ENTROPY SOLUTIONS TO NONLINEAR ELLIPTIC ...

Journal of Applied Analysis and Computation Volume 4, Number 3, August 2014

Website:http://jaac-online.com/ pp. 245–270

ENTROPY SOLUTIONS TO NONLINEAR ELLIPTIC ANISOTROPIC PROBLEM WITH VARIABLE EXPONENT Mohamed Badr Benboubker1 , Hassane Hjiaj2 and Stanislas Ouaro3,† Abstract In this work, we give an existence result of entropy solutions for nonlinear anisotropic elliptic equation of the type ( ) −div a(x, u, ∇u) + g(x, u, ∇u) + |u|p0 (x)−2 u = f − divϕ(u),

in Ω,

( ) where −div a(x, u, ∇u) is a Leray-Lions operator, ϕ ∈ C 0 (IR, IRN ). The function g(x, u, ∇u) is a nonlinear lower order term with natural growth with respect to |∇u|, satisfying the sign condition and the datum f belongs to L1 (Ω). Keywords Anisotropic Sobolev spaces, variable exponent, nonlinear elliptic problem, entropy solutions. MSC(2000) 35J20, 35J25, 35D30, 35B38, 35J60.

1. Introduction In the last decades, one of the topics from the field of partial differential equations that continuously gained interest is the one concerning the Sobolev space with variable exponents, W 1,p(.) (Ω) (where p is a function depending on x), see for example the book by Diening et als [11] and [1–3, 5]. The most studied problems was that involving the p(.)-Laplace operators ( ) p(x)−2 ∆p(.) u = div |∇u| ∇u , which generalize work wich involves the p-Laplace operators with p constant (see for example [4, 10] where the authors proved existence of entropy solution in weighted Sobolev spaces) At the same time, due to the development of the theory regarding the anisotropic Sobolev space W 1,⃗p(.) (where p⃗(.) = (p1 , ..., pN ) is a constant vector), † the

corresponding author. Email address:[email protected]; [email protected](S. Ouaro) 1 Equipe de recherche SIGL, Ecole National des Sciences Appliqu´ ees, Universit´ e Abdelmalek Essaadi, BP 2222 M’hannech T´ etouan, Maroc 2 Laboratoire d’Analyse Math´ ematique et Applications (LAMA), Facult´ e des Sciences Dhar El Mahraz, Universit´ e Sidi Mohamed Ben Abdellah, BP 1796 Atlas F` es, Maroc 3 LA boratoire de Math´ ematiques et Informatique (LAMI), UFR. Sciences Exactes et Appliqu´ ees, Universit´ e de Ouagadougou, 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso

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M.B. Benboubker, H. Hjiaj & S. Ouaro

a new operator takes its place in the mathematical literature (see [7–9, 15, 16, 19]), namely N ( ) ∑ p(x)−2 ∂xi |∂xi u| ∂xi u . ∆p⃗(.) u = i=1

In this paper, we consider Ω a bounded open subset of IRN (N ≥ 2) and pi (x) ∈ ¯ for i = 0, 1, . . . , N, with C+ (Ω) p(x) = min{pi (x),

i = 0, 1, 2, . . . , N }, ∀x ∈ Ω.

(1.1)

Our aim is to prove the existence of an entropy solution to the following nonlinear anisotropic elliptic problem  N   ∑ i − D ai (x, u, ∇u) + g(x, u, ∇u) + |u|p0 (x)−2 u = f − divϕ(u), in Ω, i=1   u = 0, on ∂Ω, (1.2) where the right-hand side f ∈ L1 (Ω). We assume that for i = 1, . . . , N the function ai : Ω × IR × IRN → IR is Carath´eodory (measurable with respect to x in Ω for every (s, ξ) in IR × IRN and continuous with respect to (s, ξ) in IR × IRN for almost every x in Ω) and satisfy the following conditions: |ai (x, s, ξ)| ≤ β (Ki (x) + |s|pi (x)−1 + |ξi |pi (x)−1 ),

for i = 1, . . . , N,

(1.3)

ai (x, s, ξ)ξi ≥ α|ξi | , ′ (ai (x, s, ξ) − ai (x, s, ξ ))(ξi − ξi′ ) > 0,

for i = 1, . . . , N, for ξi = ̸ ξi′ ,

(1.4) (1.5)

pi (x)

for a.e. x ∈ Ω and all (s, ξ) ∈ IR × IRN , where Ki (x) is a nonnegative function lying ′ in Lpi (.) (Ω) and α, β > 0. The nonlinear term g(x, s, ξ) is a Carath´eodory function which satisfies g(x, s, ξ)s ≥ 0,

(1.6) (

|g(x, s, ξ)| ≤ b(|s|) c(x) +

N ∑

)

|ξi |pi (x) ,

(1.7)

i=1

where b : IR+ → IR+ is a continuous nondecreasing function, c ∈ L1 (Ω) and f ∈ L1 (Ω) with ϕ ∈ C 0 (IR, IRN ).

(1.8)

2. Preliminary 2.1. Lebesgue and Sobolev spaces with variable exponents As the exponent pi (.) appearing in (1.3), (1.4) and (1.7) depends on the variable x, we must work with Lebesgue and Sobolev spaces with variable exponents. Set ¯ C+ (Ω) = {p ∈ C(Ω), 1 < p(x) < +∞ , for any x ∈ Ω}.

Entropy solutions to nonlinear elliptic anisotropic ....

247

For any p ∈ C+ (Ω), we define, p+ = max p(x) x∈Ω

p− = min p(x).

and

x∈Ω

We define the Lebesgue space with variable exponent Lp(·) (Ω) as the set of all measurable functions u : Ω → IR for which the convex modular ∫ ρp(x) (u) := |u|p(x) dx Ω

is finite. If the exponent is bounded, i.e. if p+ < +∞, then the expression ∥u∥p(·) = inf{λ > 0 : ρp(·) (u/λ) ≤ 1} defines a norm in Lp(·) (Ω), called the Luxemburg norm. The space (Lp(·) (Ω), ∥.∥p(·) ) is a separable Banach space. Moreover, if 1 < p− ≤ p+ < +∞, then Lp(·) (Ω) is ′ uniformly convex, hence reflexive, and its dual space is isomorphic to Lp (·) (Ω), 1 1 where + = 1. Finally, we have the H¨older type inequality: p(x) p′ (x) ∫ uv dx ≤ ( 1 + 1 )∥u∥p(·) ∥v∥p′ (·) (2.1) p− p′− Ω ′

for all u ∈ Lp(·) (Ω) and v ∈ Lp (·) (Ω). Let W 1,p(·) (Ω) = {u ∈ Lp(·) (Ω) and |∇u| ∈ Lp(·) (Ω)}, which is a Banach space equipped with the following norm ∥u∥1,p(·) = ∥u∥p(·) + ∥∇u∥p(·) . The space (W 1,p(·) (Ω), ∥.∥1,p(·) ) is a separable and reflexive Banach space. An important role in manipulating the generalized Lebesgue and Sobolev spaces is played by the modular ρp(·) of the space Lp(·) (Ω). We have the following result: Proposition 2.1. (see [12, 20]) If un , u ∈ Lp(·) (Ω) and following properties hold true: −

p+ < +∞, then the

+

(i) ∥u∥p(·) > 1 ⇒ ∥u∥pp(·) < ρp(·) (u) < ∥u∥pp(·) ; +

−

p < ρp(·) (u) < ∥u∥pp(·) ; (ii) ∥u∥p(·) < 1 ⇒ ∥u∥p(·)

(iii) ∥u∥p(·) < 1 (respectively = 1; > 1) ⇔ ρp(·) (u) < 1 (respectively = 1; > 1); (iv) ∥un ∥p(·) → 0 (respectively → +∞) ⇔ ρp(·) (un ) → 0 (respectively → +∞) ; (v) ρp(·) (u/∥u∥p(·) ) = 1. (Ω) as the closure of C0∞ (Ω) in W 1,p(·) (Ω) and  N p(·)  , for p(·) < N, p∗ (·) = N − p(·)  ∞, for p(·) ≥ N. 1,p(·)

Next, we define W0

Proposition 2.2. (see [12], [13])

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M.B. Benboubker, H. Hjiaj & S. Ouaro

¯ and q(·) < p∗ (·) for any x ∈ Ω, then the embedding W 1,p(·) (Ω) ,→,→ (i) If q ∈ C+ (Ω) Lq(·) (Ω) is compact and continuous. (ii) Poincar´e inequality: there exists a constant C > 0, such that ∥u∥p(·) ≤ C ∥∇u∥p(·) ,

1,p(·)

∀u ∈ W0

(Ω). 1,p(·)

Therefore, ∥∇u∥p(·) and ∥u∥1,p(·) are equivalent norms in W0

(Ω).

(iii) Sobolev-Poincar´e inequality: there exists another constant C > 0, such that ∥u∥p∗(·) ≤ C ∥∇u∥p(·) ,

1,p(·)

∀u ∈ W0

(Ω).

Remark 2.1. As shown by Zhikov [21, 22] the smooth functions are in general not dense in W 1,p(·) (Ω), but if the variable exponent p ∈ C+ (Ω) is log-H¨older continuous, that is, there is a constant C such that |p(x) − p(y)| ≤

1 C for every x, y ∈ Ω with |x − y| ≤ , −log|x − y| 2

(2.2)

then the smooth functions are dense in W 1,p(·) (Ω), the Sobolev embedding, conver1,p(·) gence of mollifiers regularization and identification of W0 (Ω) with W01,1 (Ω) ∩ 1,p(·) W (Ω) are established.

2.2. Anisotropic Sobolev spaces with variable exponents Now, we present the anisotropic Sobolev space with variable exponent which is used for the study of problem (1.2). Let p0 (x), p1 (x), ...., pN (x) be N + 1 variable exponents in C+ (Ω). We denote p⃗(·) = {p0 (·), . . . , pN (·)}, D0 u = u and Di u =

∂u , ∂xi

for i = 1, . . . , N

and ∀x ∈ Ω,

p(x) = min{pi (x),

i = 0, 1, 2, . . . , N }.

Then p(.) ∈ C + (Ω). The anisotropic variable exponent Sobolev space W 1,⃗p(·) (Ω) is defined as follow W 1,⃗p(·) (Ω) = {u ∈ Lp0 (·) (Ω),

Di u ∈ Lpi (·) (Ω),

i = 1, 2, . . . , N }

endowed with the norm ∥u∥1,⃗p(·) =

N ∑

∥Di u∥Lpi (·) (Ω) .

(2.3)

i=0

We define also W0 (Ω) as the closure of C0∞ (Ω) in W 1,⃗p(·) (Ω) with respect to ′ 1,⃗ p(·) the norm (2.3). The dual of W0 (Ω) is denote by W −1,⃗p(·) (Ω), where p⃗(·)′ = 1 1 ′ ′ {p0 (·), . . . , pN (·)}, p′ (·) + pi (·) = 1, (cf. [6] for the constant exponent case). ( 1,⃗pi(·) ) The space W0 (Ω), ∥u∥1,⃗p(·) is a reflexive Banach space (cf [18]). 1,⃗ p(·)

Entropy solutions to nonlinear elliptic anisotropic ....

249

Let us introduce the following notations: − − ∗ p+ − = max{p1 , ..., pN }; p− =

N N ∑

1 −1 p− i

i=1

∗ ; p−,∞ = max{p+ − , p− }.

Throughout this paper we assume that N ∑ 1 > 1. p− i=1 i

(2.4)

We have the following result (cf. [18]) Theorem 2.1. Assume Ω ⊂ IRN (N ≥ 3) is a bounded domain with smooth boundary. Assume that (2.4) is fulfilled. For any q ∈ C(Ω) verifying 1 < q(x) < p−,∞ , for all x ∈ Ω, the embedding 1,⃗ p(·)

W0

(Ω) ,→ Lq(·) (Ω) is continuous and compact. 1,⃗ p(·)

(Ω) Definition 2.1. We denote the dual of the anisotropic Sobolev space W0 −1,⃗ p(·)′ −1,⃗ p(·)′ p′i (·) by W (Ω) and for each F ∈ W (Ω) there exists fi ∈ L (Ω) for i = N ∑ 1,⃗ p(·) 0, 1, . . . , N, such that F = f0 − (Ω), we have Di fi . Moreover, for all u ∈ W0 i=1

⟨F, u⟩ =

N ∫ ∑ i=0

fi Di u dx. Ω

We define a norm on the dual space by ∥F ∥−1,⃗p(·)′ ≃

N ∑

∥fi ∥p′i (·) .

i=0

Set 1,⃗ p(.)

T0

(Ω) 1,⃗ p(.)

:={u : Ω → IR, measurable such that Tk (u) ∈ W0 1,⃗ p(.)

Proposition 2.3. Let u ∈ T0 tion vi : Ω 7−→ IR such that

(Ω), for any k > 0}.

(Ω).Then, there exists a unique measurable func-

Di Tk (u) = vi .χ{|u| 0, i ∈ {1, . . . , N }, where χA denotes the characteristic function of a measurable set A. The functions vi are called the weak partial gradients of u and are still denoted Di u. Moreover, if u belongs to W01,1 (Ω), then vi coincides with the standard distributional gradient of u, that is, vi = Di u.

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M.B. Benboubker, H. Hjiaj & S. Ouaro

3. Existence of Entropy Solutions ¯ for i = 0, 1, . . . , N, where Let pi (.) ∈ C+ (Ω) ∀x ∈ Ω,

p0 (x) ≥ max{pi (x),

i = 1, 2, . . . , N }.

(3.1)

We can give a simpler definition of an entropy solution of (1.2) as follows. Definition 3.1. A function u is called an entropy solution of the strongly nonlinear 1,⃗ p(.) anisotropic elliptic problem (1.2) if u ∈ T0 (Ω), g(x, u, ∇u) ∈ L1 (Ω), ϕi (u) ∈ ′ Lpi (.) (Ω) for i = 1, ..., N and N ∫ ∑

∫ ai (x, u, ∇u)D Tk (u − v) dx +

i=1 ∫ Ω

+ ∫ ≤

g(x, u, ∇u)Tk (u − v) dx

i

Ω

|u|p0 (x)−2 u Tk (u − v) dx N ∫ ∑ f Tk (u − v) dx + ϕi (u)Di Tk (u − v) dx,

(3.2)

Ω

Ω

i=1 1,⃗ p(.)

for every v ∈ W0

Ω

(Ω) ∩ L∞ (Ω).

Our objective is to prove the following existence theorem. Theorem 3.1. Assuming that (1.3) − (1.7) holds, f ∈ L1 (Ω) and ϕ ∈ C 0 (IR, IRN ), then the problem (1.2) has at least one entropy solution. Remark 3.1. The assumption (3.1) is essential to ensure that |ai (x, u, ∇u)| belongs N ∑ p′i (x) to L (Ω). In the case of Au = − Di ai (x, ∇u) the existence of entropy solution i=1

is guaranteed, without using this assumption. First, we give the following results which will be used in the proof of our main result.

3.1. Useful results 1 Lemma 3.1. (see [14], Theorem 13.47) ∫ Let (un )n∈I ∫ N be a sequence in L (Ω) such that un → u a.e., un , u ≥ 0 a.e. and un dx → u dx, then un → u in L1 (Ω). Ω

Ω

Lemma 3.2. (see [5]) Let g ∈ L (Ω) and gn ∈ Lp(.) (Ω) with ∥gn ∥p(.) ≤ C for 1 < p(x) < ∞. If gn (x) → g(x) a.e. in Ω, then gn ⇀ g in Lp(.) (Ω). p(.)

Lemma 3.3. Assuming that (1.3) − (1.5) hold, and let (un )n∈IN be a sequence in 1,⃗ p(.) 1,⃗ p(.) W0 (Ω) such that un ⇀ u in W0 (Ω) and ∫ (|un |p0 (x)−2 un − |u|p0 (x)−2 u)(un − u) dx Ω N ∫ ( (3.3) ) ∑ ai (x, un , ∇un ) − ai (x, un , ∇u) (Di un − Di u) dx −→ 0, + i=1

Ω 1,⃗ p(.)

then un −→ u in W0

(Ω) for a subsequence.

Entropy solutions to nonlinear elliptic anisotropic ....

Proof.

251

Let Sn

N ∑ (ai (x, un , ∇un ) − ai (x, un , ∇u))(Di un − Di u)

=

i=1

+(|un |p0 (x)−2 un − |u|p0 (x)−2 u)(un − u). Thanks to (1.5), we deduce that Sn is a positive function and, by (3.3), Sn → 0 in L1 (Ω) as n → ∞ . 1,⃗ p(.) Since un ⇀ u in W0 (Ω), using the compact embedding we obtain un → u p(.) in L (Ω), and since Sn → 0 a.e. in Ω, there exists a subset B in Ω with measure zero such that ∀x ∈ Ω\B |u(x)| < ∞, |Di u(x)| < ∞, Ki (x) < ∞,

un → u and Sn → 0.

We have Sn (x) =

N ∑ (ai (x, un , ∇un ) − ai (x, un , ∇u))(Di un − Di u) i=1

+ (|un |p0 (x)−2 un − |u|p0 (x)−2 u)(un − u) N ( ∑ = ai (x, un , ∇un )Di un + ai (x, un , ∇u)Di u i=1

) − ai (x, un , ∇u)Di un − ai (x, un , ∇un )Di u + |un |p0 (x) + |u|p0 (x) − |un |p0 (x)−2 un u − |u|p0 (x)−2 uun ≥α

N ∑

|Di un |pi (x) + α

i=0

N ∑

|Di u|pi (x)

i=0

N ∑ −β (Ki (x) + |un |pi (x)−1 + |Di u|pi (x)−1 )|Di un | i=1 N ∑ − β (Ki (x) + |un |pi (x)−1 + |Di un |pi (x)−1 )|Di u|

− ≥α

i=1 |un |p0 (x)−1 |u|

N ∑

|D un | i

pi (x)

− |u|p0 (x)−1 |un | N ∑ − Cx (1 + |Di un |pi (x)−1 + |Di un |),

i=0

i=1

with α = min(α, 1) and Cx depending only on x. It follows that Sn (x) ≥

N ∑

( |Di un |pi (x) α −

i=0

Cx i |D un |pi (x)

−

) Cx Cx − i p (x)−1 . i i |D un | |D un |

By a standard argument, (Di un )n∈IN is bounded almost everywhere in Ω for all i = 0, 1, . . . , N. Indeed, if |Di un | → ∞ in a measurable subset E ⊂ Ω then ∫ lim Sn (x) dx n→∞

≥ lim

n→∞

Ω

N ∫ ∑ i=0

( |Di un |pi (x) α − E

) Cx Cx Cx − − i dx = ∞, |D un | |Di un |pi (x)−1 |Di un |pi (x)

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M.B. Benboubker, H. Hjiaj & S. Ouaro

which is a contradiction since Sn → 0 in L1 (Ω). Let ξi∗ an accumulation point of (Di un )n∈IN for i = 1, . . . , N. We have |ξi∗ | < ∞ and by the continuity of a(x, ·, ·), we obtain (

) ai (x, un , ξ ∗ ) − ai (x, un , ∇u) (ξi∗ − Di u) = 0,

for i = 1, . . . , N.

Thanks to (1.5), we have ξ ∗ = ∇u and the uniqueness of the accumulation point implies that ∇un → ∇u a.e in Ω. ′ Since (ai (x, un , ∇un ))n∈IN is bounded in Lpi (.) (Ω) and ai (x, un , ∇un ) → ai (x, u, ∇u) a.e in Ω, by the Lemma 3.2, we can establish that ai (x, un , ∇un ) ⇀ ai (x, u, ∇u)

′

Lpi (.) (Ω),

in

for i = 1, . . . , N.

Using (3.3) and the Lemma 3.1, we deduce that |un |p0 (x) −→ |u|p0 (x) ,

L1 (Ω)

in

(3.4)

and ai (x, un , ∇un )Di un −→ ai (x, u, ∇u)Di u,

in

L1 (Ω).

(3.5)

According to the condition (1.4), we have α|Di un |pi (x) ≤ ai (x, un , ∇un )Di un , Let yni = α1 ai (x, un , ∇un )Di un and y i = Lemma, we get ∫

for i = 1, . . . , N.

1 i α ai (x, u, ∇u)D u,

in view of the Fatou’s

∫ 2y i dx ≤ lim inf

(yni + y i − |Di un − Di u|pi (x) )dx,

n→∞

Ω

Ω

∫ which implies that 0 ≤ − lim sup n→∞

|Di un − Di u|pi (x) dx and since Ω

∫

∫

0 ≤ lim inf n→∞

it follows that

|Di un − Di u|pi (x) dx ≤ lim sup n→∞

Ω

|Di un − Di u|pi (x) dx ≤ 0, Ω

∫ |Di un − Di u|pi (x) dx −→ 0 as n → ∞. Ω

Consequently Di un −→ Di u

in Lpi (.) (Ω),

for i = 1, . . . , N.

In view of (3.4) we deduce that un −→ u

in

1,⃗ p(.)

W0

(Ω).

Entropy solutions to nonlinear elliptic anisotropic ....

253

3.2. Proof of Theorem 3.1 Step 1: Approximate problems. ′

Let (fn )n∈IN be a sequence in W −1,⃗p(.) (Ω) ∩ L1 (Ω) such that fn → f in L1 (Ω) and |fn | ≤ |f |. We consider the approximate problem {

An un + gn (x, un , ∇un ) + |un |p0 (x)−2 un = fn − div ϕn (un ), 1,⃗ p(.)

u n ∈ W0

(3.6)

(Ω),

with An v = −

ϕn (s) = ϕ(Tn (s)), and

gn (x, s, ξ) =

N ∑ ∂ ai (x, Tn (v), ∇v) ∂xi i=1

g(x, s, ξ) . 1 + n1 |g(x, s, ξ)|

Note that gn (x, s, ξ)s ≥ 0,

|gn (x, s, ξ)| ≤ |g(x, s, ξ)| 1,⃗ p(.)

We define the operator Gn : W0

′

(Ω) −→ W −1,,⃗p(.) (Ω) by

∫ ⟨Gn u, v⟩ =

∀n ∈ IN ∗ .

|gn (x, s, ξ)| ≤ n,

and

∫ gn (x, u, ∇u)v dx +

Ω

|u|p0 (x)−2 uv dx,

1,⃗ p(.)

∀v ∈ W0

(Ω).

Ω 1,⃗ p(.)

Thanks to the generalized H¨older type inequality, we have for all u, v ∈ W0

(Ω),

∫ ∫ p0 (x)−2 gn (x, u, ∇u)v dx + |u| u v dx Ω Ω ) (

1 ) 1 p0 (x)−1

∥gn (x, u, ∇u) p′ (x) + ∥|u| ∥p′0 (x) v p0 (x) ≤ −+ ′ − 0 (p0 ) p0 (∫ ) ∫ 1 1 ( 1 ′ 1 ) ′ − ′ − ≤ −+ ′ − ( np0 (x) dx + 1) (p0 ) + ( |u|p0 (x) dx + 1) (p0 ) ∥v∥1,⃗p(x) (p0 ) p0 Ω Ω ) ( ∫ 1 1 ( 1 ) ′ + 1 ′ − ′ − ≤ −+ ′ − (n(p0 ) meas(Ω) + 1) (p0 ) + ( |u|p0 (x) dx + 1) (p0 ) ∥v∥1,⃗p(x) (p0 ) p0 Ω ≤C0 ∥v∥1,⃗p(.) . (3.7) 1,⃗ p(.)

Furthermore, we define the operator Rn : W0

′

(Ω) −→ W −1,⃗p(.) (Ω) by

∫ ⟨Rn (u), v⟩ = ⟨div ϕn (u), v⟩ = −

ϕn (u)∇vdx,

1,⃗ p(.)

∀u, v ∈ W0

(Ω),

Ω

with ϕn (u) = (ϕi,n (u), . . . , ϕN,n (u)). Using the generalized H¨older type inequality,

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M.B. Benboubker, H. Hjiaj & S. Ouaro

we deduce that ≤

∫

ϕn (u)∇v dx

Ω N ∑∫

|ϕi,n (u)| |Di v| dx

Ω

i=1

N ∑

( 1 1 )

ϕi,n (u) ′ Di v ≤ − + ′ )− pi (x) pi (x) (p pi i i=1 ∫ N ∑ ( 1 ) ′1 − ′ 1 )( ≤ |ϕi,n (u)|pi (x) dx + 1 (pi ) ∥v∥1,⃗p(x) − + ′ − (pi ) pi Ω i=1

≤

(3.8)

N ∑ ) ′1 − ( 1 ′ + 1 )( sup (|ϕi (s)| + 1)(pi ) meas(Ω) + 1 (pi ) ∥v∥1,⃗p(x) − + ′ − (pi ) pi |s|≤n i=1

≤C1 ∥v∥1,⃗p(.) . In view of Lemma 4.1 (see Appendix), there exists at least one solution un ∈ 1,⃗ p(.) (Ω) of the problem (3.6) (see [17]). W0

Step 2: A priori estimates. Let n large enough. We choose Tk (un ) as a test function in (3.6) to get N ∫ ∑

∫ ai (x, Tn (un ), ∇un )Di Tk (un ) dx +

gn (x, un , ∇un )Tk (un ) dx

Ω

i=1

∫

Ω

|un |p0 (x)−2 un Tk (un ) dx

+ Ω

∫ =

fn Tk (un ) dx + Ω

N ∫ ∑

(3.9)

ϕi,n (un )Di Tk (un ) dx.

Ω

i=1

From (1.4) and Young’s inequality, we obtain N ∫ ∑

α ≤

∫

N ∫ ∑ i=1

∫

≤

Ω

|un |

p0 (x)−2

un Tk (un ) dx

Ω

fn Tk (un ) dx + Ω

gn (x, un , ∇un )Tk (un ) dx

ai (x, Tn (un ), ∇un )Di Tk (un ) dx +

Ω

+ ∫

|Di Tk (un )|pi (x) dx

Ω

i=0

N ∫ ∑ i=1

Ω

ϕi,n (un )Di Tk (un ) dx.

(3.10)

Entropy solutions to nonlinear elliptic anisotropic ....

∫

255

t

ϕi,n (τ ) dτ , then Φi,n (0) = 0 and Φi,n ∈ C 1 (IR), in view of

Taking Φi,n (t) = 0

the Green formula, we obtain ∫ ϕi,n (un )Di Tk (un ) dx

∫ =

∫Ω

Ω

Di Φi,n (Tk (un )) dx Φi,n (Tk (un )) · ni dσ = 0,

=

(3.11)

∂Ω

since un = 0 on ∂Ω, with ⃗n = (n1 , n2 , . . . , nN ) the normal vector on ∂Ω. It follows from (3.10) that N ∑

p−

∥Di Tk (un )∥pi (x) ≤

i=0

k ∥f ∥1 + C2 , α

where p− = min(p(x)). Consequently, x∈Ω

1

∥Tk (un )∥1,⃗p(x) ≤ C3 k p−

for all k ≥ 1.

(3.12)

Now, we will show that (un )n∈IN is a Cauchy sequence in measure. Indeed, by Combining the generalized H¨older type inequality and (3.12), it follows that ∫ ∫ k meas{|un | > k} = |Tk (un )| dx ≤ |Tk (un )| dx {|un |>k}

Ω

( 1 1 ) ≤ − + ′ − ∥1∥p′0 (x) ∥Tk (un )∥p0 (x) (p0 ) p0 1 ( 1 1 ) ′ − ≤ − + ′ − (meas(Ω) + 1) (p0 ) ∥Tk (un )∥1,⃗p(x) (p0 ) p0

(3.13)

1

≤ C 4 k p− , which yields 1

meas{|un | > k} ≤ C4 k

1−

1 p−

−→ 0 as

k → ∞.

(3.14)

Moreover, we have, for every δ > 0, meas{|un − um | > δ} ≤meas{|un | > k} + meas{|um | > k} + meas{|Tk (un ) − Tk (um )| > δ}. Let ε > 0, using (3.14) we may choose k = k(ε) large enough such that meas{|un | > k} ≤

ε 3

and

meas{|um | > k} ≤ 1,⃗ p(.)

Moreover, since the sequence (Tk (un ))n∈IN is bounded in W0 exists a subsequence still denoted (Tk (un ))n∈IN such that Tk (un ) ⇀ ηk

1,⃗ p(.)

in W0

(Ω) as n → ∞

and by the compact embedding, we have Tk (un ) −→ ηk

in

Lp(x) (Ω) and a.e. in Ω.

ε . 3

(3.15)

(Ω), then there

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M.B. Benboubker, H. Hjiaj & S. Ouaro

Consequently, we can assume that (Tk (un ))n∈IN is a Cauchy sequence in measure. Thus, meas{|Tk (un ) − Tk (um )| > δ} ≤

ε , 3

for all m, n ≥ n0 (δ, ε).

(3.16)

Finally, from (3.15) and (3.16), we obtain that ∀δ, ε > 0,

there exists

n0 = n0 (δ, ε) such that

meas{|un − um | > δ} ≤ ε,

∀n, m ≥ n0 (δ, ε),

which proves that the sequence (un )n is a Cauchy sequence in measure and then converges almost everywhere to some measurable function u. Therefore, { 1,⃗ p(.) Tk (un ) ⇀ Tk (u), in W0 (Ω) (3.17) Tk (un ) → Tk (u), in Lp(.) (Ω) and a.e. in Ω. Step 3: Convergence of the gradient. In the sequel, we denote by εi (n), i = 1, 2, . . . a various functions of real numbers which converges to 0 as n tends to infinity. ( )2 Let φk (s) = s. exp(γs2 ), where γ = b(k) . 2α It is obvious that φ′k (s) −

b(k) 1 |φk (s)| ≥ , α 2

∀s ∈ IR.

We also consider h > k > 0 and M = 4k + h and we set ωn = T2k (un − Th (un ) + Tk (un ) − Tk (u)). By taking φk (ωn ) as a test function in (3.6), we obtain N ∫ ∑ i=1

∫

ai (x, Tn (un ), ∇un )φ′k (ωn )Di ωn dx +

∫

Ω

gn (x, un , ∇un )φk (ωn ) dx Ω

|un |p0 (x)−2 un φk (ωn ) dx

+ Ω

∫ =

fn φk (ωn ) dx + Ω

N ∫ ∑ i=1

ϕi,n (un )φ′k (ωn )Di ωn dx.

Ω

It is easy to see that Di ωn = 0 on {|un | > M } and since gn (x, un , ∇un )φk (ωn ) ≥ 0 on {|un | > k}, then N ∫ ∑ i=1

∫

+ ∫ ≤ Ω

ai (x, TM (un ), ∇TM (un ))φ′k (ωn )Di ωn dx Ω

∫ gn (x, un , ∇un )φk (ωn ) dx + |un |p0 (x)−2 un φk (ωn ) dx (3.18) {|un |≤k} {|un |≤k} ∫ fn φk (ωn ) dx + ϕi,n (TM (un ))φ′k (ωn )Di ωn dx. {|un |≤M }

Entropy solutions to nonlinear elliptic anisotropic ....

257

First estimate : We have N ∫ ∑

=

ai (x, TM (un ), ∇TM (un ))φ′k (ωn )Di ωn dx

Ω

i=1

N ∫ ∑ {|un |≤k}

i=1

+

ai (x, Tk (un ), ∇Tk (un ))φ′k (ωn )Di T2k (un − Tk (u)) dx

N ∫ ∑ {|un |>k}

i=1

(3.19)

ai (x, TM (un ), ∇TM (un ))φ′k (ωn )Di ωn dx.

On one hand, since |un − Tk (u)| ≤ 2k on {|un | ≤ k}, then N ∫ ∑

=

i=1

{|un |≤k}

i=1

Ω

N ∫ ∑

+

ai (x, Tk (un ), ∇Tk (un ))φ′k (ωn )Di T2k (un − Tk (u)) dx

ai (x, Tk (un ), ∇Tk (un ))φ′k (ωn )(Di Tk (un ) − Di Tk (u)) dx

N ∫ ∑ {|un |>k}

i=1

(3.20)

ai (x, Tk (un ), ∇Tk (un ))φ′k (ωn )Di Tk (u) dx,

since 1 ≤ φ′k (ωn ) ≤ φ′k (2k), then ∑ ∫ N i=1

{|un |>k}

≤φ′k (2k)

ai (x, Tk (un ), ∇Tk (un ))φ′k (ωn )Di Tk (u) dx

N ∫ ∑ i=1

{|un |>k}

|ai (x, Tk (un ), ∇Tk (un ))| |Di Tk (u)| dx,

′

and since (|ai (x, Tk (un ), ∇Tk (un ))|)n∈IN is bounded in Lpi (.) (Ω), then there exists ′ ′ ϑ ∈ Lpi (.) (Ω) such that |a(x, Tk (un ), ∇Tk (un ))| ⇀ ϑ in Lpi (.) (Ω). Therefore, ∫

∫ |a(x, Tk (un ), ∇Tk (un ))| |D Tk (u)| dx −→ i

{|un |>k}

{|u|>k}

ϑ |Di Tk (u)| dx = 0.

It follows that N ∫ ∑ i=1

{|un |>k}

ai (x, Tk (un ), ∇Tk (un ))φ′k (ωn )Di Tk (u) dx = ε0 (n).

(3.21)

On the other hand, for the second term on the right hand side of (3.19), taking

258

M.B. Benboubker, H. Hjiaj & S. Ouaro

zn = un − Th (un ) + Tk (un ) − Tk (u), then N ∫ ∑

=

ai (x, TM (un ), ∇TM (un ))φ′k (ωn )Di ωn dx

i=1

{|un |>k}

i=1

{|un |>k}∩{|zn |≤2k}

N ∫ ∑

ai (x, TM (un ), ∇TM (un ))φ′k (ωn )

· Di (un − Th (un ) + Tk (un ) − Tk (u)) dx N ∫ ∑ = ai (x, TM (un ), ∇TM (un ))φ′k (ωn )Di (un − Tk (u)).χ{|un |>h} dx i=1

∫

−

{|un |>k}∩{|zn |≤2k}

{|un |>k}∩{|zn |≤2k}

≥−

φ′k (2k)

ai (x, TM (un ), ∇TM (un ))φ′k (ωn )Di Tk (u).χ{|un |≤h} dx

N ∫ ∑

{|un |>k}

i=1

|ai (x, TM (un ), ∇TM (un ))| |Di Tk (u)| dx.

In the same way as for (3.21), we can prove that φ′k (2k)

N ∫ ∑ {|un |>k}

i=1

|ai (x, TM (un ), ∇TM (un ))| |Di Tk (u)| dx = ε1 (n).

(3.22)

After adding (3.19) − (3.22), we obtain N ∫ ∑

≥

ai (x, TM (un ), ∇TM (un ))φ′k (ωn )Di ωn dx

i=1

Ω

i=1

Ω

(3.23)

N ∫ ∑

ai (x, Tk (un ), ∇Tk (un ))φ′k (ωn )(Di Tk (un ) − Di Tk (u)) dx + ε2 (n),

which is equivalent to say N ∫ ∑ i=1

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

Ω

· (Di Tk (un ) − Di Tk (u)) φ′k (ωn ) dx N ∫ ∑ ≤ ai (x, TM (un ), ∇TM (un ))φ′k (ωn )Di ωn dx i=1

−

Ω

N ∫ ∑ i=1

≤

N ∫ ∑ i=1

ai (x, Tk (un ), ∇Tk (u))(Di Tk (un ) − Di Tk (u)) φ′k (ωn ) dx − ε2 (n), Ω

ai (x, TM (un ), ∇TM (un )) φ′k (ωn ) Di ωn dx

Ω

+ φ′k (2k)

N ∫ ∑ i=1

|ai (x, Tk (un ), ∇Tk (u))| |Di Tk (un ) − Di Tk (u)| dx − ε2 (n). Ω

(3.24)

Entropy solutions to nonlinear elliptic anisotropic ....

259

Concerning the second term on the right-hand side of (3.24), by applying the Lebesgue convergence theorem, we have Tk (un ) → Tk (un ) in Lp0 (x) (Ω), then, ′ |ai (x, Tk (un ), ∇Tk (u))| → |ai (x, Tk (u), ∇Tk (u))| in Lpi (x) (Ω), and since Di Tk (un ) converges to Di Tk (u) weakly in Lpi (x) (Ω), we obtain N ∫ ∑ ε3 (n) = φ′k (2k) |ai (x, Tk (un ), ∇Tk (u))||Di Tk (un ) − Di Tk (u)| dx −→ 0, i=1

Ω

as n → ∞.

(3.25)

We conclude that N ∫ ∑

(ai (x, Tk (un ), ∇Tk (un )) − ai (x, Tk (un ), ∇Tk (u))) Ω

i=1

· (Di Tk (un ) − Di Tk (u)) φ′k (ωn ) dx N ∫ ∑ ≤ ai (x, TM (un ), ∇TM (un )) φ′k (ωn ) Di ωn dx + ε4 (n).

(3.26)

Ω

i=1

Second estimate : Now, we treat the second term on the left-hand side of (3.18). From (1.7), we obtain ∫ gn (x, un , ∇un )φk (ωn ) dx ∫ ≤

{|un |≤k}

{|un |≤k}

b(|un |)(c(x) +

|Di Tk (un )|pi (x) )|φk (ωn )| dx

i=1

∫

≤b(k)

N ∑

{|un |≤k}

b(k) ∑ + α i=1 ∫ ≤b(k)

∫

b(k) ∑ + α i=1

∫

N

ai (x, Tk (un ), ∇Tk (un ))Di Tk (un )|φk (ωn )| dx Ω

{|un |≤k} N

c(x)|φk (ωn )| dx

c(x)|φk (ωn )| dx (ai (x, Tk (un ), ∇Tk (un )) − ai (x, Tk (un ), ∇Tk (u)))

Ω

· (Di Tk (un ) − Di Tk (u))|φk (ωn )| dx N ∫ b(k) ∑ + ai (x, Tk (un ), ∇Tk (u))(Di Tk (un ) − Di Tk (u))|φk (ωn )| dx α i=1 Ω N ∫ b(k) ∑ + ai (x, Tk (un ), ∇Tk (un ))Di Tk (u)|φk (ωn )| dx. α i=1 Ω Therefore, b(k) ∑ α i=1 N

∫ (ai (x, Tk (un ), ∇Tk (un )) − ai (x, Tk (un ), ∇Tk (u))) Ω

· (D Tk (un ) − D Tk (u))|φk (ωn )| dx i

i

(3.27)

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M.B. Benboubker, H. Hjiaj & S. Ouaro

∫ ≥

{|un |≤k}

∫ gn (x, un , ∇un )φk (ωn ) dx − b(k)

b(k) ∑ − α i=1 N

b(k) ∑ − α i=1 N

∫

{|un |≤k}

c(x)|φk (ωn )| dx

ai (x, Tk (un ), ∇Tk (u))(Di Tk (un ) − Di Tk (u))|φk (ωn )| dx Ω

∫ ai (x, Tk (un ), ∇Tk (un ))Di Tk (u)|φk (ωn )| dx. Ω

We have φk (ωn ) ⇀ φk (T2k (u − Th (u))) weak-⋆ in L∞ (Ω) as n → +∞, then ∫

∫ {|un |≤k}

c(x)|φk (ωn )| dx −→

{|u|≤k}

c(x)|φk (T2k (u − Th (u)))| dx = 0.

(3.28)

Concerning the third term on the right-hand side of (3.27), thanks to (3.25), we have ∑ ∫ N i i a (x, T (u ), ∇T (u))(D T (u ) − D T (u))|φ (ω )| dx i k n k k n k k n i=1

(3.29)

Ω

≤φk (2k)

N ∫ ∑ i=1

|ai (x, Tk (un ), ∇Tk (u))| |Di Tk (un ) − Di Tk (u)| dx −→ 0 as n → ∞.

Ω

For the last term of the right-hand side of (3.27), we have that ′

(ai (x, Tk (un ), ∇Tk (un )))n∈IN is bounded in Lpi (.) (Ω), ′

′

then there exists ψi ∈ Lpi (.) (Ω) such that ai (x, Tk (un ), ∇Tk (un )) ⇀ ψi in Lpi (.) (Ω) and, using the fact that Di Tk (u) |φk (ωn )| ⇀ Di Tk (u) |φk (T2k (u − Th (u)))|

in Lpi (.) (Ω),

it follows that ∫ ai (x, Tk (un ), ∇Tk (un ))Di Tk (u)|φk (ωn )| dx ∫Ω −→

(3.30) ψi Di Tk (u)|φk (T2k (u − Th (u)))| dx = 0.

Ω

By combining (3.27) − (3.30), we get b(k) ∑ α i=1 N

∫ (ai (x, Tk (un ), ∇Tk (un )) − ai (x, Tk (un ), ∇Tk (u))) Ω

· (Di Tk (un ) − Di Tk (u))|φk (ωn )| dx ∫ ≥ gn (x, un , ∇un )φk (ωn ) dx + ε5 (n). {|un |≤k}

Third estimate :

(3.31)

Entropy solutions to nonlinear elliptic anisotropic ....

261

We have ∫ ∫

{|un |≤k}

= ∫Ω

|un |p0 (x)−2 un φk (ωn ) dx

|Tk (un )|p0 (x)−2 Tk (un )(Tk (un ) − Tk (u)) exp(γωn2 ) dx

{|un |>k}

∫ ( = ∫

|Tk (un )|p0 (x)−2 Tk (un )(Tk (un ) − Tk (u)) exp(γωn2 ) dx

) |Tk (un )|p0 (x)−2 Tk (un ) − |Tk (u)|p0 (x)−2 Tk (u) (Tk (un ) − Tk (u)) exp(γωn2 ) dx

Ω

|Tk (u)|p0 (x)−2 Tk (u)(Tk (un ) − Tk (u)) exp(γωn2 ) dx

+

∫ − |Tk (un )|p0 (x)−2 Tk (un )(Tk (un ) − Tk (u)) exp(γωn2 ) dx {|u |>k} ∫ ( n ) ≥ |Tk (un )|p0 (x)−2 Tk (un ) − |Tk (u)|p0 (x)−2 Tk (u) (Tk (un ) − Tk (u)) dx Ω ∫ − exp(γ(2k)2 ) |Tk (u)|p0 (x)−1 |Tk (un ) − Tk (u)| dx Ω ∫ 2 − exp(γ(2k) ) k p0 (x)−1 |Tk (un ) − Tk (u)| dx Ω

{|un |>k}

and as Tk (un ) → Tk (u) in Lp0 (.) (Ω), then the second and the last term on the righthand side of the inequality above converges to 0 as n goes to infinity. Therefore, we get ∫ (|Tk (un )|p0 (x)−2 Tk (un ) − |Tk (u)|p0 (x)−2 Tk (u))(Tk (un ) − Tk (u)) dx Ω ∫ (3.32) ≤ |un |p0 (x)−2 un φk (ωn ) dx + ε6 (n). {|un |≤k}

Thanks to (3.26), (3.31) and (3.32), we obtain 1∑ 2 i=1 N

∫ (ai (x, Tk (un ), ∇Tk (un )) − ai (x, Tk (un ), ∇Tk (u))) Ω

· (Di Tk (un ) − Di Tk (u)) dx ∫ + (|Tk (un )|p0 (x)−2 Tk (un ) − |Tk (u)|p0 (x)−2 Tk (u))(Tk (un ) − Tk (u)) dx Ω

≤

N ∫ ∑ i=1

ai (x, TM (un ), ∇TM (un ))φ′k (ωn )Di ωn dx Ω

∫ − gn (x, un , ∇un )φk (ωn ) dx {|un |≤k} ∫ + |un |p0 (x)−2 un φk (ωn ) dx + ε7 (n), {|un |≤k}

∫ ≤

fn φk (ωn ) dx + Ω

N ∫ ∑ i=1

{|un |≤M }

ϕi,n (TM (un ))φ′k (ωn )∇ωn dx + ε7 (n).

(3.33)

262

M.B. Benboubker, H. Hjiaj & S. Ouaro

∫

We have

∫ f φk (T2k (u − Th (u))) dx + ε8 (n)

fn φk (ωn ) dx = Ω

(3.34)

Ω

and for n large enough (for example n ≥ M ), we can write ∫ ∫ ϕi,n (TM (un ))φ′k (ωn )Di ωn dx = ϕi (TM (un ))φ′k (ωn )Di ωn dx. Ω

Ω

It follows that ∫ ϕi,n (TM (un ))φ′k (ωn )Di ωn dx Ω ∫ ϕi (TM (u))φ′k (T2k (u − Th (u)))Di T2k (u − Th (u)) dx + ε9 (n). =

(3.35)

Ω

By combining (3.33), (3.34) and (3.35), we obtain N ∫ ∑ (ai (x, Tk (un ), ∇Tk (un )) − ai (x, Tk (un ), ∇Tk (u)))(Di Tk (un ) − Di Tk (u)) dx Ω

i=1

∫

+ (|Tk (un )|p0 (x)−2 Tk (un ) − |Tk (u)|p0 (x)−2 Tk (u))(Tk (un ) − Tk (u)) dx ∫ Ω ≤2 f φk (T2k (u − Th (u))) dx Ω

+2

N ∫ ∑ i=1

ϕi (TM (u))φ′k (T2k (u − Th (u)))Di T2k (u − Th (u)) dx + ε10 (n).

Ω

(3.36)

∫

t

Taking Ψi (t) =

ϕi (τ )φ′k (τ − Th (τ )) dτ , then Ψi (0) = 0 and Ψi ∈ C 1 (IR).

0

In view of the Green formula, we have ∫ ϕi (TM (u))φ′k (T2k (u − Th (u)))Di T2k (u − Th (u)) dx Ω ∫ = ϕ(u)φ′k (u − Th (u))Di u dx {h