CUBO A Mathematical Journal Vol.14, No¯ 02, (15–41). June 2012 Weak and entropy solutions for a class of nonlinear inhomogeneous Neumann boundary value problem with variable exponent Stanislas OUARO Laboratoire d’Analyse Mathématique des Equations (LAME), UFR. Sciences Exactes et Appliquées, Université de Ouagadougou, 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso email: souaro@univ-ouaga.bf, ouaro@yahoo.fr ABSTRACT We study the existence and uniqueness of weak and entropy solutions for the nonlinear inhomogeneous Neumann boundary value problem involving the p(x)-Laplace of the form − div a(x,∇u) + |u|p(x)−2 u = f in Ω, a(x,∇u).η = ϕ on ∂Ω, where Ω is a smooth bounded open domain in RN, N ≥ 3, p ∈ C(Ω) and p(x) > 1 for x ∈ Ω. We prove the existence and uniqueness of a weak solution for data ϕ ∈ L(p−) ′ (∂Ω) and f ∈ L(p−) ′ (Ω), the existence and uniqueness of an entropy solution for L1−data f and ϕ independent of u and the existence of weak solutions for f dependent on u and ϕ ∈ L(p−) ′ (Ω). RESUMEN Estudiamos la existencia y unicidad de soluciones y entroṕıa débil para el problema no lineal inhomogéneos de Neumann con valores de frontera que involucra el p(x)- Laplace de la forma − div a(x,∇u) + |u|p(x)−2 u = f en Omega, a(x,∇u).η = ϕ sobre ∂Ω, donde Omega es en un dominio abierto suave y acotado en RN, N ≥ 3, p ∈ C(Ω) y p(x) > 1 para x ∈ Ω. Probamos la existencia y unicidad de una solución débil para ϕ ∈ L(p−) ′ (∂Ω) and f ∈ L(p−) ′ (Ω), la existencia y unicidad de una solución de entroṕıa para L1−data f y ϕ independiente de u y la existencia de soluciones débiles para f dependiente sobre u y ϕ ∈ L(p−) ′ (Ω). Keywords and Phrases: Generalized Lebesgue and Sobolev spaces; Weak solution; Entropy solution; p(x)-Laplace operator. 2010 AMS Mathematics Subject Classification: 35J20, 35J25, 35D30, 35B38, 35J60. 16 Stanislas OUARO CUBO 14, 2 (2012) 1 Introduction The purpose of this paper is to study the existence and uniqueness of weak and entropy solutions to the following nonlinear inhomogeneous Neumann problem involving the p(x)-Laplace    −div a(x,∇u) + |u|p(x)−2 u = f in Ω, a(x,∇u).η = ϕ on ∂Ω, (1.1) where Ω ⊂ RN (N ≥ 3) is a bounded open domain with smooth boundary and η is the unit outward normal on ∂Ω. The study of various mathematical problems with variable exponent has recieved considerable attention in recent years (see [4,7,8-15,17,19,24-27,29,30,33,34]). These problems concern applica- tions (see [21,22,31,32,35]) and raise many difficult mathematical problems. The operator −div a(x,∇u) is called p(x)-Laplace, which becomes p-Laplace when p(x) ≡ p (a constant). It possesses more complicated nonlinearities than the p-Laplace. For related results involving the p-Laplace, see [2,3]. In [2], the authors studied the problem    −div a(x,∇u) + γ(u) ∋ φ in Ω, a(x,∇u).η + β(u) ∋ ψ on ∂Ω, (1.2) where η is the unit outward normal on ∂Ω, ψ ∈ L1(∂Ω) and φ ∈ L1(Ω). The nonlinearities γ and β are maximal monotone graphs in R2 such that 0 ∈ γ(0) and 0 ∈ β(0). They proved under a range condition the existence and uniqueness of weak and entropy solutions to the problem (1.2). Following these ideas, Ouaro and Soma [24] proved the existence and uniqueness of weak and entropy solutions for a class of homogeneous nonlinear Neumann boundary value problem of the form    −div a(x,∇u) + |u|p(x)−2 u = f in Ω, ∂u ∂ν = 0 on ∂Ω, (1.3) where Ω ⊂ RN (N ≥ 3) is a bounded open domain with smooth boundary and ∂u ∂ν is the outer unit normal derivative on ∂Ω. In this paper, our aim is to prove the existence and uniqueness of weak and entropy solutions to the nonlinear Neumann boundary value problem (1.1) in order to generalize the results in [24]. The paper is presented as follows. In section 2, we introduce some fundamental preliminary results that we use in this work. The existence and the uniqueness of weak solution for (1.1) is proved in section 3 when the data f and ϕ belongs to L(p−) ′ . In section 4, we prove some existence results of weak solution to the problem (1.1) for an f assumed to depend on u and for a boundary datum ϕ ∈ L(p−) ′ (∂Ω). Finally, in section 5, we prove the existence and the uniqueness of an entropy solution of (1.1) when the data f and ϕ belongs to L1. CUBO 14, 2 (2012) Weak and entropy solutions for a class of nonlinear ... 17 2 Assumptions and preliminaries In this work, we study the problem (1.1) for a variable exponent p(.) which is continuous, more precisely, we assume that    p(.) : Ω → R is a continuous function such that 1 < p− ≤ p+ < +∞, (2.1) where p− := ess inf x∈Ω p(x). We denote p+ := ess sup x∈Ω p(x). For the vector fields a(., .), we assume that a(x,ξ) : Ω × RN → RN is Carathéodory and is the continuous derivative with respect to ξ of the mapping A : Ω × RN → R, A = A(x,ξ), i.e. a(x,ξ) = ∇ξA(x,ξ) such that: • The following equality holds true A(x,0) = 0, (2.2) for almost every x ∈ Ω. • There exists a positive constant C1 such that |a(x,ξ)| ≤ C1(j(x) + |ξ| p(x)−1 ) (2.3) for almost every x ∈ Ω and for every ξ ∈ RN where j is a nonnegative function in Lp ′ (.)(Ω), with 1/p(x) + 1/p′(x) = 1. • There exists a positive constant C2 such that for almost every x ∈ Ω and for every ξ,η ∈ R N with ξ 6= η, (a(x,ξ) − a(x,η)).(ξ − η) > 0. (2.4) • The following inequalities hold true |ξ| p(x) ≤ a(x,ξ).ξ ≤ p(x)A(x,ξ) (2.5) for almost every x ∈ Ω and for every ξ ∈ RN. Remark 2.1. Since for almost every x ∈ Ω, a(x,.) is a gradient and is monotone then the primitive A(x,.) of a(x,.) is necessarily convex. As the exponent p(.) appearing in (2.3) and (2.5) depends on the variable x, we must work with Lebesgue and Sobolev spaces with variable exponents. 18 Stanislas OUARO CUBO 14, 2 (2012) We define the Lebesgue space with variable exponent Lp(.)(Ω) as the set of all measurable function u : Ω → R for which the convex modular ρp(.)(u) := ∫ Ω |u| p(x) dx 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 Luxembourg 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), where 1 p(x) + 1 p′(x) = 1. Finally, we have the Hölder type inequality: ∣ ∣ ∣ ∣ ∫ Ω uvdx ∣ ∣ ∣ ∣ ≤ ( 1 p− + 1 p′− ) |u|p(.) |v|p′(.) , (2.6) for all u ∈ Lp(.)(Ω) and v ∈ Lp ′ (.)(Ω). Now, let W1,p(.)(Ω) := { u ∈ Lp(.)(Ω) : |∇u| ∈ Lp(.)(Ω) } , which is a Banach space equipped with the following norm ‖u‖1,p(.) := |u|p(.) + |∇u|p(.) . The space ( W1,p(.)(Ω),‖u‖1,p(.) ) is a separable and reflexive Banach space; more details can be found in [17]. An important role in manipulating the generalized Lebesgue and Sobolev spaces is played by the modular ρp(.) of the space L p(.)(Ω). We have the following result (cf. [15]): Lemma 2.2. If un,u ∈ L p(.)(Ω) and p+ < +∞, then the following properties hold: (i) |u|p(.) > 1 ⇒ |u| p− p(.) ≤ ρp(.)(u) ≤ |u| p+ p(.) ; (ii) |u|p(.) < 1 ⇒ |u| p+ p(.) ≤ ρp(.)(u) ≤ |u| p− 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. For a measurable function u : Ω −→ R, we introduce the following notation: ρ1,p(.)(u) := ∫ Ω |u| p(x) dx + ∫ Ω |∇u| p(x) dx. We have the following lemma (cf. [33]): Lemma 2.3. If u ∈ W1,p(.)(Ω), then the following properties hold true: (i) ‖u‖1,p(.) < 1(respectively = 1;> 1) ⇔ ρ1,p(.)(u) < 1(respectively = 1;> 1); (ii) ‖u‖1,p(.) < 1 ⇔ ‖u‖ p+ 1,p(.) ≤ ρ1,p(.)(u) ≤ ‖u‖ p− 1,p(.) ; (iii) ‖u‖1,p(.) > 1 ⇔ ‖u‖ p− 1,p(.) ≤ ρ1,p(.)(u) ≤ ‖u‖ p+ 1,p(.) . CUBO 14, 2 (2012) Weak and entropy solutions for a class of nonlinear ... 19 Put p∂(x) := (p(x))∂ :=    (N−1)p(x) N−p(x) , if p(x) < N ∞, if p(x) ≥ N. We have the following useful result (cf. [13,34]). Proposition 2.4. Let p ∈ C(Ω) and p− > 1. If q ∈ C(∂Ω) satisfies the condition 1 ≤ q(x) < p∂(x), ∀x ∈ ∂Ω, then, there is a compact embedding W1,p(x)(Ω) →֒ Lq(x)(∂Ω). In particular, there is a compact embedding W1,p(x)(Ω) →֒ Lp(x)(∂Ω). Let us introduce the following notation: given two bounded measurable functions p(.),q(.) : Ω → R, we write q(.) ≪ p(.) if ess inf x∈Ω (p(x) − q(x)) > 0. 3 Weak solution In this section, we study the existence and uniqueness of a weak solution of (1.1) where the data ϕ ∈ L(p−) ′ (∂Ω) and f ∈ L(p−) ′ (Ω). The definition of weak solution is the following: Definition 3.1. A weak solution of (1.1) is a measurable function u : Ω −→ R such that u ∈ W1,p(.)(Ω), and ∫ Ω a(x,∇u).∇vdx + ∫ Ω |u|p(x)−2 uvdx − ∫ ∂Ω ϕvdσ = ∫ Ω fvdx, ∀ v ∈ W1,p(.)(Ω), (3.1) where dσ is the surface measure on ∂Ω. Let E denote the generalized Sobolev space W1,p(.)(Ω). If we denote the functional J : E → R by J(u) = ∫ Ω A(x,∇u)dx + ∫ Ω 1 p(x) |u|p(x)dx − ∫ ∂Ω ϕudσ − ∫ Ω fudx, then 〈J′(u),v〉 = ∫ Ω a(x,∇u).∇vdx + ∫ Ω |u|p(x)−2 uvdx − ∫ ∂Ω ϕvdσ − ∫ Ω fvdx, for all u,v ∈ E. Therefore, the weak solution of (1.1) corresponds to the critical point of the functional J. The main result of this section is the following: Theorem 3.2. Assume that (2.1)-(2.5) hold. Then there exists a unique weak solution of (1.1). Proof. * Existence. With the techniques that became standard by now, it is not difficult to 20 Stanislas OUARO CUBO 14, 2 (2012) verify that J is well-defined on E, is of class C1(E,R) and is weakly lower semi-continuous (see for example [6,19,24,25,26,28]). To end the proof of the existence part, we just have to prove that J is bounded from below and coercive. Using (2.5) and since E is continuously embedded in Lp−(Ω), we have J(u) = ∫ Ω A(x,∇u)dx + ∫ Ω 1 p(x) |u|p(x)dx − ∫ ∂Ω ϕudσ − ∫ Ω fudx ≥ ∫ Ω 1 p(x) |∇u|p(x)dx + ∫ Ω 1 p(x) |u|p(x)dx − ‖ϕ‖ (p−)′,∂Ω ‖u‖p−,∂Ω − ‖f‖(p−)′,Ω ‖u‖p−,Ω ≥ 1 p+ ρ1,p(.)(u) − c‖ϕ‖(p−)′,∂Ω ‖u‖1,p(.) − C‖u‖1,p(.), where ‖u‖p−,Ω = (∫ Ω |u|p−dx ) 1 p− and ‖u‖p−,∂Ω = (∫ ∂Ω |u|p−dσ ) 1 p− . As ϕ ∈ L(p−) ′ (∂Ω), then ‖ϕ‖ (p−)′,∂Ω < +∞. Also, for the coercivity of J, we will work with u such that ‖u‖1,p(.) > 1. Then, by Lemma 2.3 we obtain that J(u) ≥ 1 p+ ‖u‖ p− 1,p(.) − C3‖u‖1,p(.). As p− > 1, then J is coercive. If ‖u‖1,p(.) < 1, we have that J(u) ≥ 1 p+ ‖u‖ p+ 1,p(.) − C3‖u‖1,p(.) ≥ −C3 > −∞. Therefore, J is bounded from below. Since the functional J is proper, lower semi-continuous and coercive, then it has a minimizer which is a weak solution of (1.1). ∗ Uniqueness. Let u1 and u2 be two weak solutions of (1.1). With u1 as weak solution, we take v = u1 − u2 in (3.1) to get ∫ Ω a(x,∇u1).∇(u1−u2)dx+ ∫ Ω |u1| p(x)−2 u1(u1−u2)dx− ∫ ∂Ω ϕ(u1−u2)dσ = ∫ Ω f(x)(u1−u2)dx. (3.2) Similarly, with u2 as weak solution, we take ϕ = u2 − u1 to obtain ∫ Ω a(x,∇u2).∇(u2−u1)dx+ ∫ Ω |u2| p(x)−2 u2(u2−u1)dx− ∫ ∂Ω ϕ(u2−u1)dσ = ∫ Ω f(x)(u2−u1)dx. (3.3) After adding (3.2) and (3.3), we obtain ∫ Ω (a(x,∇u1) − a(x,∇u2)) .(∇u1 −∇u2)+ ∫ Ω ( |u1| p(x) u1 − |u2| p(x) u2 ) (u1 −u2)dx = 0. (3.4) CUBO 14, 2 (2012) Weak and entropy solutions for a class of nonlinear ... 21 Using (2.4), we deduce from (3.4) that ∫ Ω ( |u1(x)| p(x) u1(x) − |u2(x)| p(x) u2(x) ) (u1(x) − u2(x))dx = 0. (3.5) Since p− > 1, the following relation is true for any ξ,η ∈ R, ξ 6= η (cf. [14]) ( |ξ|p(x)−2ξ − |η|p(x)−2η ) (ξ − η) > 0. (3.6) Therefore, from (3.5), we get ( |u1(x)| p(x) u1(x) − |u2(x)| p(x) u2(x) ) (u1(x) − u2(x)) = 0, a.e. x ∈ Ω. (3.7) Now, we use (3.6) to get u1(x) = u2(x) a.e. x ∈ Ω. (3.8) and uniqueness is true � 4 Weak solutions for a right-hand side dependent on u In this section, we show the existence result of weak solution to some general problem. More precisely, we prove that there exists at least one weak solution to the problem    − div a(x,∇u) + |u|p(x)−2 u = f(x,u) in Ω, a(x,∇u).η = ϕ on ∂Ω, (4.1) where ϕ ∈ L(p−) ′ (∂Ω). We study (4.1) under the assumptions (2.1)-(2.5) and the following additional assumptions on f. f(x,t) : Ω × R −→ R is Carathéodory and there exists two positive constants C4, C5 such that |f(x,t)| ≤ C4 + C5|t| β(x)−1, (4.2) for every t ∈ R and for almost every x ∈ Ω with 0 ≤ β(.) ≪ p(.). Let F(x,t) = ∫t 0 f(x,s)ds. As mentioned before, we look for distributional solution of (4.1) in the following sense: Definition 4.1. A weak solution of (4.1) is a measurable function u : Ω −→ R such that u ∈ W1,p(.)(Ω) and for all v ∈ W1,p(.)(Ω) ∫ Ω a(x,∇u).∇vdx + ∫ Ω |u|p(x)−2 uvdx − ∫ ∂Ω ϕvdσ = ∫ Ω f(x,u)vdx. (4.3) 22 Stanislas OUARO CUBO 14, 2 (2012) We have the following existence result: Theorem 4.2. Assume that (2.1)-(2.5) and (4.2) hold. Then, the problem (4.1) admits at least one weak solution. Proof. Let g(u) = ∫ Ω F(x,u)dx, for all u ∈ E. The functional g is of class C1(E,R) with the derivative given by 〈 g′(u),v 〉 = ∫ Ω f(x,u)vdx, ∀u,v ∈ E. Consequently, J(u) = ∫ Ω A(x,∇u)dx + ∫ Ω 1 p(x) |u|p(x)dx − ∫ ∂Ω ϕudσ − ∫ Ω F(x,u)dx, u ∈ E is such that J is of class C1(E,R) and is lower semi-continuous. We then have to prove that J is bounded from below and coercive in order to complete the proof. From (4.2), we have |F(x,t)| ≤ C ( 1 + |t|β(x) ) and then J(u) ≥ 1 p+ ∫ Ω |∇u|p(x)dx + 1 p+ ∫ Ω |u|p(x)dx − ∫ ∂Ω ϕudσ − C ∫ Ω |u|β(x)dx − Cmeas(Ω). Let M > 1 be a fixed real number (to be chosen later) and ǫ := ess inf x∈Ω (p(x) − β(x)). We have J(u) ≥ 1 2p+ ρ1,p(.)(u) + ∫ {|u|≤M} ( 1 2p+ |u|p(x) − C|u|β(x) ) dx + ∫ {|u|>M} ( 1 2p+ |u|p(x) − C|u|β(x) ) dx − Cmeas(Ω) − C′′‖u‖1,p(.) ≥ 1 2p+ ρ1,p(.)(u) + ∫ {|u|>M} ( 1 2p+ |u|p(x) − C|u|β(x) ) dx − C′′‖u‖1,p(.) − (M β+ + 1)Cmeas(Ω) ≥ 1 2p+ ρ1,p(.)(u) + ∫ {|u|>M} |u|β(x) ( 1 2p+ |u|p(x)−β(x) − C) ) dx − C′′‖u‖1,p(.) − (M β+ + 1)Cmeas(Ω) ≥ 1 2p+ ρ1,p(.)(u) + ( 1 2p+ Mǫ − C ) ∫ {|u|>M} |u|β(x)dx − C′′‖u‖1,p(.) − (M β+ + 1)Cmeas(Ω) ≥ 1 p+ ‖u‖ p− 1,p(.) − C′′‖u‖1,p(.) − (M β+ + 1)Cmeas(Ω), For all M > max((2p+C) 1 ǫ ,1) and all u ∈ E with ‖u‖1,p(.) > 1. Since 1 < p− it follows that J(u) −→ +∞ as ‖u‖E −→ +∞. Consequently, J is bounded from below and coercive. The proof is then complete. Assume now that F+(x,t) = ∫t 0 f+(x,s)ds is such that there exists C6 > 0, C7 > 0 such that |f+(x,t)| ≤ C6 + C7|t| β(x)−1, (4.4) where 0 ≤ β(.) ≪ p(.). Then we have the following result: CUBO 14, 2 (2012) Weak and entropy solutions for a class of nonlinear ... 23 Theorem 4.3 Under assumptions (2.1)-(2.5) and (4.4), the problem (4.1) admits at least one weak solution. Proof. As f = f+ − f−, let F−(x,t) = ∫t 0 f−(x,s)ds. Then J(u) = ∫ Ω A(x,∇u)dx + ∫ Ω 1 p(x) |u|p(x)dx + ∫ Ω F−(x,u)dx − ∫ Ω F+(x,u)dx − ∫ ∂Ω ϕudσ ≥ ∫ Ω A(x,∇u)dx + ∫ Ω 1 p(x) |u|p(x)dx − ∫ Ω F+(x,u)dx − ∫ ∂Ω ϕvdσ. Therefore, similarly as in the proof of Theorem 4.2, the result of Theorem 4.3 follows immediately. 5 Entropy solutions In this section, we study the existence of entropy solution for the problem (1.1) when the data f ∈ L1(Ω) and ϕ ∈ L1(∂Ω). We first recall some notations. For any k > 0, we define the truncation function Tk by Tk(s) := max{−k,min{k,s}}. Let Ω be a bounded open subset of RN of class C1 and 1 ≤ p(.) < +∞. It is well known( see [20] or [23]) that if u ∈ W1,p(.)(Ω), it is possible to define the trace of u on ∂Ω. More precisely, there is a bounded operator τ from W1,p(.)(Ω) into Lp(.)(∂Ω) such that τ(u) = u|∂Ω whenever u ∈ C(Ω). Set T 1,p(.)(Ω) = { u : Ω −→ R, measurable such that Tk(u) ∈ W 1,p(.)(Ω), for any k > 0 } . In [1], the authors have proved the following Proposition 5.1 Let u ∈ T 1,p(.)(Ω). Then there exists a unique measurable function v : Ω −→ RN such that ∇Tk(u) = vχ{|u| 0. The function v is denoted by ∇u. Moreover if u ∈ W1,p(.)(Ω) then v ∈ ( Lp(.)(Ω) )N and v = ∇u in the usual sense. It is easy to see that, in general, it is not possible to define the trace of an element of T 1,p(.)(Ω). In demension one it is enough to consider the function u(x) = 1 x for x ∈]0,1[. Therefore, we are going to define following [2,3], the trace for the elements of a subset T 1,p(.) tr (Ω) of T 1,p(.)(Ω). T 1,p(.) tr (Ω) will be the set of functions u ∈ T 1,p(.)(Ω) such that there exists a sequence (un)n ⊂ W1,p(.)(Ω) satisfying the following conditions: (C1) un → u a.e in Ω. (C2) ∇Tk(un) → ∇Tk(u) in L1(Ω) for any k > 0. (C3) There exists a measurable function v on ∂Ω, such that un → v a.e in ∂Ω. The function v is the trace of u in the generalized sense introduced in [2,3]. In the sequel the trace of u ∈ T 1,p(.) tr (Ω) on ∂Ω will be denoted by tr(u). If u ∈ W 1,p(.)(Ω),tr(u) coincides with τ(u) in 24 Stanislas OUARO CUBO 14, 2 (2012) the usual sense. Moreover, for u ∈ T 1,p(.) tr (Ω) and for every k > 0, τ(Tk(u)) = Tk (tr(u)) and if ϕ ∈ W1,p(.)(Ω) ∩ L∞(Ω) then (u − ϕ) ∈ T 1,p(.) tr (Ω) and tr(u − ϕ) = tr(u) − tr(ϕ). We can now introduce the notion of entropy solution of (1.1). Definition 5.2. A measurable function u is an entropy solution to problem (1.1) if u ∈ T 1,p(.) tr (Ω), |u|p(x)−2 u ∈ L1(Ω) and for every k > 0, ∫ Ω a(x,∇u).∇Tk(u−v)dx+ ∫ Ω |u|p(x)−2 uTk(u−v)dx ≤ ∫ ∂Ω ϕTk(u−v)dσ+ ∫ Ω f(x)Tk(u−v)dx (5.1) for all v ∈ W1,p(.)(Ω) ∩ L∞(Ω). Our main result in this section is the following: Theorem 5.3. Assume (2.1)-(2.5), f ∈ L1(Ω) and ϕ ∈ L1(∂Ω). Then, there exists a unique entropy solution u to problem (1.1). The following propositions are useful for the proof of Theorem 5.3. Proposition 5.4. Assume (2.1)-(2.5), f ∈ L1(Ω) and ϕ ∈ L1(∂Ω). Let u be an entropy solution of (1.1). If there exists a positive constant M such that ∫ {|u|>k} kq(x)dx ≤ M (5.2) then ∫ {|∇u|α(.)>k} kq(x)dx ≤ ‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω) + M, for all k > 0, where α(.) = p(.)/(q(.) + 1). Proof. Taking v = 0 in the entropy inequality (5.1) and using (2.5), we get ∫ Ω |∇Tk(u)| p(x)dx ≤ k ( ‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω) ) for all k > 0. Therefore, defining ψ := 1 k Tk(u), we have for all k > 0, ∫ Ω kp(x)−1|∇ψ|p(x)dx = 1 k ∫ Ω |∇Tk(u)| p(x)dx ≤ ‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω). From the above inequality, from the definition of α(.) and (5.2), we get ∫ {|∇u|α(.)>k} kq(x)dx ≤ ∫ {|∇u|α(.)>k}∩{|u|≤k} kq(x)dx + ∫ {|u|>k} kq(x)dx ≤ ∫ {|u|≤k} kq(x) ( |∇u|α(x) k ) p(x) α(x) dx + M ≤ ‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω) + M, for all k > 0. Proposition 5.5. Assume (2.1)-(2.5), f ∈ L1(Ω) and ϕ ∈ L1(∂Ω). Let u be an entropy solution of (1.1), then ∫ Ω |∇Tk(u)| p(x)dx ≤ k ( ‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω) ) for all k > 0 (5.3) CUBO 14, 2 (2012) Weak and entropy solutions for a class of nonlinear ... 25 and ∥ ∥ ∥ |u|p(x)−2 u ∥ ∥ ∥ 1 = ∥ ∥ ∥ |u|p(x)−1 ∥ ∥ ∥ 1 ≤ ‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω). (5.4) Proof. The inequality (5.3) is already obtained in the proof of Proposition 5.2. Let’s prove (5.4). Taking ϕ = 0 in (5.1), we get for all k > 0, ∫ Ω |u|p(x)−2 uTk(u)dx ≤ k ( ‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω) ) , then ∫ {|u|>k} |u|p(x)−2 uTk(u)dx ≤ k ( ‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω) ) . From the inequality above, we obtain k ∫ {u>k} |u|p(x)−2 udx − k ∫ {u<−k} |u|p(x)−2 udx ≤ k ( ‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω) ) , which imply ∫ {u>k} |u|p(x)−2 udx − ∫ {u<−k} |u|p(x)−2 udx ≤ ‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω). The last inequality means ∫ {|u|>k} |u|p(x)−1dx ≤ ‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω) for all k > 0. (5.5) We use Fatou’s Lemma in (5.5) by letting k goes to 0 to obtain (5.4). Proposition 5.6. Assume that (2.1)-(2.5) hold, f ∈ L1(Ω) and ϕ ∈ L1(∂Ω). Let u be an entropy solution of (1.1), then ∫ {|u|≤k} |∇Tk(u)| p−dx ≤ C(k + 1) for all k > 0. (5.6) Proof. Note that ∫ {|u|≤k} |∇Tk(u)| p−dx = ∫ {|u|≤k,|∇u|>1} |∇Tk(u)| p−dx + ∫ {|u|≤k,|∇u|≤1} |∇Tk(u)| p−dx ≤ ∫ {|u|≤k,|∇u|>1} |∇Tk(u)| p−dx + meas(Ω) ≤ ∫ {|u|≤k} |∇Tk(u)| p(x)dx + meas(Ω). Since ∫ {|u|≤k} |∇Tk(u)| p(x)dx ≤ k ( ‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω) ) , we obtain ∫ {|u|≤k} |∇Tk(u)| p−dx ≤ k ( ‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω) ) + meas(Ω) for all k > 0. 26 Stanislas OUARO CUBO 14, 2 (2012) Proposition 5.7. Assume that (2.1)-(2.5) hold, f ∈ L1(Ω) and ϕ ∈ L1(∂Ω). Let u be an entropy solution of (1.1). Then meas{|u| > h} ≤ ‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω) hp−−1 for all h ≥ 1, (5.7) and meas{|∇u| > h} ≤ ‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω) hp−−1 for all h ≥ 1. (5.8) Proof. ∫ Ω |u|p(x)−1dx = ∫ {|u|≤h} |u|p(x)−1dx + ∫ {|u|>h} |u|p(x)−1dx ≥ ∫ {|u|>h} |u|p(x)−1dx ≥ ∫ {|u|>h} hp(x)−1dx ≥ hp−−1meas{|u| > h} since h ≥ 1. Then, by (5.4) we deduce (5.7). We next prove (5.8). For k,λ ≥ 0, set Φ(k,λ) = meas{|∇u|p− > λ, |u| > k}. We have Φ(k,0) ≤ meas{|u| > k}. For k ≥ 1, we obtain by (5.7) Φ(k,0) ≤ ( ‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω) ) k1−p−. Using the fact that the function λ 7−→ Φ(k,λ) is nonincreasing, we get for k > 0 and λ > 0, that Φ(0,λ) = 1 λ ∫λ 0 Φ(0,λ)ds ≤ 1 λ ∫λ 0 Φ(0,s)ds ≤ 1 λ ∫λ 0 [ Φ(0,s) + (Φ(k,0) − Φ(k,s)) ] ds ≤ Φ(k,0) + 1 λ ∫λ 0 (Φ(0,s) − Φ(k,s))ds. Now, let us observe that Φ(0,s) − Φ(k,s) = meas{|u| ≤ k, |∇u|p− > s}. Then, thanks to (5.6), we get ∫+∞ 0 (Φ(0,s) − Φ(k,s))ds = ∫ {|u|≤k} |∇u|p−dx ≤ C(k + 1), CUBO 14, 2 (2012) Weak and entropy solutions for a class of nonlinear ... 27 where C = max ( meas(Ω),‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω) ) . It follows that Φ(0,λ) ≤ C(k + 1) λ + ( ‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω) ) k1−p−, for all k ≥ 1,λ > 0. In particular, we have Φ(0,λ) ≤ C(k + 1) λ + ( ‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω) ) k1−p−, for all k ≥ 1,λ ≥ 1. We now set fλ(k) = C(k + 1) λ + ( ‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω) ) k1−p−, for all k ≥ 1, where λ ≥ 1 is a fixed real number. The minimization of fλ in k gives Φ(0,λ) ≤ ( ‖f‖L1(Ω) + ‖ϕ‖L1(∂Ω) ) λ−(1/(p−) ′ ), (5.9) for all λ ≥ 1. Setting λ = hp− in (5.9) gives (5.8). Proof of Theorem 5.3. ∗ Uniqueness of entropy solution. Let h > 0 and u1,u2 be two entropy solutions of (1.1). We write the entropy inequality (5.1) corresponding to the solution u1, with Th(u2) as a test function, and to the solution u2, with Th(u1) as a test function. Upon addition, we get    ∫ {|u1−Th(u2)|≤k} a(x,∇u1).∇(u1 − Th(u2))dx + ∫ {|u2−Th(u1)|≤k} a(x,∇u2).∇(u2 − Th(u1))dx + ∫ Ω |u1| p(x)−2 u1Tk(u1 − Th(u2))dx + ∫ Ω |u2| p(x)−2 u2Tk(u2 − Th(u1))dx ≤ ∫ ∂Ω ϕ ( Tk(u1 − Th(u2)) + Tk(u2 − Th(u1)) ) dσ + ∫ Ω f ( Tk(u1 − Th(u2)) + Tk(u2 − Th(u1)) ) dx. (5.10) Define now E1 := {|u1 − u2| ≤ k, |u2| ≤ h}, E2 := E1 ∩ {|u1| ≤ h}, and E3 := E1 ∩ {|u1| > h}. We start with the first integral in (5.10). By (2.5), we have 28 Stanislas OUARO CUBO 14, 2 (2012)    ∫ {|u1−Th(u2)|≤k} a(x,∇u1).∇(u1 − Th(u2))dx = ∫ {|u1−Th(u2)|≤k}∩{|u2|≤h} a(x,∇u1).∇(u1 − Th(u2))dx + ∫ {|u1−Th(u2)|≤k}∩{|u2|>h} a(x,∇u1).∇(u1 − Th(u2))dx = ∫ {|u1−Th(u2)|≤k}∩{|u2|≤h} a(x,∇u1).∇(u1 − u2)dx + ∫ {|u1−hsign(u2)|≤k}∩{|u2|>h} a(x,∇u1).∇u1dx ≥ ∫ {|u1−Th(u2)|≤k}∩{|u2|≤h} a(x,∇u1).∇(u1 − u2)dx = ∫ E1 a(x,∇u1).∇(u1 − u2)dx = ∫ E2 a(x,∇u1).∇(u1 − u2)dx + ∫ E3 a(x,∇u1).∇(u1 − u2)dx = ∫ E2 a(x,∇u1).∇(u1 − u2)dx + ∫ E3 a(x,∇u1).∇u1dx − ∫ E3 a(x,∇u1).∇u2dx ≥ ∫ E2 a(x,∇u1).∇(u1 − u2)dx − ∫ E3 a(x,∇u1).∇u2dx. (5.11) Using (2.3) and (2.6), we estimate the last integral in (5.11) as follows:    ∣ ∣ ∣ ∣ ∫ E3 a(x,∇u1).∇u2dx ∣ ∣ ∣ ∣ ≤ C1 ∫ E3 ( j(x) + |∇u1| p(x)−1 ) |∇u2|dx ≤ C1 ( |j|p′(.) + ∣ ∣ ∣ |∇u1| p(x)−1 ∣ ∣ ∣ p′(.),{h<|u1|≤h+k} ) |∇u2|p(.),{h−k<|u1|≤h}, (5.12) where ∣ ∣ ∣ |∇u1| p(x)−1 ∣ ∣ ∣ p′(.),{h<|u1|≤h+k} = ∥ ∥ ∥ |∇u1| p(x)−1 ∥ ∥ ∥ Lp ′(.)({h<|u1|≤h+k}) . The quantity C1 ( |j|p′(.) + ∣ ∣ ∣ |∇u1| p(x)−1 ∣ ∣ ∣ p′(.),{h<|u1|≤h+k} ) can be written as follows C1 ( |j|p′(.) + ∣ ∣ ∣ |∇Th+k(u1)| p(x)−1 ∣ ∣ ∣ p′(.),{h<|u1|≤h+k} ) < +∞, since Th+k(u1) ∈ W 1,p(.)(Ω) and j ∈ Lp ′ (.)(Ω). We deduce by Proposition 5.7 that C1 ( |j|p′(.) + ∣ ∣ ∣ |∇u1| p(x)−1 ∣ ∣ ∣ p′(.),{h<|u1|≤h+k} ) |∇u2|p(.),{h−k<|u1|≤h} converges to 0 as h → +∞. Therefore, from (5.11) and (5.12), we obtain ∫ {|u1−Th(u2)|≤k} a(x,∇u1).∇(u1 − Th(u2))dx ≥ Ih + ∫ E2 a(x,∇u1).∇(u1 − u2)dx, (5.13) where Ih converges to zero as h → +∞. We may adopt the same procedure to treat the second term in (5.10) to obtain ∫ {|u2−Th(u1)|≤k} a(x,∇u2).∇(u2 − Th(u1))dx ≥ Jh − ∫ E2 a(x,∇u2).∇(u1 − u2)dx, (5.14) CUBO 14, 2 (2012) Weak and entropy solutions for a class of nonlinear ... 29 where Jh converges to zero as h → +∞. Now set for all h,k > 0 Kh = ∫ Ω |u1| p(x)−2 u1Tk(u1 − Th(u2))dx + ∫ Ω |u2| p(x)−2 u2Tk(u2 − Th(u1))dx. We have |u1| p(x)−2 u1Tk(u1 − Th(u2)) −→ |u1| p(x)−2 u1Tk(u1 − u2) a.e in Ω as h → +∞, and ∣ ∣ ∣ |u1| p(x)−2 u1Tk(u1 − Th(u2)) ∣ ∣ ∣ ≤ k|u1| p(x)−1 ∈ L1(Ω). Then by Lebesgue Theorem, we deduce that lim h→+∞ ∫ Ω |u1| p(x)−2 u1Tk(u1 − Th(u2))dx = ∫ Ω |u1| p(x)−2 u1Tk(u1 − u2)dx. (5.15) Similarly, we have lim h→+∞ ∫ Ω |u2| p(x)−2 u2Tk(u2 − Th(u1))dx = ∫ Ω |u2| p(x)−2 u2Tk(u2 − u1)dx. (5.16) Using (5.15) and (5.16), we get lim h→+∞ Kh = ∫ Ω ( |u1| p(x)−2 u1 − |u2| p(x)−2 u2 ) Tk(u1 − u2)dx. (5.17) We next examine the right-hand side of (5.10). For all k > 0, f ( Tk(u1 −Th(u2))+Tk(u2 −Th(u1)) ) −→ f ( Tk(u1 −u2)+Tk(u2 −u1) ) = 0 a.e in Ω as h → +∞, ϕ ( Tk(u1−Th(u2))+Tk(u2−Th(u1)) ) −→ ϕ ( Tk(u1−u2)+Tk(u2−u1) ) = 0 a.e in ∂Ω as h → +∞, and ∣ ∣ ∣ f(x) ( Tk(u1 − Th(u2)) + Tk(u2 − Th(u1)) ) ∣ ∣ ∣ ≤ 2k|f| ∈ L1(Ω), ∣ ∣ ∣ ϕ ( Tk(u1 − Th(u2)) + Tk(u2 − Th(u1)) ) ∣ ∣ ∣ ≤ 2k|ϕ| ∈ L1(∂Ω). Lebesgue Theorem allows us to write lim h→+∞ [∫ ∂Ω ϕ ( Tk(u1 − Th(u2)) + Tk(u2 − Th(u1)) ) dσ + ∫ Ω f ( Tk(u1 − Th(u2)) + Tk(u2 − Th(u1)) ) dx ] = 0. (5.18) Using (5.13), (5.14), (5.17) and (5.18), we get    ∫ {|u1−u2|≤k} ( a(x,∇u1) − a(x,∇u2) ) . ( ∇u1 − ∇u2 ) dx + ∫ Ω ( |u1| p(x)−2 u1 − |u2| p(x)−2 u2 ) Tk(u1 − u2)dx ≤ 0. (5.19) 30 Stanislas OUARO CUBO 14, 2 (2012) Therefore ∫ Ω ( |u1| p(x)−2 u1 − |u2| p(x)−2 u2 ) Tk(u1 − u2)dx = 0. (5.20) For x fixed in Ω, s 7−→ |s|p(x)−2 s is nondecreasing and vanishes at 0. Then, ( |u1| p(x)−2 u1 − |u2| p(x)−2 u2 ) Tk(u1 − u2) ≥ 0, ∀x ∈ Ω and ∀k > 0. Now, using inequality above and (5.20), for all k ∈ R+ there exist Ωk ⊂ Ω with meas(Ωk) = 0 such that for all x ∈ Ω\Ωk, ( |u1(x)| p(x)−2 u1(x) − |u2(x)| p(x)−2 u2(x) ) Tk(u1(x) − u2(x)) = 0. Therefore, ( |u1(x)| p(x)−2 u1(x) − |u2(x)| p(x)−2 u2(x) ) (u1(x) − u2(x)) = 0, for all x ∈ Ω\ ⋃ k∈N∗ Ωk. (5.21) Now, using (5.21) and (3.6), we get u1 = u2 a.e. in Ω. ∗ Existence of entropy solution. Let fn = Tn(f) and ϕn = Tn(ϕ); then (fn)n and (ϕn)n are in L(p−) ′ (Ω) and L(p−) ′ (∂Ω) respectively and are strongly converging to f in L1(Ω) and to ϕ in L1(∂Ω) respectively. Moreover ‖fn‖L1(Ω) ≤ ‖f‖L1(Ω) and ‖ϕn‖L1(∂Ω) ≤ ‖ϕ‖L1(∂Ω), for all n ∈ N. Next, we consider the problem    −div a(x,∇un) + |un| p(x)−2 un = fn in Ω, a(x,∇un).η = ϕn on ∂Ω. (5.22) It follows from Theorem 3.2 that there exists a unique un ∈ W 1,p(.)(Ω) such that ∫ Ω a(x,∇un).∇vdx + ∫ Ω |un| p(x)−2 unvdx = ∫ ∂Ω ϕnvdσ + ∫ Ω fnvdx (5.23) for all v ∈ W1,p(.)(Ω). Our aim is to prove that these approximated solutions un tend, as n goes to infinity, to a measurable function u which is an entropy solution to the limit problem (1.1). To start with, we prove the following lemma: Lemma 5.8. For any k > 0, ‖Tk(un)‖1,p(.) ≤ 1 + C where C = C(k,ϕ,f,p−,p+,meas(Ω)) is a positive constant. Proof. By taking v = Tk(un) in (5.23), we get ∫ Ω a(x,∇un).∇Tk(un) + ∫ Ω |un| p(x)−2 unTk(un)dx = ∫ ∂Ω ϕnTk(un)dσ + ∫ Ω fnTk(un)dx. CUBO 14, 2 (2012) Weak and entropy solutions for a class of nonlinear ... 31 Since all the terms in the left-hand side of equality above are nonnegative and ∫ ∂Ω ϕnTk(un)dσ + ∫ Ω fnTk(un)dx ≤ k ( ‖ϕn‖L1(∂Ω) + ‖fn‖L1(Ω) ) ≤ k ( ‖ϕ‖L1(∂Ω) + ‖f‖L1(Ω) ) ; by using (2.5) we obtain ∫ Ω |∇Tk(un)| p(x)dx ≤ k ( ‖ϕ‖L1(∂Ω) + ‖f‖L1(Ω) ) (5.24) and ∫ Ω |un| p(x)−2 unTk(un)dx ≤ k ( ‖ϕ‖L1(∂Ω) + ‖f‖L1(Ω) ) . (5.25) The inequality (5.25) is equivalent to ∫ {|un|≤k} |Tk(un)| p(x)dx + ∫ {|un|>k} |un| p(x)−2 unTk(un)dx ≤ k ( ‖ϕ‖L1(∂Ω) + ‖f‖L1(Ω) ) . Therefore, ∫ {|un|≤k} |Tk(un)| p(x)dx ≤ k ( ‖ϕ‖L1(∂Ω) + ‖f‖L1(Ω) ) . (5.26) Furthermore ∫ {|un|>k} |Tk(un)| p(x)dx = ∫ {|un|>k} kp(x)dx ≤    kp+meas(Ω) if k ≥ 1, meas(Ω) if k < 1. This allows us to write ∫ {|un|>k} |Tk(un)| p(x)dx ≤ (1 + kp+)meas(Ω). (5.27) Relations (5.26) and (5.27) give ∫ Ω |Tk(un)| p(x)dx ≤ k ( ‖ϕ‖L1(∂Ω) + ‖f‖L1(Ω) ) + (1 + kp+)meas(Ω). (5.28) Hence, adding (5.24) and (5.28), it yields ρ1,p(.)(Tk(un)) ≤ 2k ( ‖ϕ‖L1(∂Ω) + ‖f‖L1(Ω) ) + (1 + kp+)meas(Ω) = C(k,ϕ,f,p+,meas(Ω)). (5.29) If ‖Tk(un)‖1,p(.) ≥ 1, we have ‖Tk(un)‖ p− 1,p(.) ≤ ρ1,p(.)(Tk(un)) ≤ C(k,ϕ,f,p+,meas(Ω)), which is equivalent to ‖Tk(un)‖1,p(.) ≤ ( C(k,ϕ,f,p+,meas(Ω)) ) 1 p− = C(k,ϕ,f,p−,p+,meas(Ω)). 32 Stanislas OUARO CUBO 14, 2 (2012) The above inequality gives ‖Tk(un)‖1,p(.) ≤ 1 + C(k,ϕ,f,p−,p+,meas(Ω)). Then, the proof of Lemma 5.8. is complete. From Lemma 5.8. we deduce that for any k > 0, the sequence (Tk(un)) is uniformly bounded in W1,p(.)(Ω) and so in W1,p−(Ω). Then, up to a subsequence we can assume that for any k > 0, Tk(un) converges weakly to σk in W 1,p−(Ω), and so Tk(un) converges strongly to σk in L p−(Ω). We next prove the following proposition: Proposition 5.9. Assume that (2.1)-(2.5) hold and un ∈ W 1,p(.)(Ω) is the weak solution of (5.22). Then the sequence (un)n is Cauchy in measure. In particular, there exists a measurable function u and a subsequence still denoted (un)n such that un −→ u in measure. Proof. Let s > 0 and define E1 := {|un| > k}, E2 := {|um| > k} and E3 := {|Tk(un) − Tk(um)| > s} where k > 0 is to be fixed. We note that {|un − um| > s} ⊂ E1 ∪ E2 ∪ E3, and hence meas{|un − um| > s} ≤ meas(E1) + meas(E2) + meas(E3). (5.30) Let ǫ > 0. Using Proposition 5.7, we choose k = k(ǫ) such that meas(E1) ≤ ǫ/3 and meas(E2) ≤ ǫ/3. (5.31) Since Tk(un) converges strongly in L p−(Ω), then it is a Cauchy sequence in Lp−(Ω). Thus meas(E3) ≤ 1 sp− ∫ Ω |Tk(un) − Tk(um)| p−dx ≤ ǫ 3 , (5.32) for all n,m ≥ n0(s,ǫ). Finally, from (5.30), (5.31) and (5.32), we obtain meas{|un − um| > s} ≤ ǫ for all n,m ≥ n0(s,ǫ). (5.33) Relations (5.33) mean that the sequence (un)n is Cauchy sequence in measure and the proof of Proposition 5.9. is complete. Note that as un −→ u in measure, up to a subsequence, we can assume that un −→ u a.e. in Ω. In the sequel, we need the following two technical lemmas. Lemma 5.10. ( cf.[30, Lemma 5.4] ) Let (vn)n be a sequence of measurable functions in Ω. If vn converges in measure to v and is uniformly bounded in L p(.)(Ω) for some 1 ≪ p(.) ∈ L∞(Ω), then vn −→ v strongly in L1(Ω). CUBO 14, 2 (2012) Weak and entropy solutions for a class of nonlinear ... 33 The second technical lemma is a well known result in measure theory (cf. [16] ). Lemma 5.11. Let (X,M,µ) be a measure space such that µ(X) < +∞. Consider a measurable function γ : X −→ [0,+∞] such that µ({x ∈ X : γ(x) = 0}) = 0. Then, for every ǫ > 0, there exists δ > 0, such that µ(A) < ǫ, for all A ∈ M with ∫ A γdµ < δ. We now set to prove that the function u in the Proposition 5.9 is an entropy solution of (1.1). Let v ∈ W1,p(.)(Ω) ∩ L∞(Ω). For any k > 0, choose Tk(un − v) as a test function in (5.23). We get ∫ Ω a(x,∇un).∇Tk(un − v)dx + ∫ Ω |un| p(x)−2 unTk(un − v)dx = ∫ ∂Ω ϕn(x)Tk(un − v)dσ + ∫ Ω fn(x)Tk(un − v)dx. (5.34) We have the following proposition: Proposition 5.12. Assume that (2.1)-(2.5) hold and un ∈ W 1,p(.)(Ω) be the weak solution of (5.22). Then (i) ∇un converges in measure to the weak gradient of u; (ii) For all k > 0, ∇Tk(un) converges to ∇Tk(u) in (L 1(Ω))N. (iii) For all t > 0, a(x,∇Tt(un)) converges to a(x,∇Tt(u)) in ( L1(Ω) )N strongly and in ( Lp ′ (.)(Ω) )N weakly. (iv) un converges to some function v a.e. on ∂Ω. Proof. (i) We claim that the sequence (∇un)n is Cauchy in measure. Indeed, let s > 0, and consider E1 := {|∇un| > h} ∪ {|∇um| > h}, E2 := {|un − um| > k} and E3 := {|∇un| ≤ h, |∇um| ≤ h, |un − um| ≤ k, |∇un − ∇um| > s}, where h and k will be chosen later. Note that {|∇un − ∇um| > s} ⊂ E1 ∪ E2 ∪ E3. (5.35) Let ǫ > 0. By Proposition 5.7 (relation (5.8)), we may choose h = h(ǫ) large enough such that meas(E1) ≤ ǫ/3, (5.36) 34 Stanislas OUARO CUBO 14, 2 (2012) for all n,m ≥ 0. On the other hand, by Proposition 5.9 meas(E2) ≤ ǫ/3, (5.37) for all n,m ≥ n0(k,ǫ). Moreover, since a(x,ξ) is continuous with respect to ξ for a.e every x ∈ Ω, by assumption (2.5) there exists a real valued function γ : Ω −→ [0,+∞] such that meas({x ∈ Ω : γ(x) = 0}) = 0 and (a(x,ξ) − a(x,ξ′)).(ξ − ξ′) ≥ γ(x), (5.38) for all ξ,ξ′ ∈ RN such that |ξ| ≤ h, |ξ′| ≤ h, |ξ − ξ′| ≥ s, for a.e x ∈ Ω. Let δ = δ(ǫ) be given by Lemma 5.11., replacing ǫ and A by ǫ/3 and E3 respectively. As un is a weak solution of (5.22), using Tk(un − um) as a test function, we get ∫ Ω a(x,∇un).∇Tk(un − um)dx + ∫ Ω |un| p(x)−2 unTk(un − um)dx = ∫ ∂Ω ϕnTk(un − um)dσ + ∫ Ω fnTk(un − um)dx ≤ k ( ‖ϕ‖L1(∂Ω) + ‖f‖L1(Ω) ) . Similarly for um, we have ∫ Ω a(x,∇um).∇Tk(um − un)dx + ∫ Ω |um| p(x)−2 umTk(um − un)dx = ∫ ∂Ω ϕmTk(um − un)dσ + ∫ Ω fmTk(um − un)dx ≤ k ( ‖ϕ‖L1(∂Ω) + ‖f‖L1(Ω) ) . After adding the last two inequalities, it yields    ∫ {|un−um|≤k} (a(x,∇un) − a(x,∇um)).(∇un − ∇um)dx + ∫ Ω ( |un| p(x)−2 un − |um| p(x)−2 um ) Tk(un − um)dx ≤ 2k ( ‖ϕ‖L1(∂Ω) + ‖f‖L1(Ω) ) . Since the second term of the above inequality is nonnegative, we obtain by using (5.38) ∫ E3 γ(x)dx ≤ ∫ E3 (a(x,∇un) − a(x,∇um)).(∇un − ∇um)dx ≤ 2k ( ‖ϕ‖L1(∂Ω) + ‖f‖L1(Ω) ) < δ, where k = δ/4 ( ‖ϕ‖L1(∂Ω) + ‖f‖L1(Ω) ) . From Lemma 5.11, it follows that meas(E3) ≤ ǫ/3. (5.39) CUBO 14, 2 (2012) Weak and entropy solutions for a class of nonlinear ... 35 Thus using (5.35), (5.36), (5.37) and (5.39), we get meas({|∇un − ∇um| > s}) ≤ ǫ, for all n,m ≥ n0(s,ǫ) (5.40) and then the claim is proved. Consequently, (∇un)n converges in measure to some measurable function v. In order to end the proof of (i), we need the following lemma: Lemma 5.13. (a) For a.e. t ∈ R, ∇Tt(un) converges in measure to vχ{|u| 0, {∣ ∣χ{|un| δ } ⊂ {∣ ∣χ{|un| δ } ≤ meas {|u| = t} + meas {un < t < u} + meas {u < t < un} + meas {un < −t < u} + meas {u < −t < un} . (5.41) Note that meas {|u| = t} ≤ meas {t − h < u < t + h} + meas {−t − h < u < −t + h} → 0 as h → 0 for a.e. t, since u is a fixed function. Next, meas {un < t < u} ≤ meas {t < u < t + h} + meas {|u − un| > h} , for all h > 0. Due to Proposition 5.9, we have for all fixed h > 0, meas {|u − un| > h} → 0 as n → +∞. Since meas {t < u < t + h} → 0 as h → 0, for all ǫ > 0, one can find N such that for all n > N, meas {un < t < u} < ǫ/2 + ǫ/2 = ǫ by choosing h and then N. Each of the other terms in the right-hand side of (5.41) can be treated in the same way as for meas {un < t < u}. Thus, meas {∣ ∣χ{|un| δ } → 0 as n → +∞. Finally, since ∇Tt(un) = ∇unχ{|un| 0, k > 0 and consider E4 = {|∇un − ∇um| > s, |un| ≤ k, |um| ≤ k} , E5 = {|∇um| > s, |un| > k, |um| ≤ k} , E6 = {|∇un| > s, |um| > k, |un| ≤ k} and E7 = {0 > s, |um| > k, |un| > k} . Note that {|∇Tk(un) − ∇Tk(um)| > s} ⊂ E4 ∪ E5 ∪ E6 ∪ E7. (5.42) Let ǫ > 0. By Proposition 5.7, we may choose k(ǫ) such that meas(E5) ≤ ǫ 4 ,meas(E6) ≤ ǫ 4 and meas(E7) ≤ ǫ 4 . (5.43) Therefore, using (5.40), (5.42) and (5.43), we get meas({|∇Tk(un) − ∇Tk(um)| > s}) ≤ ǫ, for all n,m ≥ n1(s,ǫ). (5.44) Consequently, ∇Tk(un) converges in measure to ∇Tk(u). Then, using lemmas 5.8 and 5.10, (ii) follows. (iii) By lemmas 5.10 and 5.13, we have that for all t > 0, a(x,∇Tt(un)) converges to a(x,∇Tt(u)) in ( L1(Ω) )N strongly and a(x,∇Tt(un)) converges to χt ∈ (L p′(.)(Ω))N in (Lp ′ (.)(Ω))N weakly. Since each of the convergences implies the weak L1-convergence, χt can be identified with a(x,∇Tt(u)); thus, a(x,∇Tt(u)) ∈ (L p′(.)(Ω))N. The proof of (iii) is then complete. (iv) As un is a weak solution of (5.22), using Tk(un) as a test function, we get ∫ Ω |Tk(un)| p(x) dx ≤ ∫ Ω |un| p(x)−2 unTk(un)dx ≤ k ( ‖ϕ‖L1(∂Ω) + ‖f‖L1(Ω) ) . (5.45) CUBO 14, 2 (2012) Weak and entropy solutions for a class of nonlinear ... 37 We deduce from (5.24) and (5.45) that ∫ Ω |Tk(un)| p− dx ≤ k ( ‖ϕ‖L1(∂Ω) + ‖f‖L1(Ω) ) + meas(Ω), (5.46) and ∫ Ω |∇Tk(un)| p− dx ≤ k ( ‖ϕ‖L1(∂Ω) + ‖f‖L1(Ω) ) + meas(Ω). (5.47) Furthermore, Tk(un) converges weakly to Tk(u) in W 1,p−(Ω) and since for every 1 ≤ p ≤ +∞, τ : W1,p(Ω) → Lp(∂Ω),u 7→ τ(u) = u|∂Ω is compact, we deduce that Tk(un) converges strongly to Tk(u) in L p−(∂Ω) and so, up to a subsequence, we can assume that Tk(un) converges to Tk(u), a.e. on ∂Ω. In other words, there exists C ⊂ ∂Ω such that Tk(un) converges to Tk(u) on ∂Ω\C with µ(C) = 0 where µ is the area measure on ∂Ω. Now, we use Hölder Inequality, (5.46) and (5.47) to get ∫ Ω |Tk(un)|dx ≤ (meas(Ω)) 1 (p−) ′ ( k ( ‖ϕ‖L1(∂Ω) + ‖f‖L1(Ω) ) + meas(Ω) ) 1 p− , (5.48) and ∫ Ω |∇Tk(un)|dx ≤ (meas(Ω)) 1 (p−) ′ ( k ( ‖ϕ‖L1(∂Ω) + ‖f‖L1(Ω) ) + meas(Ω) ) 1 p− . (5.49) By using Fatou’s Lemma in (5.48) and (5.49) we get as n goes to +∞, ∫ Ω |Tk(u)|dx ≤ (meas(Ω)) 1 (p−) ′ ( k ( ‖ϕ‖L1(∂Ω) + ‖f‖L1(Ω) ) + meas(Ω) ) 1 p− , (5.50) and ∫ Ω |∇Tk(u)|dx ≤ (meas(Ω)) 1 (p−) ′ ( k ( ‖ϕ‖L1(∂Ω) + ‖f‖L1(Ω) ) + meas(Ω) ) 1 p− . (5.51) For every k > 0, let Ak := {x ∈ ∂Ω : |Tk(u(x))| < k} and C ′ = ∂Ω\ ⋃ k>0 Ak. We have µ(C′) = 1 k ∫ C′ |Tk(u)|dx ≤ 1 k ∫ ∂Ω |Tk(u)|dx ≤ C1 k ‖Tk(u)‖W1,1(Ω) ≤ C1 k ‖Tk(u)‖L1(Ω) + C1 k ‖∇Tk(u)‖L1(Ω) . According to (5.50) and (5.51), we deduce by letting k → +∞ that µ(C′) = 0. Let us define in ∂Ω the function v by v(x) := Tk(u(x)) if x ∈ Ak. 38 Stanislas OUARO CUBO 14, 2 (2012) We take x ∈ ∂Ω\ (C ∪ C′); then there exists k > 0 such that x ∈ Ak and we have un(x) − v(x) = (un(x) − Tk(un(x))) + (Tk(un(x)) − Tk(u(x))) . Since x ∈ Ak, we have |Tk(u(x))| < k and so |Tk(un(x))| < k, from which we deduce that |un(x)| < k. Therefore un(x) − v(x) = (Tk(un(x)) − Tk(u(x))) → 0, as n → +∞. This means that un converges to v a.e. on ∂Ω. The proof of the Proposition 5.12 is then complete. We are now able to pass to the limit in the identity (5.34). For the right-hand side, the convergence is obvious since fn converges strongly to f in L 1(Ω), ϕn converges strongly to ϕ in L 1(∂Ω) and Tk(un − v) converges weakly-∗ to Tk(u − v) in L ∞(Ω) and a.e in Ω and to Tk(u − v) in L ∞(∂Ω) and a.e in ∂Ω. For the second term of (5.34), we have ∫ Ω |un| p(x)−2 unTk(un − v)dx = ∫ Ω ( |un| p(x)−2 un − |v| p(x)−2 v ) Tk(un − v)dx + ∫ Ω |v|p(x)−2 vTk(un − v)dx. The quantity ( |un| p(x)−2 un − |v| p(x)−2 v ) Tk(un − v) is nonnegative and since for all x ∈ Ω, s 7−→ |s|p(x)−2 s is continuous, we get ( |un| p(x)−2 un − |v| p(x)−2 v ) Tk(un − v) −→ ( |u|p(x)−2 u − |v|p(x)−2 v ) Tk(u − v)dx a.e in Ω. Then, it follows by Fatou’s Lemma that lim inf n→+∞ ∫ Ω ( |un| p(x)−2 un − |v| p(x)−2 v ) Tk(un−v)dx ≥ ∫ Ω ( |u|p(x)−2 u − |v|p(x)−2 v ) Tk(u−v)dx. Let us show that |v|p(x)−2 v ∈ L1(Ω). We have ∫ Ω ∣ ∣ ∣ |v|p(x)−2 v ∣ ∣ ∣ dx = ∫ Ω |v|p(x)−1dx ≤ ∫ Ω ( ‖v‖∞ )p(x)−1 dx. If ‖v‖∞ ≤ 1, then ∫ Ω ∣ ∣ ∣ |v|p(x)−2 v ∣ ∣ ∣ dx ≤ meas(Ω) < +∞. If ‖v‖∞ > 1, then ∫ Ω ∣ ∣ ∣ |v|p(x)−2 v ∣ ∣ ∣ dx ≤ ∫ Ω ( ‖v‖∞ )p+−1 dx = ( ‖v‖∞ )p+−1 meas(Ω) < +∞. CUBO 14, 2 (2012) Weak and entropy solutions for a class of nonlinear ... 39 Hence |v|p(x)−2 v ∈ L1(Ω). Since Tk(un −v) converges weakly-∗ to Tk(u−v) in L ∞(Ω) and |v|p(x)−2 v ∈ L1(Ω), it follows that lim n→+∞ ∫ Ω |v|p(x)−2 vTk(un − v)dx = ∫ Ω |v|p(x)−2 vTk(u − v)dx. Next, we write the first term in (5.34) in the following form ∫ {|un−v|≤k} a(x,∇un).∇undx − ∫ {|un−v|≤k} a(x,∇un).∇vdx. (5.52) Set l = k + ‖v‖∞, the second integral in (5.52) equals to ∫ {|un−v|≤k} a(x,∇Tl(un)).∇vdx. Since a(x,∇Tl(un)) is uniformly bounded in ( Lp ′ (.)(Ω) )N (by (2.3) and (5.24) ), by Proposition 5.12−(iii), it converges weakly to a(x,∇Tl(u)) in ( Lp ′ (.)(Ω) )N . Therefore lim n→+∞ ∫ {|un−v|≤k} a(x,∇Tl(un)).∇vdx = ∫ {|u−v|≤k} a(x,∇Tl(u)).∇vdx. Moreover a(x,∇un).∇un is nonnegative and converges a.e in Ω to a(x,∇u).∇u. Thanks to Fatou’s Lemma, we obtain lim inf n→+∞ ∫ {|un−v|≤k} a(x,∇un).∇undx ≥ ∫ {|u−v|≤k} a(x,∇u).∇udx. Gathering results, we obtain ∫ Ω a(x,∇u).∇Tk(u − v)dx + ∫ Ω |u|p(x)−2 uTk(u − v)dx ≤ ∫ ∂Ω ϕTk(u − v)dσ + ∫ Ω fTk(u − v)dx. We conclude that u is an entropy solution of (1.1). Acknowledgment The author want to express is deepest thanks to the editor and anonymous referee’s for comments on the paper. Received: January 2011. Revised: September 2011. 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Zhikov; On passing to the limit in nonlinear variational problem. Math. Sb. 183 (1992), 47-84. Introduction Assumptions and preliminaries Weak solution Weak solutions for a right-hand side dependent on u Entropy solutions