CONTINUOUS IMBEDDING IN MUSIELAK SPACES WITH AN APPLICATION TO ANISOTROPIC NONLINEAR NEUMANN PROBLEMS

. We prove a continuous embedding that allows us to obtain a boundary trace imbedding result for anisotropic Musielak-Orlicz spaces, which we then apply to obtain an existence result for Neumann problems with non-linearities on the boundary associated to some anisotropic nonlinear elliptic equations in Musielak-Orlicz spaces constructed from Musielak-Orlicz functions on which and on their conjugates we do not assume the ∆ 2 -condition. The uniqueness of weak solutions is also studied.


Introduction
Let Ω be an open bounded subset of R N , (N ≥ 2). We denote by φ : Ω × R + → R N the vector function φ = (φ 1 , . . . , φ N ) where for every i ∈ {1, . . . , N }, φ i is a Musielak-Orlicz function differentiable with respect to its second argument whose complementary Musielak-Orlicz function is denoted by φ * i (see preliminaries). We consider the problem a i (x, ∂ xi u)ν i = g(x, u) on ∂Ω, (1.1) where ∂ xi = ∂ ∂x i and for every i = 1, . . . , N , we denote by ν i the i th component of the outer normal unit vector and a i : Ω × R → R is a Carathéodory function such that there exist a locally integrable Musielak-Orlicz function (see definition 1.1 below) P i : Ω × R + → R + with P i φ i , a positive constant c i and a nonnegative function d i ∈ E φ * i (Ω) satisfying for all s, t ∈ R and for almost every x ∈ Ω the following assumptions
Defining p max (x) = max i∈{1,...,N } p i (x) and p min (x) = min i∈{1,...,N } p i (x), one has φ max (x, t) = |t| p M (x) and then ϕ max (x, t) = p M (x)|t| p M (x)−2 t, where p M is p max or p min according to whether |t| ≥ 1 or |t| ≤ 1 and then the space W 1 L φ (Ω) is nothing but the anisotropic space with variable exponent W 1, p(·) (Ω), where p(·) = (p 1 (·), . . . , p N (·)) (see [7] for more details on this space). Therefore, the problem (1.1) can be rewritten as (1. 10) where b 1 (x) = p M (x)b(x). Boureanu and Rǎdulescu [2] have proved the existence and uniqueness of the weak solution of (1.10). They prove an imbedding and a trace results which they use together with a classical minimization existence result for functional reflexive framework (see [22,Theorem 1.2]). Problem (1.10) with Dirichlet boundary condition and b 1 (x) = 0 was treated in [15]. The authors proved that if f (·, u) = f (·) ∈ L ∞ (Ω) then (1.10) admits a unique solution by using [22,Theorem 1.2]. The problem (1.10) with for all i = 1, . . . , N with p ∈ C 1 (Ω) and b 1 = g = 0 was treated in [12], where the authors proved the three nontrivial smooth solutions, two of which have constant sign (one positive, the other negative). In connection with Neumann problems, the authors [21] studied the problem − div a(∇u(z)) + ζ(z) + λ u(z) p−1 = f (z, u(z)) in Ω, ∂u ∂n = 0 on ∂Ω u > 0, λ > 0, 1 < p < +∞, (1.11) where the function a : R N → R N is strictly monotone, continuous and satisfies certain other regularity and growth conditions. The function ζ involved in (1.11) changes its sign and is such that ζ ∈ L ∞ (Ω). The reaction term f (z, x) is a Carathéodory function. They proved the existence of a critical parameter value λ * > 0 such that if λ > λ * problem (1.11) has at least two positive solutions, if λ = λ * (1.11) has at least a positive solution and if λ ∈ (0, λ * ) problem (1.11) has no positive solution. Let us mention some related results in the framework of Orlicz-Sobolev spaces. Le and Schmitt [17] proved an existence result for the boundary value problem − div(A(|∇u| 2 )∇u) + F (x, u) = 0, in Ω, where φ(s) = A(|s| 2 )s and F is a Carathéodory function satisfying some growth conditions. This result extends the one obtained in [11] with F (x, u) = −λψ(u), where ψ is an odd increasing homeomorphism of R onto R. In [11,17] the authors assume that the N -function φ * complementary to the N -function φ satisfies the ∆ 2 condition, which is used to prove that the functional u → Ω Φ(|∇u|) dx is coercive and of class C 1 , where Φ is the antiderivative of φ vanishing at the origin.
Here we are interested in proving the existence and uniqueness of the weak solutions for problem (1.1) without any additional condition on the Musielak-Orlicz function φ i or its complementary φ * i for i = 1, . . . , N . Therefore, the resulting Musielak-Orlicz spaces L φi (Ω) are neither reflexive nor separable and thus classical existence results can not be applied.
The approach we use consists in proving first a continuous imbedding and a trace result which we then apply to solve the problem (1.1). The results we prove extend to the anisotropic Musielak-Orlicz-Sobolev spaces the continuous imbedding obtained in [6] under some extra conditions and the trace result proved in [18]. The imbedding result we obtain extends to Musielak spaces a part of the one obtained in [19] in the anisotropic case and that of Fan [9] in the isotropic case (see Remark 3.2). In the variable exponent Sobolev space W 1,p(x) (Ω) where 1 < p + = sup x∈Ω p(x) < N , other imbedding results can be found for instance in [3,4,16] while the case 1 ≤ p − ≤ p + ≤ N was investigated in [13].
To the best of our knowledge, the trace result we obtain here is new and does not exist in the literature. The main difficulty we found when we deal with problem (1.1) is the coercivity of the energy functional. We overcome this by using both our continuous imbedding and trace results. Then we prove the boundedness of a minimization sequence and by a compactness argument, we are led to obtain a minimizer which is a weak solution of problem (1.1).
Let Ω be an open subset of R N , (N ≥ 2). We say that a Musielak-Orlicz function φ is locally-integrable, if for every compact subset K of Ω and every constant c > 0, we have The article is organized as follows: Section 2 contains some definitions. In Section 3, we give and prove our main results, which we then apply in Section 4 to solve problem (1.1). In the last section we give an appendix which contains some important lemmas that are necessary for the accomplishment of the proofs of the results.

Anisotropic Musielak-Orlicz-Sobolev spaces.
Let Ω be an open subset of R N . A real function φ : Ω × R + → R + will be called a Musielak-Orlicz function if it satisfies the following conditions (i) φ(·, t) is a measurable function on Ω.
(ii) φ(x, ·) is an N -function, that is a convex nondecreasing function with φ(x, t) = 0 if only if t = 0, φ(x, t) > 0 for all t > 0 and for almost every x ∈ Ω, We will extend these Musielak-Orlicz functions into even functions on all Ω × R.
The complementary function φ * of the Musilek-Orlicz function φ is defined by It can be checked that φ * is also a Musielak-Orlicz function (see [20]). Moreover, for every t, s ≥ 0 and a.e. x ∈ Ω we have the so-called Young inequality (see [20]) For any function h : R → R the second complementary function h * * = (h * ) * (cf. (2.1)), is convex and satisfies 2) with equality when h is convex. Roughly speaking, h * * is a convex envelope of h, that is the biggest convex function smaller or equal to h.
Let φ and ψ be two Musielak-Orlicz functions. We say that ψ grows essentially more slowly than φ, denote ψ φ, if for every constant c > 0 and for almost every x ∈ Ω. We point out that if ψ : Ω × R + → R + is locally integrable then ψ φ implies that for all c > 0 there exists a nonnegative function h ∈ L 1 (Ω) such that ψ(x, t) ≤ φ(x, ct) + h(x), for all t ∈ R and for a.e. x ∈ Ω.
Endowed with the so-called Luxemborg norm is a Banach space. Observe that since lim t→+∞ inf x∈Ω φ(x,t) t = +∞ and if Ω has finite measure then we have the following continuous imbedding We will also use the space Observe that for every u ∈ L φ (Ω) the following inequality holds For two complementary Musielak-Orlicz functions φ and φ * , Hölder's inequality (see [20]) holds for every u ∈ L φ (Ω) and v ∈ L φ * (Ω). Define φ * −1 for every s ≥ 0 by Then, for almost every x ∈ Ω and for every s ∈ R we have φ(x, s) ≤ s ∂φ(x, s) ∂s ≤ φ(x, 2s). (2.8) where for every i ∈ {1, . . . , N }, φ i is a Musielak-Orlicz function. We define the anisotropic Musielak-Orlicz-Sobolev space by By the continuous imbedding (2.3), we obtain that W 1 L φ (Ω) is a Banach space with respect to the following norm Moreover, we have the continuous embedding W 1 L φ (Ω) → W 1,1 (Ω).

Main results
In this section we prove an imbedding theorem and a trace result. Let us assume the conditions (3.1) Thus, we define the Sobolev conjugate (φ * * min ) * It may readily be checked that (φ * * min ) * is a Musielak-Orlicz function. We assume that there exist two positive constants ν < 1 N and c 0 > 0 such that for all t ∈ R and for almost every x ∈ Ω, provided that for every i = 1, . . . , N the derivative 3.1. Imbedding theorem. is Lipschitz continuous on Ω and φ max is locally integrable. Then, there is a continuous embedding . Some remarks about Theorem 3.1 are in order. We discuss how Theorem 3.1 include some previous results known in the literature when reducing to some particular Musielak-Orlicz functions.
is Lipschitz continuous on Ω, with 1 < p − = inf x∈Ω p(x) ≤ p(x) ≤ p + = sup x∈Ω p(x) < N . Since M (·, t) and m(·, t) are continuous on Ω, we can use Lemma 5.8 (given in Appendix) to define the following Musielak-Orlicz function where t 1 > 1 and α > 1 are two constants mentioned in the proof of Lemma 5.8. Let us now consider the particular case where for all i = 1, . . . , N , It is worth pointing out that since Ω is of finite Lebesgue measure, it can be seen easily that provided that α < N . Now we shall prove that (φ * * min ) * satisfies (3.3) and our imbedding result include some previous result known in the literature. For every t ∈ R and for almost every x ∈ Ω we have Since p(·) is Lipschitz continuous on Ω there exists a constant C 1 > 0 satisfying Since p(·) is Lipschitz continuous on Ω, it can be seen easily that p * (·) is also Lipschitz continuous on Ω. Then, there exists a constant C 2 > 0 satisfying ∂p * ∂xi (x) ≤ C 2 . So that we have Let 0 < < 1/N . For all t > 0 we can easily check that Now, since the Musielak-Orlicz function (φ * * min ) * has a superlinear growth, we can choose A > 0 for which there exists t 0 > max{t 1 , e} (not depending on x) such that (3.7) Therefore, from (3.5), (3.7) and (3.8), we obtain that for every t ≥ 0 and for almost every x ∈ Ω, there is a constant c 0 > 0 such that Before we show that our imbedding result includes some previous known results in the literature, we remark that the proof of Theorem 3.1 relies to the application of Lemma 5.4 in Appendix for the function g(x, t) = ((φ * * min ) * (x, t)) α , α ∈ (0, 1), where we have used the fact that Ω is bounded to ensure that max x∈Ω g(x, t) < ∞ for some t > 0. In the case of the variable exponent Sobolev space W 1,p(·) (Ω) built upon the Musielak-Orlicz function given in (3.4), we do not need Ω to be bounded, since Therefore, the embedding result in Theorem 3.1 can be seen as an extension to the Musielak-Orlicz framework of the one obtained in [9, Theorem 1.1].
(2) Let us consider the particular case where, for i ∈ {1, . . . , N }, Since Ω is of finite Lebesgue measure, it can be seen easily that . Therefore, by (3.9) the result we obtain can be found in [19,Theorem 1].
(3) Let us now consider the case where for x ∈ Ω and t ≥ t 0 . Choosing this t 0 > 0 in Lemma 5.8 given in Appendix, we can take t 1 > t 0 + 1 obtaining On the other hand, for t ≤ t 1 we have Since p m (·) is Lipschitz continuous on Ω there exists a constant C 3 > 0 satisfying Therefore, by (3.10) and (3.11) the function (φ * * min ) * satisfies the assertions of Theorem 3.1 and then we obtain the continuous embedding dx, for s > 0, one has lim s→0 + f (s) = +∞ and lim s→∞ f (s) = 0. Since f is continuous on (0, +∞), there exists λ > 0 such that f (λ) = 1. Then by the definition of the Luxemburg norm, we obtain On the other hand, So that by (3.12) and (3.13), we obtain λ = u (φ * * min ) * and From (3.2) we can easily check that (φ * * min ) * satisfies the differential equation Hence, by (2.7) we obtain the inequality Since (φ * * min ) * (·, t) is Lipschitz continuous on Ω and (φ * * min ) * (x, ·) is locally Lipschitz continuous on R + , the function h is Lipschitz continuous on Ω. Hence, we can compute using Lemma 5.6 (given in Appendix) for f = h, obtaining for a.e. x ∈ Ω, where we have set |∂ xi u(x)|, Now we estimate the two integrals Ω I 1 (x) dx and Ω I 2 (x) dx. By (3.15), we can write From (2.5), (3.18) and (3.19) it follows that Recalling the definition of φ min and (2.2), we obtain ∂ xi u(x) φ * * min ≤ ∂ xi u(x) φi , so that (3.20) implies Using (3.3) we can write is continuous on Ω and ν < 1 N , we can apply Lemma 5.4 (given in Appendix) with the functions g(x, t) = Using again Lemma 5.4 with the functions g(x, t) = and = 1 8c1c * , we obtain by substituting t by |u(x)| λ (φ * * min ) * x, where c * is the constant in the continuous embedding By (3.22) and (3.23), we obtain Putting together (3.21) and (3.25) in (3.17) we obtain where c 2 is the constant in the continuous embedding (2.3). Then it follows that with c 3 = max{2, 2K 0 c 1 c 2 }. Now, using again Lemma 5.4 (in Appendix) with the Thus, by (3.26) and (3.27) we obtain where c 4 = c 3 + K 0 c 2 , which shows that h ∈ W 1,1 (Ω) and which together with Having in mind (3.14), we obtain So that one has We now extend the estimate (3.28) to an arbitrary u ∈ W 1 L φ (Ω). Let T n , n > 0, be the truncation function at levels ±n defined on R by T n (s) = min{n, max{s, −n}}.
By (3.2), we obtain Being the inverse of a Musielak-Orlicz function, it is clear that (φ * * min ) −1 satisfies It follows that ∂ ∂t (ψ min ) −1 (x, t) is positive and decreases monotonically from +∞ to 0 as t increases from 0 to +∞ and thus ψ min is a Musielak-Orlicz function.
As above we can prove that (φ * * min ) * satisfies the conditions of Theorem 3.4 and then our trace result is an extension to Musielak-Orlicz framework of the one proved by Fan in [8].

Finally, in both cases there exists a constant c > 0 such that
For an arbitrary u ∈ W 1 L φ (Ω), we proceed as in the proof of Theorem 3.1 by truncating the function u.

Application to anisotropic elliptic equations
In this section, we apply the above results to obtain the existence and uniqueness of the weak solution for the problem (1.1).

4.1.
Properties of the energy functional.
We note that all the terms in (4.1) make sense. Indeed, for the first term in the right hand side in (4.1), we can write by using (2.8) where P i is the Musielak-Orlicz function given in (1.2) and p i (x, s) = ∂Pi ∂s (x, s). Since P i is locally integrable and P i φ i , we can use Lemma 5.7 (see the Appendix) obtaining p i (·, ∂ xi u(·)) ∈ L P * i (Ω). So that by Hölder's inequality (2.5), we obtain Since v ∈ C ∞ (Ω) and φ max is locally integrable, then v ∈ W 1 L φ (Ω). So we can use the growth condition (1.2) and again the Hölder inequality (2.5) to write For the second term, the inequality (2.8) enables us to write where R is the Musielak-Orlicz function given in (1.5) and r(x, s) = ∂R(x,s) ∂s . Since R is locally integrable and R φ max , Lemma 5.7 (in Appendix) gives which shows that ϕ max (·, u(·)) ∈ L φ * max (Ω). Thus, We now turn to the third term in the right hand side in (4.1). By using (1.6) and the Hölder inequality (2.5), one has Since M is locally integrable and M φ * * min , then M φ max and Lemma 5.7 ensures that | Ω f (x, u)v dx| < ∞. For the last term in the right hand side in (4.1), using (1.7) to have ∂Ω g(x, u)v ds ≤ k 2 ∂Ω |h(x, u)v| ds.
Since the primitive H of h is a locally integrable function satisfying H φ * * min we can use a similar way as in Lemma 5.7 to obtain h(·, u) ∈ L H * (∂Ω) and since ∂Ω is a bounded set, the imbedding (2.3) gives h(x, u) ∈ L 1 (∂Ω). On the other hand, since v ∈ C ∞ (Ω) one has v ∈ L ∞ (∂Ω). Therefore, Define the functional I :  (ii) The functional I has a Gâteaux derivative I (u) for every u ∈ W 1 L φ (Ω). Moreover, for every v ∈ W 1 L φ (Ω) So that, the critical points of I are weak solutions to the problem (1.1).
Regarding the last term in the right hand side in (4.5), we can use (1.7) and (2.8) to have Since the function H is locally integrable and satisfies H φ * * min , it follows that H φ max and in a similar manner as in Lemma 5.7 (in Appendix) we obtain h(·, u) ∈ L H * (∂Ω). Applying the Hölder inequality (2.5) one has ∂Ω |G(x, u)| ds < ∞.
(ii) For i = 1, . . . , N we define the functional Λ i : We denote by B, L 1 , L 2 : Observe that for u ∈ W 1 L φ (Ω), v ∈ C ∞ (Ω), and r > 0, we have as r → 0 for almost every x ∈ Ω. On the other hand, by the mean value theorem there exists ν ∈ [0, 1] such that Hence, by using this equality and (1.2) we obtain Next, by Hölder's inequality (2.5) we obtain The dominated convergence theorem can be applied to obtain for i = 1, . . . , N . By a similar calculus as in above, we can show that

Existence of solutions.
Our main existence result reads as follows. Proof. We divide the proof into three steps.
Step 1: Weak * lower semicontinuity property of I. We define the functional J : First, we claim that J is sequentially weakly lower semicontinuous. Indeed, since u → φ max (x, u) is continuous, it is sufficient to show that the functional is sequentially weakly− * lower semicontinuous. To do this, let u n By the definitions of φ min and φ max , (4.6) and (4.7) hold for every ϕ ∈ E φ * min (Ω). Being φ * min locally integrable, one has L ∞ (Ω) ⊂ E φ * min (Ω). Therefore, for every i ∈ {1, . . . , N } ∂ i u n ∂ i u and u n u in L 1 (Ω), (4.8) for the weak topology σ(L 1 , L ∞ ). Since the embedding W 1 L φ (Ω) → W 1,1 (Ω) is continuous and the embedding W 1,1 (Ω) → L 1 (Ω) is compact, we conclude that the sequence {u n } is relatively compact in L 1 (Ω). Therefore, there exist a subsequence still indexed by n and a function v ∈ L 1 (Ω) such that u n → v strongly in L 1 (Ω). In view of (4.8), we have v = u almost everywhere on Ω and u n → u in L 1 (Ω).
Passing once more to a subsequence, we can have u n → u almost everywhere on Ω.
Recall that ζ → A i (x, ζ) is a convex function, so by (1.3) we can use [5, Theorem 2.1, Chapter 8] obtaining We shall now prove that L 1 and L 2 are continuous. Since M φ * * min , it follows that M (φ * * min ) * , then by Corollary 3.3 we have u n → u in L M (Ω). Thus, there exists n 0 such that for every n ≥ n 0 , u n − u M < 1 2 . By (1.6), we obtain Let θ n = u n − u M . By the convexity of M , we can write Hence, Moreover, Since M is locally integrable and M φ max , there exists a nonnegative function h ∈ L 1 (Ω) such that Thus, the Lebesgue dominated convergence theorem yields and therefore by (4.9) we have lim sup In addition, by Fatou's Lemma we obtain Gathering the two inequalities above, we obtain Applying [14,Theorem 13.47] we obtain M (x, u n (x)) → M (x, u(x)) strongly in L 1 (Ω) which in turn implies that M (x, u n (x)) is equi-integrable and then so is F (x, u n (x)). Since F (x, u n ) → F (x, u) almost everywhere on Ω by Vitali's theorem we have L 1 (u n ) → L 1 (u). Similarly, we can show that L 2 (u n ) → L 2 (u). That is to say that L 1 and L 2 are continuous. Since J is weakly− * lower semicontinuous, we conclude that I is weakly- * lower semicontinuous.
Step 2: Coercivity of the functional I. By (1.3), (1.9) and (2.4) we can write By (1.6) and (1.7) we obtain As M (φ * * min ) * and H ψ min , by Theorems 3.1 and 3.4 there exist two positive constant C 1 > 0 and C 2 > 0 such that u L M (Ω) ≤ C 1 u W 1 L φ (Ω) and u L H (∂Ω) ≤ C 2 u W 1 L φ (Ω) . Since M and H satisfy the ∆ 2 -condition, there exist two positive constants r 1 > 0 and r 2 > 0 and two nonnegative functions h 1 ∈ L 1 (Ω) and h 2 ∈ L 1 (∂Ω) such that Step 3: Existence of a weak solution. Let λ > 0 be arbitrary. Since I is coercive there exists R > 0 such that We define E λ = {u ∈ W 1 L φ (Ω) : I(u) ≤ λ} and denote by B R (0) the closed ball in W 1 L φ (Ω) of radius R centered at the origin. We claim that α = inf v∈W 1 L φ (Ω) I(v) > −∞. If not, for all n > 0 there is a sequence u n ∈ E λ such that I(u n ) < −n. As E λ ⊂ B R (0), by the Banach-Alaoglu-Bourbaki theorem there exists u ∈ B R (0) such that, passing to a subsequence if necessary, we can assume that u n u weak * in W 1 L φ (Ω). So that the weak- * lower semicontinuity of I implies I(u) = −∞ which contradicts the fact that I is well defined on W 1 L φ (Ω). Therefore, for every n > 0 there exists a sequence u n ∈ E λ such that I(u n ) ≤ α + 1 n . Thus, there exists u ∈ B R (0) such that for a subsequence still indexed by n, u n u weak− * in W 1 L φ (Ω). Since I is weakly− * lower semicontinuous we obtain Note that u belongs also to E λ , which yields I(u) = α ≤ λ. This shows that I(u) = min{I(v) : v ∈ W 1 L φ (Ω)}. Moreover, inserting v = −u − as test function in (4.1) and then using (2.8), we obtain u ≥ 0. This ends the proof of Theorem 4.3.

Uniqueness.
To prove the uniqueness of the weak solution we need the following monotonicity assumptions: f (x, s) − f (x, t) s − t < 0 for a.e. x ∈ Ω, ∀s, t ∈ R with s = t, (4.10) g(x, s) − g(x, t) s − t < 0 for a.e. x ∈ Ω, ∀s, t ∈ R with s = t, (4.11) ϕ max (x, s) − ϕ max (x, t) s − t > 0 for a.e. x ∈ Ω, ∀s, t ∈ R with s = t. Proof. Suppose that there exists another weak solution w of problem (1.1). We choose v = u − w as a test function in (4.1) obtaining Then choosing v = w − u as a test function in the weak formulation of solution (4.1) solved by w, we obtain Combining the previous two equalities, we obtain In view of (1.4), (4.10), (4.11) and (4.12) we obtain u = w a.e. in Ω.

Appendix
We present some important results that are necessary for the accomplishment of the proofs of the above results.
belongs to the dual space of E φ (Ω), denoted E φ (Ω) * , and its norm F η satisfies

2)
where F η = sup{|F η (u)| : u L M (Ω) ≤ 1}. Proof. According to Lemma 5.2 any element η ∈ L φ * (Ω) defines a bounded linear functional F η on L φ (Ω) and also on E φ (Ω) which is given by (5.1). It remains to show that every bounded linear functional on E φ (Ω) is of the form F η for some η ∈ L φ * (Ω). The proof of this claim is done in [23]. For the convenience of the reader we give it here. Given F ∈ E φ (Ω) * we define the complex measure λ by setting λ(E) = F (χ E ), where E is a measurable subset of Ω having a finite measure and χ E stands for the characteristic function of E. Due to the fact that φ is locally integrable, the measurable function φ ·, φ −1 (x 0 , 1 2|E| )χ E (·) belongs to L 1 (Ω) for any x 0 ∈ Ω. Hence, there is an > 0 such that for any measurable subset Ω of Ω, one has As φ(·, s) is measurable on E, Luzin's theorem implies that for > 0 there exists a closed set K ⊂ E, with |E \ K | < , such that the restriction of φ(·, s) to K is continuous. Let k be the point where the maximum of φ(·, s) is reached in the set K . dx.
For the first term in the right-hand side of the equality, we can write Since |E \ K | < , the second term can be estimated as Thus, we obtain It follows that As the right-hand side tends to zero when |E| converges to zero, the measure λ is absolutely continuous with respect to the Lebesgue measure and so by Radon-Nikodym's Theorem (see for instance [1, Theorem 1.52]), it can be expressed in the form for some nonnegative function η ∈ L 1 (Ω) unique up to sets of Lebesgue measure zero. Thus, holds for every measurable simple function ζ. Note first that since φ is locally integrable any measurable simple function lies in E φ (Ω) and the set of measurable simple functions is dense in (E φ (Ω), · φ ). Indeed, for nonnegative ζ ∈ E φ (Ω), there exists a sequence of increasing measurable simple functions ζ j converging almost everywhere to ζ and |ζ j (x)| ≤ |ζ(x)| on Ω. By the theorem of dominated convergence one has ζ j → ζ in E φ (Ω). For an arbitrary ζ ∈ E φ (Ω), we obtain the same result splitting ζ into positive and negative parts.
Let ζ ∈ E φ (Ω) and let ζ j be a sequence of measurable simple functions converging to ζ in E φ (Ω). By Fatou's Lemma and the inequality (5.2) we can write which implies that η ∈ L φ * (Ω). Thus, the linear functional F η (ζ) = Ω ζ(x)η(x) dx and F are both defined on E φ (Ω) and have the same values on the set of measurable simple functions, so by a density argument they agree on E φ (Ω). g(x,t) = +∞, then for all > 0, there exists a positive constant K 0 such that Proof. Let > 0 be arbitrary. There exists t 0 > 0 such that t ≥ t 0 implies g(x, t) ≤ f (x, t). Then, for all t ≥ 0, where K(x) = sup t∈(0,t0) g(x, t). Being g(·, t) continuous on Ω, one has g(x, t) ≤ f (x, t) + K 0 with K 0 = max x∈Ω K(x). Proof. Let us fix > 0. Defining v j,k (x) = uj (x)−u k (x) , we shall prove that {u j } is a Cauchy sequence in the Banach space L B (Ω). Clearly {v j,k } is bounded in L A (Ω), say v j,k A ≤ K for all j and k and for some positive constant K. Since B A there exists a positive number t 0 such that for t ≥ t 0 one has On the other hand, since B(·, t) is continuous on Ω we denote x 0 the point where the maximum of B(·, t) is reached in Ω. Let δ = 1 4B(x0,t0) and set .

(5.3)
For the first term in the right-hand side of (5.3), we can write Since B A, the second term in the right hand side of (5.3) can be estimated as follows while for the third term in the right hand side of (5.3), we obtain Finally, putting all the above estimates in (5.3), we arrive at Ω B(x, |v j,k (x)|) dx ≤ 1, for every j, k ≥ N 0 , which yields u j − u k B ≤ . Thus, {u j } converges in the Banach space L B (Ω).
Proof. Let ϕ ∈ D(Ω) and let {e j } N j=1 be the standard basis in R N . We can write h→0 Ω F (x + he j , u(x + he j )) − F (x, u(x + he j )) h ϕ(x) dx
Let φ : Ω × R + → R + be a real function such that the partial function φ(x, ·) is convex. The function φ is called the principal part of the Musielak-Orlicz function M if M (x, t) = φ(x, t) for large values of the argument t.
We define the function The function M (x, t) is a Musielak-Olicz function inasmuch as its derivative, is a function which is positive for t > 0, right-continuous for t ≥ 0 non-decreasing, and lim t→+∞ ∂M (x,t) ∂t = +∞.