GENERALIZED P ( X )-ELLIPTIC SYSTEM WITH NONLINEAR PHYSICAL DATA

This paper considers the following Dirichlet problem of the form −div ( Φ(Du−Θ(u) ) = v(x) + f(x, u) + div ( g(x, u) ) , which corresponds to a diffusion problem with a source v in moving and dissolving substance, the motion is described by g and the dissolution by f . By the theory of Young measure we will prove the existence result in variable exponent Sobolev spaces W 1,p(x) 0 (Ω;R).


Introduction and main result
Let Ω be a bounded open domain in R n , n ≥ 2. Throughout this paper, we denote by M m×n the set of real m by n matrices equipped with the inner product ξ : η = i,j ξ ij η ij . For ξ ∈ M m×n , |ξ| is the norm of ξ when regarded as a vector of R mn . In [6], the following quasilinear elliptic system was considered    −div Φ(Du − Θ(u)) = v in Ω, u = 0 on ∂Ω, (1.1) where Du is the symmetric part of the gradient of u : Ω → R m , v belongs to W −1,p ′ (Ω; R m ), 1 p + 1 p ′ = 1, Φ(ξ) = |ξ| p−2 ξ for all ξ ∈ M m×n and 1 < p < ∞ (a constant). And Θ : R m → M m×n is a continuous function satisfying where the positive constant c satisfies When the exponent p is not constant, but depends on x, i.e. p ≡ p(x), Azroul and Balaadich [8] established the existence result for (1.1), where v belongs to W −1,p ′ (x) (Ω; R m ) and the constant c in (1.2) is assumed to satisfy c < 1 diam(Ω) Here p + is the essential sup of the variable exponent p(x) for x ∈ Ω. They used also the tool of Young measure in establishing their result.
The operator −div |Du| p(x)−2 Du is said to be the p(x)-Laplacian and becomes p-Laplacian when p(x) ≡ p. When p(x) > 2 (resp. 1 < p(x) < 2), then the problem (1.1) is nonlinear degenerate (resp. singular) elliptic systems. Partial differential equations with nonlinearities involving nonconstant exponents have attracted an increasing amont of attention in recent years. The impulse for this maybe comes from the sound physical applications in play, or perhaps it is just the thrill of developping a mathematical theory where PDEs again meet functional analysis in a truly two-way street.
As we know, the p(x)-Laplacian is inhomogeneous. This implies that it posesses more complicated nonlinearities than the case of p constant. Problems with variable exponent appear in several domains. For example; in the mathematical modeling of stationary thermorheological viscous of the process filtration of an ideal barotropic gas through a porous medium (cf. [2,3]). In image processing [18], to outline the borders of a true image and to elliminate possible noise, the variable nonlinearity find its applications. For the case of calculus of variations, the reader can see [1,14] and references therein.
The authors in [10] considered the following Here ∇×u is the curl of u = (u 1 , u 2 , u 3 ). They studied the existence and nonexistence of solutions. Note that the above system is arising in electromagnetism. E. Azroul et al. [4] investigated a class of nonlinear p(x)-Laplacian problems, in the scalar case, of the form where the source term f was assumed to belong to L 1 (Ω). They used the techniques of entropy solutions to prove the existence of a solution. See also [5,13]. In this paper we shall prove the existence of weak solutions for the following problem which is motivated by physics or geometry: supplemented with the Dirichlet boundary condition u = 0 on ∂Ω.
Here v belongs to for all ξ ∈ M m×n and Θ : R m → M m×n is a continuous function such that where c is a positive constant that satisfies c < 1 diam(Ω) 1 2 1 p + . Moreover, f and g satisfy the following continuity and growth conditions: (G0)(Continuity) g : Ω × R m −→ M m×n is a Carathéodory function in the sense of (F0).
1) The strict bound p(x) − 1 for γ(x) and q(x) in the growth conditions (F1) and (G1) ensures the coercivity of the operator T introduced in Section 3.
2) The function f may depend even on the Jacobien matrix Du := 1 2 ∇u+(∇u) t and linear with respect to its variable ξ ∈ M m×n , see Appendix.
Let u : Ω → R m be a vector-valued function.
We shall prove the following existence theorem: The outline of the present paper is as follows: In section 2, we introduce the functional space and its properties, and a brief review on the theory of Young measure. Section 3 is devoted to construct the approximating solutions by the Galerkin method and we give the proof of the main result. This paper ends with an appendix.

Variable exponent Lebesgue and Sobolev spaces
In order to discuss the solutions to the Dirichlet problem (1.4), we need some theories and basic properties of variable exponent Lebesgue and Sobolev spaces (Ω; R m ) respectively (see [16,18,21] and other references therein).
Let Ω be a bounded open domain of R n . We denote C + (Ω) the following set Throughout this paper, for every p ∈ C + (Ω). We define the modular of a measurable function u : Ω → R m by The variable exponent Lebesgue space L p(x) (Ω; R m ) is a Banach space that is the set of all measurable functions u : Ω → R m such that its modular In these spaces, a version of Hölder's inequality For the relation between the modular ρ p(.) (.) and the norm . p(x) , we recall the following properties: if We define the generalized Lebesgue-Sobolev space This space is again a special case of Sobolev-Orlicz spaces and inherits many of the properties of L p(x) (Ω; R m ). In particular, As in [19], the following Poincaré's inequality: there exists a positive constant α = diam(Ω) such that together with Hölder's inequality, are central in this paper. Let us summarize the above properties in the following proposition:

Young measures
The weak convergence is a basic tool of modern nonlinear analysis, because it has the same compactness properties as the convergence in finite dimensional spaces as discussed in the paper of Evans [15]. However this convergence, sometimes, does not behave as we desire with respect to nonlinear functionals and operators. To solve this difficulties, we can use the tool of Young measure which we will use in this paper to prove the needed result.
For convenience of the reader not familiar with this concept, we recall some basic notions and properties (see eg. [11,15,17,20] and references therein).
By C 0 (R m ) we denote the set of functions g ∈ C(R m ) satisfying lim |λ|→∞ g(λ) = 0. Its dual is the well known space of signed Radon measures with finite mass and denoted by M(R m ). The duality pairing of these spaces is defined for ν : Ω −→ M(R m ) as Then there exist a subsequence (still denoted by {w k }) and a Borel probability measure ν x on R m for a.e. x ∈ Ω, such that for each We call ν = {ν x } x∈Ω the family of Young measure associated with the subsequence {w k }. In [11] it is shown that if for all R > 0 weakly in L 1 (Ω ′ ), for any Carathéodory function h : The Young measure associated to the sequence (y k , w k ) is given by if y k → y in measure and if ν x is the Young measure associated to w k .

Lemma 2.2 ( [8])
. Let (w k ) be the sequence defined as above. Then the Young measure ν x generated by Dw k in L p(x) (Ω; M m×n ) has the following properties: (i) ν x is a probability measure, i.e. ν x M(M m×n ) = 1 for almost every x ∈ Ω.
(ii) The weak L 1 -limit of Dw k is given by ν x , id = M m×n λdν x (λ).

Approximating solutions
To construct the approximating solutions, we will use the Galerkin method. To this purpose, we consider the following map T : W (Ω; R m ). As a consequence, our problem (1.4) is then equivalent to find u ∈ W 1,p(x) 0 (Ω; R m ) such that T (u), ϕ = 0 for all ϕ ∈ W 1,p(x) 0 (Ω; R m ).

Proof. For arbitrary
Note that since |a + b| r ≤ 2 r−1 |a| r + |b| r for r > 1, it follows by Hölder's inequality that The generalized Hölder inequality implies that On the other hand, it follows from the growth condition (F1) (without loss of generality, we can assume that γ(x) = p(x) − 1), that Finally, the growth condition (G1) (without loss of generality, we may assume that q(x) = p(x) − 1) allows to estimate (by application of the Hölder inequality) By virtual of the Poincaré inequality, the I i for i = 1, .., 4 are finite, then T (u) is well defined. Moreover, for all ϕ ∈ W 1,p(x) 0 (Ω; R m ) and this implies that T (u) is bounded.

Lemma 3.2. The restriction of T to a finite dimensional linear subspace
Proof. Let r be the dimension of V and (e i ) r i=1 a basis of V . Let (u j = a i j e i ) be a sequence in V which converges to u = a i e i in V (with conventional summation). Then u j → u and Du j → Du almost everywhere. The continuity of Θ, f and g implies that According to [12] (Chapter IV, Section 3, Theorem 3) there exist a subsequence of {u j } still denoted by {u j } and h 1 , h 2 ∈ L 1 (Ω) such that |u j − u| p(x) ≤ h 1 , |Du j − Du| p(x) ≤ h 2 . By vertue of the Eq (3.1), we can write from which (similarly) we get |Du j | p(x) ≤ 2 p + −1 h 2 + |Du| p(x) . Consequently, the sequences u j p(x) and Du j p(x) are bounded by a constant denoted C. Now, if Ω ′ ⊂ Ω is a measurable subset and ϕ ∈ W 1,p(x) 0 (Ω; R m ), then by Poincaré's inequality where the small c is the constant in (1.5), and (without loss of generality, we can assume that γ( and Therefore, the sequences Φ(Du j − Θ(u j )) : Dϕ , f (x, u j )ϕ and g(x, u j ) : Dϕ are equiintegrable, since Ω ′ |Dϕ| p(x) dx is arbirary small if the measure of Ω ′ is chosen small enough. Applying the Vitali Theorem, it follows for all ϕ ∈ W 1,p(x) 0 (Ω; R m ) that lim j→∞ T (u j ), ϕ = T (u), ϕ as we desire.

Lemma 3.3. The mapping T is coercive.
Proof. Taking ϕ = u in the definition of T , then then for ξ = Du − Θ(u) and η = −Θ(u), we get On the other hand, by (3.1) by definition of the function Θ and Poincaré's inequality. Next the Hölder inequality implies that Finally, it follows from the growth conditions (F1) and (G1) that and From these estimations it follows that (Ω; R m ) be a sequence of finite dimensional subspaces with the property that ., e r is a basis of V k for a fixed k. To construct the approximating solution, we define the map 3) The sequence constructed in 2) is uniformly bounded in W 1,p(x) 0 (Ω; R m ), i.e. there exists a constant R > 0 such that (3.4) Proof. 1) Since T restricted to V k is continuous (see Lemma 3.2), then S is continuous.
2) Let a ∈ R r and u = a i e i ∈ V k . Then S(a).a = T (u), u . Notice that a R r → +∞ is equivalent to u 1,p(x) → +∞. It follows by Lemma 3) We have by Lemma 3.3, that T (u), u → +∞ as u 1,p(x) → +∞. It result then the existence of R > 0 with the property, that T (u), u > 1 whenever u 1,p(x) > R. Taking this into consideration and the sequence of Galerkin approximations u k ∈ V k which satisfy T (u k ), u k = 0 (by 2)), it follows that (u k ) is uniformly bounded. Before we pass to the limit in the approximating sequences and so to prove Theorem 1.1, notice that since (u k ) is bounded in W Proof of Theorem 1.1. To apply the convergence described in Lemma 2.2 to our approximating problem, we need the convergence in measure of u k to u. To this purpose, consider (Ω; R m ), then for a subsequence still denoted u k , u k → u in L p(x) (Ω; R m ). Therefore which implies that Hence the sequence u k converges in measure to u on Ω. On the other hand, since {Du k −Θ(u k )} is equiintegrable by the condition (1.5) and the boundedness of (u k ), it result that weakly in L 1 (Ω), where we have used Lemma 2.2. Further, from the reflexivity of L p ′ (x) (Ω) and the boundedness of {Φ(Du k − Θ(u k ))}, we deduce that Φ(Du k − Θ(u k )) converges in L p ′ (x) (Ω) and its weak L p ′ (x) -limit is given by Φ(Du − Θ(u)). Consequently Moreover, since u k → u in measure for k → ∞, we may infer that, after extraction of a suitable subsequence, if necessary, u k −→ u almost everywhere for k → ∞.