u_n)* T_k(u) weakly in W₀^1,p(Ω, w), furtheru is a solution of the unilateral problem(P) (withF = 0).

Proof. We give the proof brievely.

Step 1. A priori estimates

We proceed as previous, we takev=ψ⁺ as test function in(P_n⁰), we get Z

Ω N

X

i=1

w_i|∂T_k(u_n)

∂x_i |^pdx≤C₁. (4.25) Hence, by the same method used in the first step in the proof of Theorem 3.1 there exists a function u (with T_k(u) ∈ W₀^1,p(Ω, w) ∀ k > 0) and a subsequence still denoted byu_n such that

T_k(u_n)* T_k(u) weakly in W₀^1,p(Ω, w), ∀k >0.

Step 2. Strong convergence of truncation

The choice ofv=T_h(u_n−ηφ(w_n)), h >kψ⁺k∞ as test function in (P_n⁰), we get,∀ l >0

Z

Ω

a(x, u_n,∇u_n)∇T_l(u_n−T_h(u_n−ηφ(w_n))) +

Z

Ω

g(x, u_n,∇u_n)T_l(u_n−T_h(u_n−ηφ(w_n))dx

≤ Z

Ω

f_nT_l(u_n−T_h(u_n−ηφ(w_n)))dx.

Which implies that Z

{|un−ηφ(wn)|≤h}

a(x, u_n,∇u_n)∇T_l(ηφ(w_n)) +

Z

Ω

g(x, u_n,∇u_n)T_l(u_n−T_h(u_n−ηφ(w_n)))dx

≤ Z

Ω

f_nT_l(u_n−T_h(u_n−ηφ(w_n)))dx.

Lettingh tends to infinity and choosingllarge enough, we deduce Z

Ω

a(x, u_n,∇u_n)∇φ(w_n) + Z

Ω

g(x, u_n,∇u_n)φ(w_n)dx≤ Z

Ω

f_nφ(w_n)dx, the rest of the proof of this step is the same as in step 2 of the proof of Theorem 3.1.

Step 3. Passing to the limit

This step is similarly to the step3of the proof of Theorem3.1, by using the Egorov’s theorem in the last term of(P_n⁰).

Remark 4.4: In the case where F = 0, if we suppose that the second mumber are nonnegative, then we obtain a nonnegative solution.

Indeed. If we take v=T_h(u⁺) (withh≥ kψk∞) in(P), we have Z

Ω

a(x, u,∇u)∇T_k(u−T_h(u⁺))dx+ Z

Ω

g(x, u,∇u)T_k(u−T_h(u⁺))dx

≤ Z

Ω

f T_k(u−T_h(u⁺))dx.

Sinceg(x, u,∇u)T_k(u−T_h(u⁺))≥0, we deduce Z

Ω

a(x, u,∇u)∇T_k(u−T_h(u⁺))dx≤ Z

Ω

f T_k(u−T_h(u⁺))dx, we remark also, using f ≥0

Z

Ω

f T_k(u−T_h(u⁺))dx≤ Z

{u≥h}

f T_k(u−T_h(u))dx.

On the other hand, by using(3.3), we conclude α

Z

Ω N

X

i=1

w_i|∂T_k(u⁻)

∂x_i |^pdx≤ Z

{u≥h}

f T_k(u−T_h(u))dx.

Lettinghtend to infinity, we can easily conclude that, T_k(u⁻) = 0, ∀k >0, which impliesu≥0.

References

[1] Y. Akdim, E. Azroul, and A. Benkirane. Existence of solutions for quasilinear degenerated elliptic equations. Electronic J. Diff. Eqns., 2001(71):1–19, 2001.

[2] Y. Akdim, E. Azroul, and A. Benkirane. Existence of solution for quasi- linear degenerated elliptic unilateral problems. Annale Mathématique Blaise Pascal, 10:1–20, 2003.

[3] E. Azroul, A. Benkirane, and O. Filali. Strongly nonlinear degenerated unilateral problems withL¹data.Electronic J. Diff. Eqns., pages 46–64, Conf. 09. 2002.

[4] P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre, and J. L.

Vazquez. AnL¹-theory of existence and uniqueness of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa, 22:240–273, 1995.

[5] L. Boccardo and T. Gallouët. Non-linear elliptic equations with right hand side measures. commun. In partial Differential Equations, 17:641–

655, 1992.

[6] L. Boccardo and T. Gallouët. Strongly non-linear elliptic equations having natural growth andL¹data. Nonlinear Anal., 19:573–578, 1992.

[7] L. Boccardo, T. Gallouët, and F. Murat. A unified presentation of tow existence results for problems with natural growth. in : Progress in Partial Differential Equations : The Metz Surveys 2, M. Chipot (ed), Pitman Res. Notes Math. Ser. 296, Longman, pages 127–137, 1993.

[8] L. Boccardo, F. Murat, and J.-P. Puel. Existence of bounded solu- tions for nonlinear elliptic unilateral problems. Ann. Math. Pura Appl., 152:183–196, 1988.

[9] G. Dalmaso, F. Murat, L. Orsina, and A. Prignet. Renormalized so- lutions of elliptic equations with general measure data. Ann. Scuola Norm. Sup Pisa Cl. Sci, 12(4):741–808, 1999.

[10] P. Drabek, A. Kufner, and F. Nicolosi. Nonlinear elliptic equations, singular and degenerate cases. University of West Bohemia, 1996.

[11] A. Elmahi and D. Meskine. Unilateral elleptic problems in L¹ with natural growth terms. To appear Nonlinear and convex analysis.

[12] A. Porretta. Existence for elliptic equations in L¹ having lower order terms with natural growth. Portugal. Math., 57:179–190, 2000.

Lahsen Aharouch Youssef Akdim

Faculté des Sciences Dhar-Mahraz

Faculté des Sciences Dhar-Mahraz

Dép. de Math. et Informatique Dép. de Math. et Informatique B.P 1796 Atlas Fès. B.P 1796 Atlas Fès.

Fès Fès

MAROC MAROC

lahrouche@caramail.com akdimyoussef@yahoo.fr