Existence of Solution for Quasilinear Degenerated Elliptic Unilateral Problems

BLAISE PASCAL

Youssef Akdim, Elhoussine Azroul, Abdelmoujib Benkirane

Existence of Solution for Quasilinear Degenerated Elliptic Unilateral Problems

Volume 10, n^o1 (2003), p. 1-20.

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Existence of Solution for Quasilinear Degenerated Elliptic Unilateral Problems

Youssef Akdim Elhoussine Azroul Abdelmoujib Benkirane

Abstract

An existence theorem is proved, for a quasilinear degenerated ellip- tic inequality involving nonlinear operators of the formAu+g(x, u,∇u), whereAis a Leray-Lions operator fromW₀^1,p(Ω, w)into its dual, while g(x, s, ξ)is a nonlinear term which has a growth condition with respect toξ and no growth with respect tos, but it satisfies a sign condition ons, the second term belongs toW^−1,p0(Ω, w^∗).

1 Introduction

LetΩbe a bounded open set ofR^N,pbe a real number such that1< p <∞ andw ={w_i(x), 0≤i≤N}be a vector of weight functions on Ω, i.e. each w_i(x)is a measurablea.e.strictly positive onΩ, satisfying some integrability conditions (see section 2). This paper is concerned with the existence of solution of unilateral degenerate problems associated to a nonlinear operator of the form

Au+g(x, u,∇u).

The principal partAis a differential operator of second order in divergence form of Leray-Lions type acting fromW₀^1,p(Ω, w)into it’s dualW^−1,p0(Ω, w^∗), i.e.

Au=−div(a(x, u,∇u)) (1.1) andg is a nonlinear lower order term having natural growth with respect to

|∇u|, with respect to |u| we do not assume any growth restrictions, but we assume the sign-condition. Bensoussan, Boccardo and Murat have proved in the first part of[3], the existence of a solution for the problem

Au+g(x, u,∇u) =f,

where f ∈ W^−1,p0(Ω). In the second part of [3], the authors have extended the last result to variational inequalities, more precisely, they have proved the existence of at least one solution of the following unilateral problem:











hAu, v−ui+R

Ωg(x, u,∇u)(v−u)dx≥ hf, v−ui for all v∈K_ψ

u∈W₀^1,p(Ω) u≥ψ a.e.in Ω

g(x, u,∇u)∈L¹(Ω) g(x, u,∇u)u∈L¹(Ω),

whereK_ψ ={v∈W₀^1,p(Ω)∩L^∞(Ω), v≥ψ a.e.in Ω},withψ a measurable function on Ω such that ψ⁺ ∈ W₀^1,p(Ω)∩L^∞(Ω). The same result is also proved in[2] wheref ∈L¹(Ω).

It is our purpose in this paper, to study the variational degenerated inequal- ities. More precisely, we prove the existence of solution to the problem (P) (see section 4), in the framework of weighted Sobolev space. We obtain the existence results by proving that the positive part u⁺_ε (resp. negative part u⁻_ε) of u_ε strongly converges to u⁺(resp. u⁻) in W₀^1,p(Ω, w), where u_ε is a solution of the approximate problem (P_ε) (see section 4). Let us point out, that another work in this direction can be found in[6]and [1] in the case of equation.

Note that, this paper can be seen as a generalization of [3] in weighted case and as a continuation of[1]where the case of equation is treated. This paper is organized as follows: Section 2 contains some preliminaries, section 3 is concerned with the basic assumptions and some technical lemmas, in section 4 we state and prove main results.

2 Preliminaries

Let Ω be a bounded open subset of R^N (N ≥ 1), let 1 < p < ∞, and let w = {w_i(x), 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. in Ω. Further, we suppose in all our considerations that

w_i∈L¹_loc(Ω) (2.1)

and

w⁻

1 p−1

i ∈L¹_loc(Ω) (2.2)

for any0≤i≤N.

We define the weighted spaceL^p(Ω, γ), whereγ is a weight function onΩby, L^p(Ω, γ) ={u=u(x), uγ^1p ∈L^p(Ω)}

with the norm

kuk_p,γ= Z

Ω

|u(x)|^pγ(x)dx ¹_p

.

Now, we denote by W^1,p(Ω, w) the space of all real-valued functions u ∈ L^p(Ω, w₀) such that the derivatives in the sense of distributions fulfil

∂u

∂x_i ∈L^p(Ω, w_i)for all i= 1, ..., N, which is a Banach space under the norm

kuk_1,p,w = Z

Ω

|u(x)|^pw₀(x)dx+

N

X

i=1

Z

Ω

|∂u(x)

∂x_i |^pw_i(x)dx

!¹_p

. (2.3) Since we shall deal with the Dirichlet problem, we shall use the space

X =W₀^1,p(Ω, w) (2.4)

defined as the closure ofC₀^∞(Ω)with respect to the norm (2.3). 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 weighted Sobolev spacesW₀^1,p(Ω, w)is equiv- alent to W^−1,p0(Ω, w^∗),where w^∗= {w_i^∗=w_i^1−p0, ∀i = 0, ..., N}, where p⁰ is the conjugate ofp i.e. p⁰= _p−1^p ( for more details we refer to[5]).

Definition: Let Y be a separable reflexive Banach space, the operator B from Y to its dual Y^∗ is called of the calculus of variations type, if B is bounded and is of the form,

B(u) =B(u, u), (2.5)

where (u, v) −→ B(u, v) is an operator from Y ×Y into Y^∗ satisfying the following properties:

∀u∈Y, v→B(u, v) is bounded hemicontinuous fromY intoY^∗ and (B(u, u)−B(u, v), u−v)≥0,

(2.6)

∀v∈Y, u→B(u, v) is bounded hemicontinuous fromY intoY^∗, (2.7) if u_n* uweakly in Y and if(B(u_n, u_n)−B(u_n, u), u_n−u)→0

then, B(u_n, v)* B(u, v)weakly in Y^∗, ∀v∈Y,

(2.8) if u_n* uweakly in Y and ifB(u_n, v)* ψweakly in Y^∗,

then, (B(u_n, v), u_n)→(ψ, u). (2.9)

Definition: LetY be a reflexive Banach space, a bounded mappingBfrom Y toY^∗ is called pseudo-monotone if for any sequenceu_n∈Y withu_n* u weakly in Y andlim sup

n→∞

hBu_n, u_n−ui ≤0, one has lim inf

n→∞ hBu_n, u_n−vi ≥ hBu, u−vi for allv∈Y.

3 Basic assumption and some technical lem- mas

We start by the following assumptions.

Assumption(H₁) The expression

k|u|k_X =

N

X

i=1

Z

Ω

|∂u(x)

∂x_i |^pw_i(x)dx

!¹_p

is a norm defined on X and it’s equivalent to the norm(2.3).

And there exist a weight function σonΩ and a parameterq, such that 1< q < p+p⁰, (3.1) and

σ^1−q0 ∈L¹_loc(Ω), (3.2)

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

Ω

|u(x)|^qσ(x)dx ¹_q

≤c

N

X

i=1

Z

Ω

|∂u(x)

∂x_i |^pw_i(x)dx

!¹_p

, (3.3)

holds for everyu∈X with a constantc >0independent ofu, and moreover, the imbedding

X ,→,→L^q(Ω, σ), (3.4)

expressed by the inequality(3.3) is compact.

Notice that (X,k|.|k_X) is a uniformly convex (and thus reflexive ) Banach space.

Remark: If we assume that w₀(x) ≡ 1 and in addition the integrability condition:

there existsν∈]N

p,∞[∩[ 1

p−1,∞[ such thatw^−ν_i ∈L¹(Ω) ∀i= 1, ..., N, which is stronger than(2.2). Then

k|u|k_X =

N

X

i=1

Z

Ω

|∂u(x)

∂x_i |^pw_i(x)dx

!¹_p ,

is a norm defined onW₀^1,p(Ω, w) and it’s equivalent to(2.3), moreover W₀^1,p(Ω, w),→,→L^q(Ω),

for all 1 ≤ q < p^∗₁ if pν < N(ν + 1) and for all q ≥ 1 if pν ≥ N(ν+ 1), where p₁ = _ν+1^pν and p^∗₁ = _N−p^{N p1}

1 = _{N(ν+1)−pν}^{N pν} is the Sobolev conjugate of p₁ (see [5]). Thus the hypotheses (H₁) are verified for σ ≡ 1 and for all 1 < q < min{p^∗₁, p+p⁰} if pν < N(ν+ 1) and for all 1 < q < p+p⁰ if pν≥N(ν+ 1).

LetAbe a nonlinear operator fromW₀^1,p(Ω, w)into its dualW^−1,p0(Ω, w^∗) defined by(1.1), i.e.,

Au=−div(a(x, u,∇u)),

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

Assumption(H₂)

|a_i(x, s, ξ)| ≤βw

1 p

i(x)[k(x) +σ^p10|s|^pq0 +

N

X

j=1

w

1 p0

j (x)|ξ_j|^p−1] for alli= 1, ..., N, (3.5)

[a(x, s, ξ)−a(x, s, η)](ξ−η)>0, for allξ6=η∈R^N, (3.6) a(x, s, ξ).ξ≥α

N

X

i=1

w_i|ξ_i|^p, (3.7)

where k(x) is a positive function in L^p0(Ω) and α, β are strictly positive constants.

Assumption(H₃)

Let g(x, s, ξ) be a Carathéodory function satisfying the following assump- tions:

g(x, s, ξ)s≥0 (3.8)

|g(x, s, ξ)| ≤b(|s|)

N

X

i=1

w_i|ξ_i|^p+c(x)

!

, (3.9)

whereb:R⁺−→R⁺is a continuous increasing function andc(x)is a positive function which lies inL¹(Ω).

We consider,

f ∈W^−1,p0(Ω, w^∗). (3.10) Now we recall some lemmas introduced in [1] which will be used later.

Lemma 3.1: (cf.[1])Letg∈L^r(Ω, γ)and letg_n∈L^r(Ω, γ), withkg_nk_r,γ ≤ c (1< r <∞). Ifg_n(x)−→g(x)a.e.inΩ, theng_n* gweakly inL^r(Ω, γ), where γ is a weight function onΩ.

Lemma 3.2: (cf.[1])Assume that(H₁)holds. LetF :R−→Rbe uniformly 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)6∈D}

0 a.e. in{x∈Ω : u(x)∈D}.

Lemma 3.3: (cf. [1])Assume that (H₁)holds. Let u∈W₀^1,p(Ω, w), and let T_k(u), k ∈R⁺, be the usual truncation thenT_k(u)∈W₀^1,p(Ω, w). Moreover, we have

T_k(u)−→u strongly inW₀^1,p(Ω, w).

Lemma 3.4: (cf. [1]) Assume that (H₁) holds. Let u ∈ W₀^1,p(Ω, w), then u⁺= max(u,0) and u⁻= max(−u,0)lie inW₀^1,p(Ω, w). Moreover, we have

∂(u⁺)

∂x_i = _∂u

∂xi , ifu >0 0 , ifu≤0

∂(u⁻)

∂x_i =

0 , ifu≥0

−_∂x^∂u

i , ifu <0.

Lemma 3.5: (cf. [1]) Assume that (H₁) holds. Let (u_n) be a sequence of W₀^1,p(Ω, w) such thatu_n * u weakly in W₀^1,p(Ω, w). Then, u⁺_n * u⁺ weakly inW₀^1,p(Ω, w) andu⁻_n * u⁻ weakly in W₀^1,p(Ω, w).

Lemma 3.6: (cf.[1])Assume that (H₁)and(H₂)are satisfied, and let(u_n) be a sequence of W₀^1,p(Ω, w) such that

u_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−→ustrongly in W₀^1,p(Ω, w).

4 Main general result

Letψ be a measurable function with values in Rsuch that,

ψ⁺∈W₀^1,p(Ω, w)∩L^∞(Ω). (4.1) Set

K_ψ ={v∈W₀^1,p(Ω, w)∩L^∞(Ω) v≥ψ a.e.inΩ}. (4.2) Remark that (4.1) implies thatK_ψ 6=∅.

Consider the nonlinear problem with Dirichlet boundary condition,

(P)











hAu, v−ui+R

Ωg(x, u,∇u)(v−u)dx≥ hf, v−ui for allv∈K_ψ

u∈W₀^1,p(Ω, w) u≥ψ a.e.inΩ

g(x, u,∇u)∈L¹(Ω) g(x, u,∇u)u∈L¹(Ω),

hereuis the solution of the problem (P).

Our main result is the following.

Theorem 4.1: Assume that the assumption (H₁)−(H₃),(3.10) and (4.1) hold, then, there exists at least one solution of(P).

Remarks:

1) The statement of Theorem 4.1, generalizes in weighted case the analo- gous one in [3].

2) If we take ψ = −∞, we obtain the existence result for the equation case (see [1]).

Proof of Theorem 4.1

Step (1) The approximate problem and a priori estimate.

LetΩ_ε be a sequence of compact subsets ofΩ such thatΩ_ε increase to Ωas ε→0.

We consider the sequence of approximate problems:

(P_ε)







hAu_ε, v−u_εi+R

Ωg_ε(x, u_ε,∇u_ε)(v−u_ε)dx≥ hf, v−u_εi v∈W₀^1,p(Ω, w) v≥ψ a.e.in Ω

u_ε∈W₀^1,p(Ω, w) u_ε ≥ψ a.e.inΩ, where

g_ε(x, s, ξ) = g(x, s, ξ)

1 +ε|g(x, s, ξ)|χ_Ωε(x), and whereχ_Ωε is the characteristic function ofΩ_ε. Note thatg_ε(x, s, ξ) satisfies the following conditions,

g_ε(x, s, ξ)s≥0, |g_ε(x, s, ξ)| ≤ |g(x, s, ξ)| and|g_ε(x, s, ξ)| ≤ 1 ε.

We define the operator G_ε : X −→X^∗ by, hG_εu, vi=

Z

Ω

g_ε(x, u,∇u)v dx.

Thanks to Hölder’s inequality we have for allu∈X and v∈X,

| Z

Ω

g_ε(x, u,∇u)v dx| ≤ Z

Ω

|g_ε(x, u,∇u)|^q0σ^−q

0 q dx

_q¹0 Z

Ω

|v|^qσ dx ¹_q

≤ 1 ε

Z

Ωε

σ^1−q0 dx _q¹0

kvk_q,σ

≤ c_εk|v|k.

(4.3) The last inequality is due to (3.2) and (3.4).

Lemma 4.2: The operator B_ε=A+G_ε fromX into its dualX^∗ is pseudo- monotone. Moreover,B_ε is coercive, in the following sense:

( there existsv₀∈K_ψ such that

hBεv,v−v0i

k|v|k →+∞ ifk|v|k → ∞, v∈K_ψ. This lemma will be proved below.

In view of lemma 4.2,(P_ε)has a solution by the classical result (cf. theorem 8.2 chapter 2[7]).

Letv=ψ⁺ as test function in (P_ε), we easily deduce that R

Ωg_ε(x, u_ε,∇u_ε)(u_ε−ψ⁺)≥0, then,hAu_ε, u_εi ≤ hf, u_ε−ψ⁺i+hAu_ε, ψ⁺i, i.e.

Z

Ω

a(x, u_ε,∇u_ε)∇u_ε dx≤ hf, u_ε−ψ⁺i+

N

X

i=1

Z

Ω

a_i(x, u_ε,∇u_ε)∂ψ⁺

∂x_i dx,

then, α

N

X

i=1

Z

Ω

w_i|∂u_ε

∂x_i|^pdx = αk|u_ε|k^p≤ kfk_X^∗(k|u_ε|k+k|ψ⁺|k) + +

N

X

i=1

Z

Ω

|a_i(x, u_ε,∇u_ε)|^p0w_i^1−p0 dx _p¹₀Z

Ω

|∂ψ⁺

∂x_i|^pw_idx ¹_p

≤ kfk_X∗(k|u_ε|k+k|ψ⁺|k) + + c

N

X

i=1

Z

Ω

(k^p0+|u_ε|^qσ+

N

X

j=1

|∂u_ε

∂x_j|^pw_j)dx

!_p¹0

k|ψ⁺|k.

Using (3.4) the last inequality becomes

αk|u_ε|k^p≤c₁k|u_ε|k+c₂k|u_ε|k^pq0 +c₃k|u_ε|k^p−1+c₄

wherec_iare various positive constants. Then thanks to (3.1), we can deduce thatu_ε remains bounded inW₀^1,p(Ω, w), i.e.

k|u_ε|k ≤β₀, (4.4)

where β₀is a positive constant.

Extracting a subsequence (still denoted byu_ε) we get u_ε* u weakly inX and a.e.in Ω.

Note thatu≥ψ a.e.inΩ.

Step(2) We study the convergence of the positive part of u_ε. Letk >0. Define u⁺_k = min{u⁺, k}.

We shall fix k, and use the notation,

z_ε=u⁺_ε −u⁺_k. (4.5)

Assertion(i)We claim that, lim sup

ε→0

Z

Ω

[a(x, u_ε,∇u⁺_ε)−a(x, u_ε,∇u⁺_k)]∇(u⁺_ε −u⁺_k)⁺dx≤Q_k, (4.6) where Q_k →0, ifk→+∞.

Indeed,Consider the test functionv_ε=u_ε−z⁺_ε. By lemma 3.3 and lemma 3.4, we havez_ε ∈W₀^1,p(Ω, w) and z_ε⁺∈W₀^1,p(Ω, w). And since k−→ ∞ and ψ⁺ ∈ L^∞(Ω) we can assume that k ≥ ψ a.e.in Ω, by the choice of k, the above test function is admissible for(P_ε). Multiplying(P_ε)byv_ε we obtain,

hAu_ε, z_ε⁺i+ Z

Ω

g_ε(x, u_ε,∇u_ε)z_ε⁺dx≤ hf, z_ε⁺i. (4.7) Ifz_ε⁺>0, we haveu_ε>0and from(3.8),g_ε(x, u_ε,∇u_ε)≥0, then,

Z

Ω

a(x, u_ε,∇u_ε)∇z⁺_ε dx≤ hf, z_ε⁺i.

Sinceu_ε =u⁺_ε in {x∈Ω / z_ε⁺>0}, then, Z

Ω

a(x, u_ε,∇u⁺_ε)∇z_ε⁺dx≤ hf, z_ε⁺i,

which implies that, Z

Ω

[a(x, u_ε,∇u⁺_ε)−a(x, u_ε,∇u⁺_k)]∇(u⁺_ε −u⁺_k)⁺dx≤

≤ − Z

Ω

a(x, u_ε,∇u⁺_k)∇(u⁺_ε −u⁺_k)⁺dx+hf, z_ε⁺i.

(4.8) Asε →0, we havez⁺_ε −→(u⁺−u⁺_k)⁺a.e. in Ω, moreover z_ε⁺ is bounded in W₀^1,p(Ω, w), hence, we have,

z_ε⁺*(u⁺−u⁺_k)⁺ weakly in W₀^1,p(Ω, w).

Sincea(x, u_ε,∇u⁺_k)−→a(x, u,∇u⁺_k)inQN

i=1L^p0(Ω, w^∗_i),we obtain by passing to the limit inε in (4.8)the inequality (4.6) withQ_k defined by,

Q_k =− Z

Ω

a(x, u,∇u⁺_k)∇(u⁺−u⁺_k)⁺dx+hf,(u⁺−u⁺_k)⁺i. (4.9) Because,(u⁺−u⁺_k)⁺ −→0 inW₀^1,p(Ω, w) ask −→ ∞, we haveQ_k −→0as k−→ ∞.

Assertion(ii)Let us show that, lim sup

ε→0

Z

Ω

−[a(x, u_ε,∇u⁺_ε)−a(x, u_ε,∇u⁺_k)]∇(u⁺_ε −u⁺_k)⁻dx≤0. (4.10) Indeed. We consider for that, the test function v_ε = u_ε+ϕ_λ(z_ε⁻), where ϕ_λ(s) = se^λs2. We have, 0 ≤ z⁻_ε ≤ k, i.e., z_ε⁻ ∈ L^∞(Ω) and since z⁻_ε ∈ W₀^1,p(Ω, w), hence using lemma 3.2 we havev_ε ∈W₀^1,p(Ω, w), then clearly, v_ε is an admissible test function.

Multiplying(P_ε)byv_ε we obtain, Z

Ω

a(x, u_ε,∇u_ε)∇z_ε⁻ϕ⁰_λ(z⁻_ε)dx+ Z

Ω

g_ε(x, u_ε,∇u_ε)ϕ_λ(z_ε⁻)dx≥ hf, ϕ_λ(z_ε⁻)i.

(4.11) Define,

E_ε={x∈Ω, u⁺_ε(x)≤u⁺_k(x)}and F_ε={x∈Ω, 0≤u_ε(x)≤u⁺_k(x)}.

Sinceϕ_λ(z_ε⁻) = 0 inE_ε^c, we have, Z

Ω

g_ε(x, u_ε,∇u_ε)ϕ_λ(z_ε⁻)dx= Z

Eε

g_ε(x, u_ε,∇u_ε)ϕ_λ(z_ε⁻)dx. (4.12)

Whenu_ε≤0, we have g_ε(x, u_ε,∇u_ε)≤0and sinceϕ_λ(z_ε⁻)≥0, we obtain Z

Eε

g_ε(x, u_ε,∇u_ε)ϕ_λ(z_ε⁻)dx ≤ Z

Fε

g_ε(x, u_ε,∇u_ε)ϕ_λ(z_ε⁻)dx

≤ Z

Fε

b(|u_ε|)[

N

X

i=1

w_i|∂u_ε

∂x_i|^p+c(x)]ϕ_λ(z⁻_ε)dx

≤ b(k) Z

Fε

[

N

X

i=1

w_i|∂u_ε

∂x_i|^p+c(x)]ϕ_λ(z⁻_ε)dx

≤b(k) α

Z

Fε

a(x, u_ε,∇u_ε)∇u_εϕ_λ(z_ε⁻)dx+b(k) Z

Fε

c(x)ϕ_λ(z_ε⁻)dx. (4.13) As in the proof of theorem 1.1 in [3], we can show that,

−1 2

Z

Ω

[a(x, u_ε,∇u⁺_ε)−a(x, u_ε,∇u⁺_k)]∇(u⁺_ε −u⁺_k)⁻dx≤

≤ Z

Ω

[a(x, u_ε,∇u_ε)−a(x, u_ε,∇u⁺_ε)]∇u⁺_kϕ⁰_λ(u⁺_k)dx+h−f, ϕ_λ(z⁻_ε)i +

Z

Ω

a(x, u_ε,∇u⁺_k)∇z_ε⁻ϕ⁰_λ(z_ε⁻)dx+ b(k) α

Z

Ω

a(x, u_ε,∇u⁺_ε)∇u⁺_kϕ_λ(z_ε⁻)dx+

+b(k) α

Z

Ω

a(x, u_ε,∇u⁺_k)∇(u⁺_ε −u⁺_k)ϕ_λ(z_ε⁻)dx+b(k) Z

Ω

c(x)ϕ_λ(z_ε⁻)dx, (4.14) forλ= ^b(k)2

4α².

For short notation, we rewrite the above inequality as,

I_εk ≤I_εk¹ +I_εk² +I_εk³ +I_εk⁴ +I_εk⁵. (4.15) Extracting a subsequence such that,







a(x, u_ε,∇u_ε)* γ₁weakly in QN

i=1L^p0(Ω, w^∗_i) and

a(x, u_ε,∇u⁺_ε)* γ₂ weakly in QN

i=1L^p0(Ω, w_i^∗).

(4.16)

Lemma 4.3:(cf. [1]) For k fixed and letting ε→0, we claim that, 1)I_εk¹ −→I_k¹=

Z

Ω

[γ₁−γ₂]∇u⁺_kϕ⁰_λ(u⁺_k)dx+h−f, ϕ_λ((u⁺−u⁺_k)⁻)i.

2)I_εk² −→I_k²= Z

Ω

a(x, u,∇u⁺_k)∇((u⁺−u⁺_k)⁻)ϕ⁰_λ((u⁺−u⁺_k)⁻)dx.

3)I_εk³ −→I_k³= b(k) α

Z

Ω

γ₂∇u⁺_kϕ_λ((u⁺−u⁺_k)⁻)dx.

4)I_εk⁴ −→I_k⁴= b(k) α

Z

Ω

a(x, u,∇u⁺_k)∇(u⁺−u⁺_k)ϕ_λ((u⁺−u⁺_k)⁻)dx.

5)I_εk⁵ −→I_k⁵=b(k) Z

Ω

c(x)ϕ_λ((u⁺−u⁺_k)⁻)dx.

In view of lemma 4.3 and (u⁺−u⁺_k)⁻= 0and ϕ_λ(0) = 0 we have, lim sup

ε→0

I_εk≤I_k¹+I_k²+I_k³+I_k⁴+I_k⁵= Z

Ω

[γ₁(x)−γ₂(x)]∇u⁺_kϕ⁰_λ(u⁺_k)dx.

Moreover, ifu_ε<0we have (u_ε)⁺_k = 0, hence,

(a(x, u_ε,∇u_ε)−a(x, u_ε,∇u⁺_ε))(u_ε)⁺_k = 0a.e.in Ω which implies that(γ₁(x)−γ₂(x))u⁺_k = 0, and so that,

lim sup

ε→0

I_εk ≤0, thus, (4.10) follows.

Assertion(iii)Let us show that,

u⁺_ε −→u⁺ strongly in W₀^1,p(Ω, w). (4.17) From (4.6) and (4.10) we have (as in the proof of theorem 1.1 in[3]),

lim sup

ε→0

Z

Ω

[a(x, u_ε,∇u⁺_ε)−a(x, u_ε,∇u⁺)]∇(u⁺_ε −u⁺)dx≤

≤Q_k+ Z

Ω

[γ₂(x)−a(x, u,∇u⁺_k)]∇(u⁺_k −u⁺)dx.

Now lettingk−→ ∞and using lemma 3.6 we obtain (4.17).

Step(3) We study the convergence of the negative part of u_ε Similarly to the preceding step, we shall prove that

u⁻_ε −→u⁻ strongly inW₀^1,p(Ω, w). (4.18) Assertion (j)Let us show that,

lim sup

ε→0

Z

Ω

−[a(x, u_ε,−∇u⁻_ε)−a(x, u_ε,−∇u⁻_k)]∇(u⁻_ε −u⁻_k)⁺dx≤Q˜_k, (4.19)

where Q˜_k →0, ifk→+∞.

Indeed. We define

u⁻_k = min{u⁻, k} and y_ε =u⁻_ε −u⁻_k.

Consider the test functionv_ε =u_ε+y⁺_ε in (P_ε), it is clearly admissible.

Multiplying(P_ε) byv_ε, we have, Z

Ω

a(x, u_ε,∇u_ε)∇y⁺_ε dx+ Z

Ω

g_ε(x, u_ε,∇u_ε)y⁺_ε dx≥ hf, y⁺_εi.

Since y_ε⁺ > 0 implies u_ε < 0, then from (3.8), we have g_ε(x, u_ε,∇u_ε) ≤ 0, henceg_ε(x, u_ε,∇u_ε)y⁺_ε ≤0a.e.in Ω, then,

Z

Ω

a(x, u_ε,∇u_ε)∇y⁺_ε dx≥ hf, y⁺_εi.

Sinceu_ε =−u⁻_ε on the set{x∈Ω, y⁺_ε >0} we can also write R

Ωa(x, u_ε,−∇u⁻_ε)∇y⁺_ε dx≥ hf, y⁺_εi, which implies that,

− Z

Ω

[a(x, u_ε,−∇u⁻_ε)−a(x, u_ε,−∇u⁻_k)]∇(u⁻_ε −u⁻_k)⁺dx≤ Z

Ω

a(x, u_ε,−∇u⁻_k)∇(u⁻_ε −u⁻_k)⁺dx− hf, y_ε⁺i.

As ε→ 0we have y⁺_ε −→(u⁻−u⁻_k)⁺a.e. in Ω, and sincey_ε⁺ is bounded in W₀^1,p(Ω, w), then y_ε⁺*(u⁻−u⁻_k)⁺ weakly in W₀^1,p(Ω, w)(fork fixed).

Passing to the limit in εwe obtain (4.19), with Q˜_k defined by, Q˜_k=

Z

Ω

a(x, u,−∇u⁻_k)∇(u⁻−u⁻_k)⁺dx− hf,(u⁻−u⁻_k)⁺i.

Because(u⁻−u⁻_k)⁺−→0strongly inW₀^1,p(Ω, w)ask−→ ∞we obtain that Q˜_k −→0 ask−→ ∞.

Assertion (jj)Let us show that, lim sup

ε→0

Z

Ω

[a(x, u_ε,−∇u⁻_ε)−a(x, u_ε,−∇u⁻_k)]∇(u⁻_ε −u⁻_k)⁻dx≤0. (4.20) Indeed. Considering the following test function v_ε = u_ε−δ_εϕ_λ(y⁻_ε) where δ_ε>0such thatδ_εe^λ(yε−)2≤1this function is admissible (cf. [3]), then,

hAu_ε,−δ_εϕ_λ(y⁻_ε)i −δ_ε Z

Ω

g_ε(x, u_ε,∇u_ε)ϕ_λ(y_ε⁻)dx≥ −hf, δ_εϕ_λ(y⁻_ε)i,

i.e.,

hAu_ε, ϕ_λ(y⁻_ε)i+ Z

Ω

g_ε(x, u_ε,∇u_ε)ϕ_λ(y_ε⁻)dx≤ hf, ϕ_λ(y_ε⁻)i, with this choice (4.20) follows as in (4.10).

Finally combining (4.19) and (4.20), we deduce as in (4.17) the assertion (4.18).

Step (4) Convergence of u_ε.

From (4.17) and (4.18) we deduce that for a subsequence

u_ε−→ustrongly in W₀^1,p(Ω, w) and a.e.in Ω (4.21)

∇u_ε−→ ∇u a.e.inΩ, (4.22) which implies that,

g_ε(x, u_ε,∇u_ε)−→g(x, u,∇u)a.ein Ω

g_ε(x, u_ε,∇u_ε)u_ε−→g(x, u,∇u)u a.e.inΩ. (4.23) On the other hand, multiplying(P_ε)byu_ε and using(3.7),(3.8),(4.3),(4.4), we obtain

0≤ Z

Ω

g_ε(x, u_ε,∇u_ε)u_εdx≤β,˜ (4.24) whereβ˜is some positive constant.

For any measurable subsetE ofΩ and anym >0we have, Z

E

|g_ε(x, u_ε,∇u_ε)|dx= Z

E∩X^ε_m

|g_ε(x, u_ε,∇u_ε)|dx+ Z

E∩Y_m^ε

|g_ε(x, u_ε,∇u_ε)|dx where

X_m^ε ={x∈Ω : |u_ε(x)| ≤m}

Y_m^ε ={x∈Ω : |u_ε(x)|> m}. (4.25) From(3.9),(4.24),(4.25)we have,

Z

E

|g_ε(x, u_ε,∇u_ε)|dx ≤ Z

E∩X_m^ε

|g_ε(x, u_ε,∇u_ε)|dx+ 1 m

Z

Ω

g_ε(x, u_ε,∇u_ε)u_εdx

≤ b(m) Z

E

(

N

X

i=1

w_i|∂u_ε

∂x_i|^p+c(x))dx+ ˜β1 m.

(4.26)

Since the sequence (∇u_ε) strongly converges in QN

i=1L^p(Ω, w_i), then (4.26) implies the equi-integrability of g_ε(x, u_ε,∇u_ε).

Thanks to (4.23) and Vitali’s theorem one easily has

g_ε(x, u_ε,∇u_ε)−→g(x, u,∇u) strongly in L¹(Ω). (4.27) Moreover, sinceg_ε(x, u_ε,∇u_ε)u_ε ≥0 a.e. in Ω, then by using (4.23), (4.24) and Fatou’s lemma we have

g(x, u,∇u)u∈L¹(Ω). (4.28) From (4.21) and (4.27) we can pass to the limit in

hAu_ε, v−u_εi+ Z

Ω

g_ε(x, u_ε,∇u_ε)(v−u_ε)dx≥ hf, v−u_εi and we obtain,







hAu, v−ui+ Z

Ω

g(x, u,∇u)(v−u)dx≥ hf, v−ui for anyv∈W₀^1,p(Ω, w)∩L^∞(Ω)v≥ψ p.p.inΩ.

(4.29)

Proof of lemma 4.2

By the proposition 2.6 chapter 2 [7], it is sufficient to show that B_ε is of the calculus of variations type. Indeed put,

b₁(u, v,w) =˜

N

X

i=1

Z

Ω

a_i(x, u,∇v)∇w dx˜

b₂(u,w) =˜ Z

Ω

g_ε(x, u,∇u) ˜w dx.

The formw˜−→b₁(u, v,w) +˜ b₂(u,w)˜ is continuous inX. Then,

b₁(u, v,w) +˜ b₂(u,w) =˜ b(u, v,w) =˜ hB_ε(u, v),wi, B˜ _ε(u, v)∈W^−1,p0(Ω, w^∗) and we have

B_ε(u, u) =B_εu.

Using (3.5) and Hölder’s inequality we can show that Ais bounded [4], and thanks to (4.3),B_ε is bounded. Then, it is sufficient to check(2.6)−(2.9).

Show that(2.6)and (2.7)are true.

By(3.6)we have,

(B_ε(u, u)−B_ε(u, v), u−v) =b₁(u, u, u−v)−b₁(u, v, u−v)≥0.

The operatorv→B_ε(u, v)is bounded hemicontinuous. Indeed, we have a_i(x, u,∇(v₁+λv₂))−→a_i(x, u,∇v₁)strongly in L^p0(Ω, w_i^∗) asλ→0.

(4.30) On the other hand,(g_ε(x, u₁+λu₂,∇(u₁+λu₂)))_λis bounded inL^q0(Ω, σ^1−q0) and g_ε(x, u₁+λu₂,∇(u₁+λu₂))−→g_ε(x, u₁,∇u₁) a.e. in Ω hence lemma 3.1 gives

g_ε(x, u₁+λu₂,∇(u₁+λu₂))* g_ε(x, u₁,∇u₁)weakly in L^q0(Ω, σ^1−q0)as λ→0.

(4.31) Using (4.30) and (4.31), we can write

b(u, v₁+λv₂,w)˜ −→b(u, v₁,w)˜ asλ→0 ∀u, v_i,w˜∈X.

Similarly we can prove (2.7).

Proof of assertion (2.8). Assume thatu_n* u weakly in Xand(B(u_n, u_n)− B(u_n, u), u_n−u)→0. We have: (B(u_n, u_n)−B(u_n, u), u_n−u)

=

N

X

i=1

Z

Ω

(a_i(x, u_n,∇u_n)−a_i(x, u_n,∇u))∇(u_n−u)dx→0, then, by lemma 3.6 we have,

u_n−→ustrongly in X, which gives

b(u_n, v,w)˜ −→b(u, v,w)˜ ∀w˜∈X, i.e.,

B_ε(u_n, v)* B_ε(u, v)weakly in X^∗. It remains to prove (2.9). Assume that

u_n* u weakly in X (4.32)

and that

B(u_n, v)* ψweakly in X^∗. (4.33)

Thanks to (3.4), (3.5) and (4.32), we obtain

a_i(x, u_n,∇v)→a_i(x, u,∇v) strongly inL^p0(Ω, w^∗_i)as n→ ∞, then,

b₁(u_n, v, u_n)−→b₁(u, v, u). (4.34) On the other hand, by Hölder’s inequality,

|b₂(u_n, u_n−u)| ≤ Z

Ω

|g_ε(x, u_n,∇u_n)|^q0σ^−q

0 q dx

_q0¹ Z

Ω

|u_n−u|^qσ dx ¹_q

≤ 1 ε

Z

Ωε

σ^−q

0 q dx

_q¹₀

ku_n−uk_L^q_(Ω,σ) →0as n→ ∞, i.e.,

b₂(u_n, u_n−u)−→0asn→ ∞, (4.35) but in view of (4.33) and (4.34), we obtain

b₂(u_n, u) = (B_ε(u_n, v), u)−b₁(u_n, v, u)−→(ψ, u)−b₁(u, v, u) and from (4.35) we have,

b₂(u_n, u_n)−→(ψ, u)−b₁(u, v, u).

Then,

(B_ε(u_n, v), u_n) =b₁(u_n, v, u_n) +b₂(u_n, u_n)−→(ψ, u).

Now, let us show thatB_ε is coercive. Let v₀∈K_ψ.

From Hölder’s inequality, the growth condition (3.5) and the compact imbed- ding (3.4) we have,

hAv, v₀i =

N

X

i=1

Z

Ω

a_i(x, v,∇v)∂v₀

∂x_i dx

≤

N

X

i=1

Z

Ω

|a_i(x, v,∇v)|^p0w

−p0 p

i dx

p0¹ Z

Ω

|∂v₀

∂x_i|^pw_idx ¹_p

≤ c₁k|v₀|k Z

Ω

k(x)^p0+|v|^qσ+

N

X

j=1

|∂v

∂x_j|^pw_jdx

!_p¹0

≤ c₂(c₃+k|v|k^pq0 +k|v|k^p−1),

wherec_i are various constants, thanks to (3.7) we obtain,

hAv, vi

k|v|k −hAv, v₀i

k|v|k ≥αk|v|k^p−1− k|v|k^p−2− k|v|k^pq0−1− c k|v|k. In view of (3.1) we havep−1> _p^q₀−1. Then,

hAv, v−v₀i

k|v|k −→ ∞as k|v|k −→ ∞, sincehG_εv, vi ≥0 andhG_εv, v₀i is bounded we have

hB_εv, v−v₀i

k|v|k ≥hAv, v−v₀i

k|v|k −hG_εv, v₀i

k|v|k −→ ∞as k|v|k −→ ∞.

Remark: The assumption (3.1) appears necessary, in order to prove the boundedness of (u_ε)_ε in W₀^1,p(Ω, w) and the coercivity of the operator B_ε. While the assumption(3.2) is necessary to prove the boundedness of G_ε in W₀^1,p(Ω, w). Thus, wheng≡0, we don’t need to suppose(3.2).

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] A. Benkirane and A. Elmahi. Strongly nonlinear elliptic unilateral prob- lem having natural growth terms andL¹data. Rendiconti di Matematica, 18:289–303, 1998.

[3] A. Bensoussan, L. Boccardo, and F. Murat. On a non linear partial dif- ferential equation having natural growth terms and unbounded solution.

Ann. Inst. Henri Poincaré, 5(4):347–364, 1988.

[4] P. Drabek, A. Kufner, and V. Mustonen. Pseudo-monotonicity and de- generated or singular elliptic operators.Bull. Austral. Math. Soc., 58:213–

221, 1998.

[5] P. Drabek, A. Kufner, and F. Nicolosi.Quasilinear elliptic equations with degenerations and singularities. De Gruyter Series in Nonlinear Analysis and Applications, New York, 1997.

[6] P. Drabek and F. Nicolosi. Existence of bounded solutions for some degenerated quasilinear elliptic equations. Annali di Mathematica pura ed applicata, CLXV:217–238, 1993.

[7] J.L. Lions. Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris, 1969.

Youssef Akdim Elhoussine Azroul

Faculté des Sciences Dhar-Mahraz

Faculté des Sciences Dhar-Mahraz

Département de Mathématiques Département de Mathématiques et Informatique et Informatique

B.P 1796 Atlas Fès. B.P 1796 Atlas Fès.

Fès Fès

MAROC MAROC

y-akdim@caramail.com elazroul@caramail.com

Abdelmoujib Benkirane

Faculté des Sciences Dhar-Mahraz Département de Mathématiques

et Informatique B.P 1796 Atlas Fès.

Fès MAROC

abdelmoujib@iam.net.ma