CUBO, A Mathematical Journal Vol. 23, no. 01, pp. 21–62, April 2021 DOI: 10.4067/S0719-06462021000100021 Anisotropic problem with non-local boundary con- ditions and measure data A. Kaboré S. Ouaro Laboratoire de Mathematiques et Informatiques (LAMI), UFR. Sciences Exactes et Appliquées, Université Joseph KI-ZERBO, 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso. kaboreadama59@yahoo.fr; ouaro@yahoo.fr ABSTRACT We study a nonlinear anisotropic elliptic problem with non- local boundary conditions and measure data. We prove an existence and uniqueness result of entropy solution. RESUMEN Estudiamos un problema eĺıptico nolineal anisotrópico con condiciones de borde no-locales y data de medida. Probamos un resultado de existencia y unicidad de la solución de en- troṕıa. Keywords and Phrases: Entropy solution, non-local boundary conditions, Leray-Lions operator, bounded Radon diffuse measure, Marcinkiewicz spaces. 2020 AMS Mathematics Subject Classification: 35J05, 35J25, 35J60, 35J66. Accepted: 06 January, 2021 Received: 13 November, 2019 ©2021 A. Kaboré et al. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/S0719-06462021000100021 https://orcid.org/0000-0003-0671-2378 22 A. Kaboré & S. Ouaro CUBO 23, 1 (2021) 1 Introduction and assumptions Let Ω be a bounded domain in RN (N ≥ 3) such that ∂Ω is Lipschitz and ∂Ω = ΓD ∪ ΓNe with ΓD ∩ ΓNe = ∅. Our aim is to study the following problem. P(ρ,µ,d)   − N∑ i=1 ∂ ∂xi ai ( x, ∂ ∂xi u ) + |u|pM (x)−2u = µ in Ω u = 0 on ΓD ρ(u) + N∑ i=1 ∫ ΓNe ai ( x, ∂ ∂xi u ) ηi = d u ≡ constant   on ΓNe, (1.1) where the right-hand side µ is a bounded Radon diffuse measure (that is µ does not charge the sets of zero pm(.)-capacity), ρ : R → R a surjective, continuous and non-decreasing function, with ρ(0) = 0, d ∈ R and ηi, i ∈{1, ...,N} are the components of the outer normal unit vector. For any Ω ⊂ RN , we set C+(Ω̄) = {h ∈ C(Ω̄) : inf x∈Ω h(x) > 1} (1.2) and we denote h+ = sup x∈Ω h(x), h− = inf x∈Ω h(x). (1.3) For the exponents, ~p(.) : Ω̄ → RN , ~p(.) = (p1(.), ...,pN (.)) with pi ∈ C+(Ω̄) for every i ∈{1, ...,N} and for all x ∈ Ω̄. We put pM (x) = max{p1(x), ...,pN (x)} and pm(x) = min{p1(x), ...,pN (x)} . We assume that for i = 1, ...,N, the function ai : Ω × R → R is Carathéodory and satisfies the following conditions. • (H1): ai(x,ξ) is the continuous derivative with respect to ξ of the mapping Ai = Ai(x,ξ), that is, ai(x,ξ) = ∂ ∂ξ Ai(x,ξ) such that the following equality holds. Ai(x, 0) = 0, (1.4) for almost every x ∈ Ω. • (H2) : There exists a positive constant C1 such that |ai(x,ξ)| ≤ C1(ji(x) + |ξ|pi(x)−1), (1.5) for almost every x ∈ Ω and for every ξ ∈ R, where ji is a non-negative function in Lp ′ i(.)(Ω), with 1 pi(x) + 1 p′i(x) = 1. CUBO 23, 1 (2021) Anisotropic Problem with Non-local Boundary Conditions and Measure Data 23 • (H3) : there exists a positive constant C2 such that (ai(x,ξ) −ai(x,η)).(ξ −η) ≥   C2|ξ −η|pi(x) if |ξ −η| ≥ 1, C2|ξ −η|p − i if |ξ −η| < 1, (1.6) for almost every x ∈ Ω and for every ξ, η ∈ R, with ξ 6= η. • (H4) : For almost every x ∈ Ω and for every ξ ∈ R, |ξ|pi(x) ≤ ai(x,ξ).ξ ≤ pi(x)Ai(x,ξ). (1.7) • (H5) : The variable exponents pi(.) : Ω̄ → [2,N) are continuous functions for all i = 1, ...,N such that p̄(N − 1) N(p̄− 1) < p−i < p̄(N − 1) N − p̄ , N∑ i=1 1 p−i > 1 and p+i −p − i − 1 p−i < p̄−N p̄(N − 1) , (1.8) where 1 p̄ = 1 N N∑ i=1 1 p−i . As examples under assumptions (H1) -(H5), we can give the following. (1) Set Ai(x,ξ) = ( 1 pi(x) )|ξ|pi(x) and ai(x,ξ) = |ξ|pi(x)−2ξ , where 2 ≤ pi(x) < N. (2) Ai(x,ξ) = ( 1 pi(x) )((1 + |ξ|2) pi(x) 2 − 1) and ai(x,ξ) = (1 + |ξ|2) pi(x)−2 2 ξ , where 2 ≤ pi(x) < N. We put for all x ∈ ∂Ω, p∂(x) =   (N − 1)p(x) N −p(x) if p(x) < N, ∞ if p(x) ≥ N. We introduce the numbers q = N(p̄− 1) N − 1 , q∗ = Nq N −q = N(p̄− 1) N − p̄ . (1.9) We denote by Mb(Ω) the space of bounded Radon measure in Ω, equipped with its standard norm ‖.‖Mb(Ω). Note that, if u belongs to Mb(Ω), then |µ|(Ω) (the total variation of µ) is a bounded positive measure on Ω. Given µ ∈ Mb(Ω), we say that µ is diffuse with respect to the capacity W 1,p(.) 0 (Ω) (p(.)-capacity for short) if µ(A) = 0, for every set A such that Capp(.)(A, Ω) = 0. For every A ⊂ Ω, we denote Sp(.)(A) = {u ∈ W 1,p(.) 0 (Ω) ∩C0(Ω) : u = 1 on A,u ≥ 0 on Ω}. 24 A. Kaboré & S. Ouaro CUBO 23, 1 (2021) The p(.)-capacity of every subset A with respect to Ω is defined by Capp(.)(A, Ω) = inf u∈Sp(.)(A) { ∫ Ω |∇u|p(x)dx}. In the case Sp(.)(A) = ∅, we set Capp(.)(A, Ω) = ∞. The set of bounded Radon diffuse measure in the variable exponent setting is denoted by Mp(.)b (Ω). We use the following result of decomposition of bounded Radon diffuse measure proved by Nyan- quini et al. (see [31]). Theorem 1.1. Let p(.) : Ω̄ → (1,∞) be a continuous function and µ ∈Mb(Ω). Then µ ∈M p(.) b (Ω) if and only if µ ∈ L1(Ω) + W−1,p ′(.)(Ω). Remark 1.2. Since µ ∈ Mpm(.)b (Ω), the Theorem 1.1 implies that there exist f ∈ L 1(Ω) and F ∈ (Lp ′ m(.)(Ω))N such that µ = f − divF, (1.10) where 1 pm(x) + 1 p′m(x) = 1, ∀x ∈ Ω. The study of nonlinear elliptic equations involving the p-Laplace operator is based on the the- ory of standard Sobolev spaces Wm,p(Ω) in order to find weak solutions. For the nonhomogeneous p(.)-Laplace operators, the natural setting for this approach is the use of the variable exponent Lebesgue and Sobolev spaces Lp(.)(Ω) and Wm,p(.)(Ω). Variable exponent Lebesgue spaces appeared in the literature for the first time in a article by Orlicz in 1931. In the 1950’s, this study was carred on by Nakano who made the first systematic study of spaces with variable exponent (called modular spaces). Nakano explicitly mentioned variable expo- nent Lebesgue spaces as an example of more general spaces he considered (see [30], p. 284). Later, the polish mathematicians investigated the modular function spaces (see [29]). Note also that H. Hudzik [18] investigated the variable exponent Sobolev spaces. Variable exponent Lebesgue spaces on the real line have been independently developed by Russian researchers, notably Sharapudinov [40] and Tsenov [42]. The next major step in the investigation of variable exponent Lebesgue and Sobolev spaces was the comprehensive paper by O. Kovacik and J. Rakosnik in the early 90’s [23]. This paper established many of basic properties of Lebesgue and Sobolev spaces with variables exponent. Variable Sobolev spaces have been used in the last decades to model various phenomena. In [9], Chen, Levine and Rao proposed a framework for image restoration based on a Laplacian variable exponent. Another application which uses nonhomogeneous Laplace operators is related to the modelling of electrorheological fluids see [38]. The first major discovery in electrorheological fluids was due to Winslow in 1949 (cf. [43]). These fluids have the interesting property that their viscosity depends on the electric field in the fluid. They can raise the viscosity by as much as five orders of magnitude. This phenomenon is known as the Winslow effect. For some technical applications, we refer the readers to the work by Pfeiffer et al [33]. Electrorheological fluids have been used in robotics and space technology. The experimental research has been done mainly in CUBO 23, 1 (2021) Anisotropic Problem with Non-local Boundary Conditions and Measure Data 25 the USA, for instance in NASA laboratories. For more information on properties, modelling and the application of variable exponent spaces to these fluids, we refer to Diening [11], Rajagopal and Ruzicka [35], and Ruzicka [36]. In this paper, the operator involved in (1.1) is more general than the p(.)-Laplace operator. Thus, the variable exponent Sobolev space W 1,p(.)(Ω) is not adequate to study nonlinear problems of this type. This leads us to seek entropy solutions for problems (1.1) in a more general variable exponent Sobolev space which was introduced for the first time by Mihäılescu et al. [28], see also [34, 26, 27]. The need for such theory comes naturally every time we want to consider materials with inho- mogeneities that have different behavior on different space directions. Non-local boundary value problems of various kinds for partial differential equations are of great interest by now in several fields of application. In a typical non-local problem, the partial differential equation (resp. bound- ary conditions) for an unknown function u at any point in a domain Ω involves not only the local behavior of u in a neighborhood of that point but also the non-local behavior of u elsewhere in Ω. For example, at any point in Ω the partial differential equation and/or the boundary conditions may contains integrals of the unknown u over parts of Ω, values of u elsewhere in D or, generally speaking, some non-local operator on u. Beside the mathematical interest of nonlocal conditions, it seems that this type of boundary condition appears in petroleum engineering model for well modeling in a 3D stratified petroleum reservoir with arbitrary geometry (see [12] and [15]). A lot of papers ( see [34], [24], [25], [2], [19], [1]) on problems like (1.1) considered cases of generally boundary value condition. In [6], Bonzi et al. studied the following problems.   − N∑ i=1 ∂ ∂xi ai ( x, ∂ ∂xi u ) + |u|pM (x)−2u = f in Ω N∑ i=1 ai ( x, ∂ ∂xi u ) ηi = −|u|r(x)−2u on ∂Ω, (1.11) which correspond to the Robin type boundary condition. The authors used minimization tech- niques used in [8] to prove the existence and uniqueness of entropy solution. By the same tech- niques, Koné and al. proved the existence and uniqueness of entropy solution for the following problem.   − N∑ i=1 ∂ ∂xi ai ( x, ∂ ∂xi u ) + |u|pM (x)−2u = f in Ω N∑ i=1 ai ( x, ∂ ∂xi u ) ηi + λu = g on ∂Ω, (1.12) which correspond to the Fourier type boundary condition. In a recent paper we studied a nonlinear elliptic anisotropic problem involving non- local conditions. We also considered variable exponent and general maximal monotone graph datum at the boundary 26 A. Kaboré & S. Ouaro CUBO 23, 1 (2021) and proved existence and uniqueness of weak solution to the following problem. S(ρ,µ,d)   − N∑ i=1 ∂ ∂xi ai ( x, ∂ ∂xi u ) + |u|pM (x)−2u = f in Ω u = 0 on ΓD ρ(u) + N∑ i=1 ∫ ΓNe ai ( x, ∂ ∂xi u ) ηi 3 d u ≡ constant   on ΓNe, where the right-hand side f ∈ L∞(Ω) and ρ a maximal monotone graph on R such that D(ρ) = Im(ρ) = R and 0 ∈ ρ(0), d ∈ R, by using the technique of monotone operators in Banach spaces (see [21]) and approximation methods. There are two difficulties associated with the study of problem P(ρ,µ,d). The first is to give a sense to the partial derivative of u which appear in the term ai ( x, ∂ ∂xi u ) . As µ is a measure (even if µ is a integrable function), then we cannot take the partial derivative of u in the usual distribution sense. The idea consists in considering troncatures of the solution u (see [5]). The second difficulty appears with the question of uniqueness of solutons. We obtain existence and uniqueness of a special class of solutions of problem P(ρ,µ,d) that satisfy an extra condition that we call the entropy condition (see formula (2.9)). An alternative notion of solution which can leads to existence and uniqueness of solution to problem P(ρ,µ,d) is the notion of renormalized solution. But in this work, we consider the notion of entropy solution. The paper is organized as follows. Section 2 is devoted to mathematical preliminaries including, among other things, a brief discussion on variable exponent Lebesgue, Sobolev, anisotropic and Marcinkiewicz spaces. In Section 3, we study an approximated problem and in Section 4, we prove by using the results of the Section 3, the existence and uniqueness of entropy solution of problem P(ρ,µ,d). 2 Preliminary This part is related to anisotropic Lebesgue and Sobolev spaces with variable exponent and some of their properties. Given a measurable function p(.) : Ω → [1,∞). We define the Lebesgue space with variable exponent Lp(.)(Ω) as the set of all measurable functions 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 } CUBO 23, 1 (2021) Anisotropic Problem with Non-local Boundary Conditions and Measure Data 27 defines a norm in Lp(.)(Ω), called the Luxembourg norm. The space (Lp(.)(Ω), |.|p(.)) is a separable Banach space. 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, for all x ∈ Ω. We have the following properties (see [13]) on the modular ρp(.). If u,un ∈ Lp(.)(Ω) and p+ < ∞, then |u|p(.) < 1 ⇒|u| p+ p(.) ≤ ρp(.)(u) ≤ |u| p− p(.) , (2.1) |u|p(.) > 1 ⇒|u| p− p(.) ≤ ρp(.)(u) ≤ |u| p+ p(.) , (2.2) |u|p(.) < 1(= 1; > 1) ⇒ ρp(.)(u) < 1(= 1; > 1), (2.3) and |un|p(.) → 0 (|un|p(.) →∞) ⇔ ρp(.)(un) → 0 (ρp(.)(un) →∞). (2.4) If in addition, (un)n∈N ⊂ Lp(.)(Ω), then limn→∞ |un −u|p(.) = 0 ⇔ limn→∞ρp(.)(un −u) = 0 ⇔ (un)n∈N converges to u in measure and limn→∞ρp(.)(un) = ρp(.)(u). We introduce the definition of the isotropic Sobolev space with variable exponent, W 1,p(.)(Ω) := { u ∈ Lp(.)(Ω) : |∇u| ∈ Lp(.)(Ω) } , which is a Banach space equipped with the norm ‖u‖1,p(.) := |u|p(.) + |∇u|p(.). Now, we present the anisotropic Sobolev space with variable exponent which is used for the study of P(ρ,µ,d). The anisotropic variable exponent Sobolev space W 1,~p(.)(Ω) is defined as follow. W 1,~p(.)(Ω) := { u ∈ LpM (.)(Ω) : ∂u ∂xi ∈ Lpi(.)(Ω), for all i ∈{1, ...,N} } . Endowed with the norm ‖u‖~p(.) := |u|pM (.) + N∑ i=1 ∣∣∣∣ ∂u∂xi ∣∣∣∣ pi(.) , the space ( W 1,~p(.)(Ω),‖.‖~p(.) ) is a reflexive Banach space (see [14], Theorem 2.1 and Theorem 2.2). As consequence, we have the following. Theorem 2.1. (see [14]) Let Ω ⊂ RN (N ≥ 3) be a bounded open set and for all i ∈{1, ...,N}, pi ∈ L∞(Ω), pi(x) ≥ 1 a.e. in Ω. Then, for any r ∈ L∞(Ω) with r(x) ≥ 1 a.e. in Ω such that ess inf x∈Ω (pM (x) −r(x)) > 0, we have the compact embedding W 1,~p(.)(Ω) ↪→ Lr(.)(Ω). 28 A. Kaboré & S. Ouaro CUBO 23, 1 (2021) We also need the following trace theorem due to [7]. Theorem 2.2. Let Ω ⊂ RN (N ≥ 2) be a bounded open set with smooth boundary and let ~p(.) ∈ C(Ω̄) satisfy the condition 1 ≤ r(x) < min x∈∂Ω {p∂1 (x), ...,p ∂ N (x)}, ∀x ∈ ∂Ω. (2.5) Then, there is a compact boundary trace embedding W 1,~p(.)(Ω) ↪→ Lr(.)(∂Ω). Let us introduce the following notation: ~p− = (p − 1 , ...,p − N ). We will use in this paper, the Marcinkiewicz spaces Mq(Ω) (1 < q < ∞) with constant exponent. Note that the Marcinkiewicz spaces Mq(.)(Ω) in the variable exponent setting was introduced for the first time by Sanchon and Urbano (see [37]). Marcinkiewicz spaces Mq(Ω) (1 < q < ∞) contain all measurable function h : Ω → R for which the distribution function λh(γ) := meas({x ∈ Ω : |h(x)| > γ}), γ ≥ 0, satisfies an estimate of the form λh(γ) ≤ Cγ−q, for some finite constant C > 0. The space Mq(Ω) is a Banach space under the norm ‖h‖∗Mq(Ω) = sup t>0 t 1 q ( 1 t ∫ t 0 h∗(s)ds ) , where h∗ denotes the nonincreasing rearrangement of h. h∗(t) := inf { C : λh(γ) ≤ Cγ−q, ∀γ > 0 } , which is equivalent to the norm ‖h‖∗Mq(Ω) (see [3]). We need the following Lemma (see [4], Lemma A-2). Lemma 2.3. Let 1 ≤ q < p < ∞. Then, for every measurable function u on Ω, (i) (p− 1)p pp+1 ‖u‖pMp(Ω) ≤ sup λ>0 {λpmeas[x ∈ Ω : |u| > λ]}≤‖u‖pMp(Ω). Moreover, (ii) ∫ K |u|qdx ≤ p p−q ( p q ) q p‖u‖qMp(Ω)(meas(K)) p−q p , for every measurable subset K ⊂ Ω. In particular, Mp(Ω) ⊂ Lqloc(Ω), with continuous embedding and u ∈ M p(Ω) implies |u|q ∈ M p q (Ω). CUBO 23, 1 (2021) Anisotropic Problem with Non-local Boundary Conditions and Measure Data 29 The following result is due to Troisi (see [39]). Theorem 2.4. Let p1, ...,pN ∈ [1,∞), ~p = (p1, ...,pN ); g ∈ W 1,~p(Ω), and let  q = p̄∗ if p̄∗ < N, q ∈ [1,∞) if p̄∗ ≥ N; (2.6) where p∗ = N∑N i=1 1 pi − 1 , ∑N i=1 1 pi > 1 and p̄∗ = Np̄ N − p̄ . Then, there exists a constant C > 0 depending on N, p1, ...,pN if p̄ < N and also on q and meas(Ω) if p̄ ≥ N such that ‖g‖Lq(Ω) ≤ c N∏ i=1 [ ‖g‖LpM (Ω) + ‖ ∂g ∂xi ‖Lpi(Ω) ] 1 N , (2.7) where pM = max{p1, ...,pN} and 1p̄ = 1 N ∑N i=1 1 pi . In particular, if u ∈ W 1,~p0 (Ω), we have ‖g‖Lq(Ω) ≤ c N∏ i=1 [∥∥∥∥ ∂g∂xi ∥∥∥∥ Lpi(Ω) ] 1 N . (2.8) In the sequel, we consider the following spaces. W 1,~p(.) D (Ω) = {ξ ∈ W 1,~p(.)(Ω) : ξ = 0 on ΓD} and W 1,~p(.) Ne (Ω) = {ξ ∈ W 1,~p(.) D (Ω) : ξ ≡ constant on ΓNe}. T 1,~p(.)D (Ω) = {ξ measurable on Ω such that ∀k > 0, Tk(ξ) ∈ W 1,~p(.) D (Ω)} and T 1,~p(.)Ne (Ω) = {ξ measurable on Ω such that ∀k > 0, Tk(ξ) ∈ W 1,~p(.) Ne (Ω)}, where Tk is a truncation function defined by Tk(s) =   k if s > k, s if |s| ≤ k, −k if s < −k. For any v ∈ W 1,~p(.)Ne (Ω), we set vN = vNe := v|ΓNe. Definition 2.5. A measurable function u : Ω → R is an entropy solution of P(ρ,µ,d) if u ∈ T 1,~p(.)Ne (Ω) and for every k > 0,  ∫ Ω ( N∑ i=1 ai ( x, ∂ ∂xi u ) ∂ ∂xi Tk(u− ξ) ) dx + ∫ Ω |u|pM (x)−2uTk(u− ξ)dx ≤∫ Ω Tk(u− ξ)dµ + (d−ρ(uNe))Tk(uNe − ξ), (2.9) for all ξ ∈ W 1,~p(.)Ne (Ω) ∩L ∞(Ω). 30 A. Kaboré & S. Ouaro CUBO 23, 1 (2021) Our main result in this paper is the following theorem. Theorem 2.6. Assume (H1)-(H5). Then for any (µ,d) ∈M pm(.) b (Ω) × R, the problem P(ρ,µ,d) admits a unique entropy solution u. 3 The approximated problem corresponding to P(ρ,µ,d) We define a new bounded domain Ω̃ in RN as follow. We fix θ > 0 and we set Ω̃ = Ω ∪{x ∈ RN/dist(x, ΓNe) < θ}. Then, ∂Ω̃ = ΓD ∪ Γ̃Ne is Lipschitz with ΓD ∩ Γ̃Ne = ∅. Figure 1: Domains representation Let us consider ãi(x,ξ) (to be defined later) Carathéodory and satisfying (1.4), (1.5), (1.6) and (1.7), for all x ∈ Ω̃. We also consider a function d̃ in L1(Γ̃Ne) such that∫ Γ̃Ne d̃dσ = d. (3.1) For any � > 0, we set µ� = f� − divF, where f� = T1 � (f) ∈ L∞(Ω) . Note that f� → f as � → 0 in L1(Ω) and ‖f�‖1 ≤‖f‖1. We set µ̃� = f�χΩ − divFχΩ, d̃� = T1 � (d̃) and we consider the problem P(ρ̃, µ̃�, d̃�)   − N∑ i=1 ∂ ∂xi ãi(x, ∂ ∂xi u�) + |u�|pM (x)−2u�χΩ(x) = µ̃� in Ω̃ u� = 0 on ΓD ρ̃(u�) + N∑ i=1 ãi(x, ∂ ∂xi u�)ηi = d̃� on Γ̃Ne, (3.2) where the function ρ̃ is defined as follow. CUBO 23, 1 (2021) Anisotropic Problem with Non-local Boundary Conditions and Measure Data 31 • ρ̃(s) = 1 |Γ̃Ne| ρ(s), where |Γ̃Ne| denotes the Hausdorff measure of Γ̃Ne. We obviously have ∀� > 0, d̃� ∈ L∞(Γ̃Ne). The following definition gives the notion of solution for the problem P�(ρ̃, µ̃�, d̃�). Definition 3.1. A measurable function u� : Ω̃ → R is a solution to problem P�(ρ̃, µ̃�, d̃�) if u� ∈ W 1,~p(.) D (Ω̃) and∫ Ω̃ N∑ i=1 ãi(x, ∂ ∂xi u�) ∂ ∂xi ξ̃dx + ∫ Ω |u�|pM (x)−2u�ξ̃dx = ∫ Ω f�ξ̃dx + ∫ Ω F.∇ξ̃ + ∫ Γ̃Ne (d̃� − ρ̃(u�))ξ̃dσ, (3.3) for any ξ̃ ∈ W 1,~p(.)D (Ω̃) ∩L ∞(Ω). Theorem 3.2. The problem P�(ρ̃, µ̃�, d̃�) admits at least one solution in the sense of Definition 3.1. Step 1: Approximated problem we study an existence result to the following problem. For any k > 0 we consider P�,k(ρ̃, µ̃�, d̃�)   − N∑ i=1 ∂ ∂xi ãi(x, ∂ ∂xi u�,k) + Tk(b(u�,k))χΩ(x) = µ̃� in Ω̃ u�,k = 0 on ΓD Tk(ρ̃(u�,k)) + N∑ i=1 ãi(x, ∂ ∂xi u�,k)ηi = d̃� on Γ̃Ne, (3.4) where b(u) = |u|pM (x)−2u. We have to prove that P�,k(ρ̃, µ̃�, d̃�) admits at least one solution in the following sense.  u�,k ∈ W 1,~p(.) D (Ω̃) and for all ξ̃ ∈ W 1,~p(.) D (Ω̃),∫ Ω̃ N∑ i=1 ãi(x, ∂ ∂xi u�,k) ∂ ∂xi ξ̃dx + ∫ Ω Tk(b(u�,k))ξ̃dx = ∫ Ω ξ̃dµ� + ∫ Γ̃Ne (d̃� −Tk(ρ̃(u�,k)))ξ̃dσ. (3.5) For any k > 0, let us introduce the operator Λk : W 1,~p(.) D (Ω̃) → (W 1,~p(.) D (Ω̃)) ′ such that for any (u,v) ∈ W 1,~p(.)D (Ω̃) ×W 1,~p(.) D (Ω̃), 〈Λk(u),v〉 = ∫ Ω̃ ( N∑ i=1 ãi(x, ∂ ∂xi u) ∂ ∂xi v ) dx + ∫ Ω Tk(b(u))vdx + ∫ Γ̃Ne Tk(ρ̃(u))vdσ. (3.6) We need to prove that for any k > 0, the operator Λk is bounded, coercive, of type M and therefore, surjective. (i) Boundedness of Λk. Let (u,v) ∈ F ×W 1,~p(.) D (Ω̃) with F a bounded subset of W 1,~p(.) D (Ω̃) . 32 A. Kaboré & S. Ouaro CUBO 23, 1 (2021) We have  |〈Λk(u),v〉| ≤ N∑ i=1 (∫ Ω̃ ∣∣∣∣ãi(x, ∂∂xiu) ∣∣∣∣ ∣∣∣∣ ∂∂xiv ∣∣∣∣dx ) + ∫ Ω̃ |Tk(b(u))||v|dx + ∫ Γ̃Ne |Tk(ρ̃(u))||v|dσ = I1 + I2 + I3, where we denote by I1, I2 and I3 the three terms on the right hand side of the first inequality. By (H2) and the Hölder type inequality, we have  I1 ≤ C1 N∑ i=1 (∫ Ω̃ |ji(x)| ∣∣∣∣ ∂∂xiv ∣∣∣∣dx + ∫ Ω̃ ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x)−1 ∣∣∣∣ ∂∂xiv ∣∣∣∣dx ) ≤ C1 N∑ i=1 ( 1 p′−i + 1 p−i ) |ji|p′ i (.) ∣∣∣∣ ∂∂xiv ∣∣∣∣ pi(.) + N∑ i=1 ( 1 p′−i + 1 p−i )∣∣∣∣∣ ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x)−1 ∣∣∣∣∣ p′ i (.) ∣∣∣∣ ∂∂xiv ∣∣∣∣ pi(.) . As u ∈ F, ∀ i ∈{1, ...,N}, there exists a constant M > 0 such that N∑ i=1 ∣∣∣∣∣ ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x)−1 ∣∣∣∣∣ p′ i (.) < M; so ∣∣∣∣∣ ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x)−1 ∣∣∣∣∣ p′ i (.) < M, ∀ i ∈{1, ...,N}. Let C4 = max i=1,...,N   ∣∣∣∣∣ ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x)−1 ∣∣∣∣∣ p′ i (.)   . As ji ∈ Lp ′ i(.)(Ω̃), we have I1 ≤ C5(C1,p−i , (p ′ i) −,C3(ji)) N∑ i=1 ∣∣∣∣ ∂∂xiv ∣∣∣∣ pi(.) + C6(C1,p − i , (p ′ i) −,C4) N∑ i=1 ∣∣∣∣ ∂∂xiv ∣∣∣∣ pi(.) . It is easy to see that I2 ≤ k ∫ Ω̃ |v|dx. Using Theorem 2.1, we have ‖v‖L1(Ω̃) ≤ C7‖v‖W1,~p(.) D (Ω̃) . So, I2 ≤ kC7‖v‖W1,~p(.) D (Ω̃) . Similarly, by using Theorem 2.2, we have I3 ≤ kC8‖v‖W1,~p(.) D (Ω̃) � Therefore, Λk maps bounded subsets of W 1,~p(.) D (Ω̃) into bounded subsets of (W 1,~p(.) D (Ω̃)) ′. Thus, Λk is bounded on W 1,~p(.) D (Ω̃). CUBO 23, 1 (2021) Anisotropic Problem with Non-local Boundary Conditions and Measure Data 33 (ii) Coerciveness of Λk. We have to show that for any k > 0, 〈Λk(u),u〉 ‖u‖ W 1,~p(.) D (Ω̃) → ∞ as ‖u‖ W 1,~p(.) D (Ω̃) →∞. For any u ∈ W 1,~p(.)D (Ω̃), we have 〈Λk(u),u〉 = 〈Λ(u),u〉 + ∫ Ω Tk(b(u))udx + ∫ Γ̃Ne Tk(ρ̃(u))udσ, (3.7) where 〈Λ(u),u〉 = N∑ i=1 (∫ Ω̃ ãi(x, ∂ ∂xi u) ∂ ∂xi udx ) . The last two terms on the right-hand side of (3.7) are non-negative by the monotonicity of Tk, b and ρ̃. We can assert that  〈Λk(u),u〉≥ 〈Λ(u),u〉 ≥ 1 Np − m−1 ‖u‖p − m W 1,~p(.) D (Ω̃) −N. Indeed, since ∫ Ω̃ |Tk(b(u))||u|dx + ∫ Γ̃Ne |Tk(ρ̃(u))||u|dσ ≥ 0, for all u ∈ W 1,~p(.) D (Ω̃), we have 〈Λk(u),u〉≥ 〈Λ(u),u〉. So, 〈Λk(u),u〉 ≥ N∑ i=1 (∫ Ω̃ ãi(x, ∂ ∂xi u) ∂ ∂xi udx ) ≥ N∑ i=1 (∫ Ω̃ ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x) dx ) . We make the following notations: I = { i ∈{1, ...,N} : ∣∣∣∣ ∂∂xiu ∣∣∣∣ pi(.) ≤ 1 } and J = { i ∈{1, ...,N} : ∣∣∣∣ ∂∂xiu ∣∣∣∣ pi(.) > 1 } . We have 〈Λk(u),u〉 ≥ ∑ i∈I (∫ Ω̃ ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x) dx ) + ∑ i∈J (∫ Ω̃ ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x) dx ) ≥ ∑ i∈I (∣∣∣∣ ∂∂xiu ∣∣∣∣p + i pi(.) ) + ∑ i∈J (∣∣∣∣ ∂∂xiu ∣∣∣∣p − i pi(.) ) ≥ ∑ i∈J (∣∣∣∣ ∂∂xiu ∣∣∣∣p − i pi(.) ) ≥ ∑ i∈J (∣∣∣∣ ∂∂xiu ∣∣∣∣p − m pi(.) ) ≥ N∑ i=1 (∣∣∣∣ ∂∂xiu ∣∣∣∣p − m pi(.) ) − ∑ i∈I (∣∣∣∣ ∂∂xiu ∣∣∣∣p − m pi(.) ) ≥ N∑ i=1 (∣∣∣∣ ∂∂xiu ∣∣∣∣p − m pi(.) ) −N. 34 A. Kaboré & S. Ouaro CUBO 23, 1 (2021) We now use Jensen’s inequality on the convex function Z : R+ → R+, Z(t) = tp − m, p−m > 1 to get   〈Λk(u),u〉≥ 〈Λ(u),u〉 ≥ 1 Np − m−1 ‖u‖p − m W 1,~p(.) D (Ω̃) −N. Hence, Λk is coercive (as p − m > 1). (iii) The operator Λk is of type M. Lemma 3.3. (cf [41]) Let A and B be two operators. If A is of type M and B is monotone and weakly continuous, then A + B is of type M. Now , we set 〈Au,v〉 := 〈Λ(u),v〉 and 〈Bku,v〉 := ∫ Ω Tk(b(u))vdx + ∫ Γ̃Ne Tk(ρ̃(u))vdσ. Then, for every k > 0, we have Λk = A + Bk. We now have to show that for every k > 0, Bk is monotone and weakly continuous, because it is well-known that A is of type M. For the monotonicity of Bk, we have to show that 〈Bku−Bkv,u−v〉≥ 0 for all (u,v) ∈ W 1,~p(.) D (Ω̃) ×W 1,~p(.) D (Ω̃). We have 〈Bku−Bkv,u−v〉 = ∫ Ω (Tk(b(u)) −Tk(b(v)))(u−v)dx + ∫ Γ̃Ne (Tk(ρ̃(u)) −Tk(ρ̃(v)))(u−v)dσ. From the monotonicity of b, ρ̃ and the map Tk, we conclude that 〈Bku−Bkv,u−v〉≥ 0. (3.8) We need now to prove that for each k > 0 the operator Bk is weakly continuous, that is, for all sequences (un)n∈N ⊂ W 1,~p(.) D (Ω̃) such that un ⇀ u in W 1,~p(.) D (Ω̃), we have Bkun ⇀ Bku as n →∞. For all φ ∈ W 1,~p(.)D (Ω̃), we have 〈Bkun,φ〉 := ∫ Ω Tk(b(un))φdx + ∫ Γ̃Ne Tk(ρ̃(un))φdσ. (3.9) Passing to the limit in (3.9) as n goes to ∞ and using the Lebesgue dominated convergence theorem, since un ⇀ u in W 1,~p(.) D (Ω̃); up to a subsequence, we have un → u in L 1(Ω̃) and a.e. in Ω̃. As |Tk(b(un))φ| ≤ k|φ| and φ ∈ W 1,~p(.) D (Ω̃) ↪→ L 1(Ω̃), for the first term on the right-hand side of (3.9), we obtain lim n→∞ ∫ Ω Tk(b(un))φdx = ∫ Ω Tk(b(u))φdx. (3.10) Furthermore, since un ⇀ u in W 1,~p(.) D (Ω̃); up to a subsequence, we have un → u in L 1(∂Ω̃) and a.e. on ∂Ω̃ . As |Tk(ρ̃(un))φ| ≤ k|φ| and φ ∈ W 1,~p(.) D (Ω̃) ↪→ L 1(∂Ω̃), we deduce by the Lebesgue dominated convergence theorem that lim n→∞ ∫ Γ̃Ne Tk(ρ̃(un))φdx = ∫ Γ̃Ne Tk(ρ̃(u))φdx. (3.11) CUBO 23, 1 (2021) Anisotropic Problem with Non-local Boundary Conditions and Measure Data 35 From (3.10) and (3.11) we conclude that for every k > 0, Bk(un) →Bk(u) as n →∞. The operator A is type M and as Bk is monotone and weakly continuous, thanks to Lemma 3.3, we conclude that the operator Λk is of type M. Then for any L ∈ (W 1,~p(.) D (Ω̃)) ′, there exists u�,k ∈ W 1,~p(.) D (Ω̃), such that Λk(u�,k) = L. We now consider L ∈ (W 1,~p(.)D (Ω̃)) ′ defined by L(v) = ∫ Ω vdµ� + ∫ Γ̃Ne d̃�vdσ, for v ∈ W 1,~p(.) D (Ω̃) and we obtain (3.5)� Step 2: A priori estimates Lemma 3.4. Let u�,k a solution of P�,k(ρ̃, µ̃�, d̃�). Then  |ρ̃(u�,k)| ≤ k1 := max{‖d̃�‖∞, (ρ̃� ◦ b−1)(‖µ�‖∞)} a.e. on Γ̃Ne, |b(u�,k)| ≤ k2 := max{|µ�‖∞; (b◦ρ−10 )(|Γ̃Ne|‖d̃�‖∞)} a.e. in Ω. (3.12) Proof. For any τ > 0, let us introduce the function Hτ : R → R by Hτ (s) =   0 if s < 0, s τ if 0 ≤ s ≤ τ, 1 if s > τ. In (3.5) we set ξ̃ = Hτ (u�,k −M), where M > 0 is to be fixed later. We get  ∫ Ω̃ N∑ i=1 ãi(x, ∂ ∂xi u�,k) ∂ ∂xi Hτ (u�,k −M)dx + ∫ Ω Tk(b(u�,k))Hτ (u�,k −M)dx =∫ Ω Hτ (u�,k −M)dµ� + ∫ Γ̃Ne (d̃� −Tk(ρ̃(u�,k)))Hτ (u�,k −M)dσ. (3.13) The first term in (3.13) is non-negative. Indeed,∫ Ω̃ N∑ i=1 ãi(x, ∂ ∂xi u�,k) ∂ ∂xi Hτ (u�,k −M)dx = 1 τ ∫ {0≤u�,k−M≤τ} N∑ i=1 ãi(x, ∂ ∂xi u�,k) ∂ ∂xi u�,kdx ≥ 0. From (3.13) we obtain∫ Ω Tk(b(u�,k))Hτ (u�,k −M)dx ≤ ∫ Ω Hτ (u�,k −M)dµ� + ∫ Γ̃Ne (d̃� −Tk(ρ̃(u�,k)))Hτ (u�,k −M)dσ. Then, one has  ∫ Ω (Tkb(u�,k) −Tk(b(M)))Hτ (u�,k −M)dx + ∫ Γ̃Ne (Tk(ρ̃(u�,k)) −Tk(ρ̃(M)))Hτ (u�,k −M)dx ≤∫ Ω (µ� −Tk(b(M)))Hτ (u�,k −M)dx + ∫ Γ̃Ne (d̃� −Tk(ρ̃(M)))Hτ (u�,k −M)dσ. Letting τ go to 0 in the inequality above, we get  ∫ Ω (Tk(b(u�,k)) −Tk(b(M)))+dx + ∫ Γ̃Ne (Tk(ρ̃(u�,k)) −Tk(ρ̃(M)))+dσ ≤∫ Ω (µ� −Tk(b(M)))sign+0 (uk −M)dx + ∫ Γ̃Ne (d̃� −Tk(ρ̃(M)))sign+0 (u�,k −M)dσ. 36 A. Kaboré & S. Ouaro CUBO 23, 1 (2021) As Im(b) = Im(ρ) = R, we can fix M = M0 = max{b−1(‖µ�‖∞),ρ−10 (|Γ̃Ne|‖d̃�‖∞)}. From the above inequality we obtain  ∫ Ω (Tk(b(u�,k)) −Tk(b(M0)))+dx + ∫ Γ̃Ne (Tk(ρ̃(u�,k) −Tk(ρ̃(M0)))+dσ ≤∫ Ω (µ� −Tk(‖µ�‖∞))sign+0 (u�,k −M0)dx + ∫ Γ̃Ne (d̃−Tk(‖d̃�‖∞))sign+0 (u�,k −M0)dσ. For k > k0 := max{‖µ�‖,‖d̃�‖∞}, it follows that∫ Ω (Tk(b(u�,k)) −Tk(b(M0)))+dx + ∫ Γ̃Ne (Tk(ρ̃(u�,k)) −Tk(ρ̃(M0)))+dσ ≤ 0. (3.14) From (3.14), we deduce that  Tk(ρ̃(u�,k)) ≤ Tk(ρ̃(M0)) a.e. on Γ̃Ne, Tk(b(u�,k)) ≤ Tk(b(M0)) a.e. in Ω. (3.15) From (3.15), we deduce that for every k > k1 := max{‖d̃�‖∞,‖µ�‖∞, b(M0), ρ̃(M0)}, ρ̃(u�,k) ≤ ρ̃(M0) a.e. on Γ̃Ne and b(u�,k) ≤ b(M0) a.e. in Ω. Note that with the choice of M0 and the fact that D(ρ) = D(b) = R, for every k > k1 := max{‖d̃�‖∞,‖µ�‖∞, b(M0), ρ̃(M0)}, we have  b(u�,k) ≤ max{‖µ�‖∞,b◦ρ−10 (|Γ̃Ne|‖d̃�‖∞) } a.e. in Ω, ρ̃(u�,k) ≤ max{‖d̃�‖∞, (ρ̃◦ b−1)(‖µ�‖∞)} a.e. on Γ̃Ne. (3.16) We need to show that for any k large enough,  b(u�,k) ≥−max{‖µ�‖∞,b◦ρ−10 (|Γ̃Ne|‖d̃�‖∞)} a.e. in Ω, ρ̃(u�,k) ≥−max{‖d̃�‖∞, (ρ̃◦ b−1)(‖µ�‖∞)} a.e. on Γ̃Ne. (3.17) It is easy to see that if (u�,k) is a solution of P�,k(ρ̃, µ̃�, d̃�), then (−u�,k) is a solution of P�,k(ρ̂, µ̂�, d̂�)   − N∑ i=1 ∂ ∂xi âi(x, ∂ ∂xi u�,k) + Tk(b̂(u�,k))χΩ(x) = µ̂� in Ω̃ u�,k = 0 on ΓD Tk(ρ̂(u�,k)) + N∑ i=1 âi(x, ∂ ∂xi u�,k)ηi = d̂� on Γ̃Ne, where âi(x,ξ) = −ãi(x,−ξ), ρ̂(s) = −ρ̃(−s), b̂(s) = −b(−s), µ̂� = −µ̃� and d̂ = −d̃�. Then for every k > k2 := max{‖d̃�‖∞,‖µ�‖∞, −b(−M0), −ρ̃(−M0)}, we have  −b(u�,k) ≤ max{‖µ�‖∞,b◦ρ−10 (|Γ̃Ne|‖d̃�‖∞)} a.e. in Ω, −ρ̃(u�,k) ≤ max{‖d̃�‖∞, (ρ̃◦ b−1)(‖µ�‖∞)} a.e. on Γ̃Ne, CUBO 23, 1 (2021) Anisotropic Problem with Non-local Boundary Conditions and Measure Data 37 which implies (3.17). From (3.16) and (3.17), we deduce (3.12). Step 3. Convergence Since u�,k is a solution of P�,k(ρ̃, µ̃�, d̃�), thanks to Lemma 3.4 and the fact that Ω is bounded, we have ρ̃(u�,k) ∈ L1(Γ̃Ne) and b(u�,k) ∈ L1(Ω). For k = 1 + max(k1,k2) fixed, by Lemma 3.4, one sees that problem P�(ρ̃, µ̃�, d̃�) admits at least one solution u� � Remark 3.5. Using the relation (3.12) and the fact that the functions b and ρ are non-decreasing, it follows that for k large enough, the solution of the problem P(ρ̃, µ̃�, d̃�) belongs to L ∞(Ω) ∩ L∞(Γ̃Ne) and |u�| ≤ c(b,k1) a.e. in Ω and |u�| ≤ c(ρ,k2) a.e. on Γ̃Ne. Now, we set ãi(x,ξ) = ai(x,ξ)χΩ(x) + 1 �pi(x) |ξ|pi(x)−2ξχΩ̃\Ω(x) for all (x,ξ) ∈ Ω̃ × R N and we consider the following problem. P�(ρ̃, µ̃�, d̃�)  − N∑ i=1 ∂ ∂xi ( ai ( x, ∂ ∂xi u� ) χΩ(x) + 1 �pi(x) ∣∣∣∣ ∂∂xiu� ∣∣∣∣pi(x)−2 ∂∂xiu�χΩ̃\Ω(x) ) + |u�|pM (x)−2u�χΩ = µ̃� in Ω̃ u� = 0 on ΓD ρ̃(u�) + N∑ i=1 ãi(x, ∂ ∂xi u�)ηi = d̃� on Γ̃Ne. (3.18) Thanks to Theorem 3.2, P�(ρ̃, µ̃�, d̃�) has at least one solution. So, there exists at least one measurable function u� : Ω̃ → R such that  N∑ i=1 ∫ Ω ai ( x, ∂ ∂xi u� ) ∂ ∂xi ξ̃dx + N∑ i=1 ∫ Ω̃\Ω ( 1 �pi(x) | ∂ ∂xi u�|pi(x)−2 ∂ ∂xi u�. ∂ ∂xi ξ̃ ) dx + ∫ Ω |u�|pM (x)−2u�ξ̃dx = ∫ Ω ξ̃dµ� + ∫ Γ̃Ne (d̃� − ρ̃(u�)ξ̃dσ, (3.19) where u� ∈ W 1,~p(.) D (Ω̃) and ξ̃ ∈ W 1,~p(.) D (Ω̃) ∩L ∞(Ω). Moreover u� ∈ L∞(Ω) ∩L∞(Γ̃Ne). Our aim is to prove that these approximated solutions u� tend, as � goes to 0, to a measurable function u which is an entropy solution of the problem P(ρ̃, µ̃, d̃). To start with, we establish some a priori estimates. Proposition 3.6. Let u� be a solution of the problem P�(ρ̃, µ̃�, d̃�). Then, the following statements hold. (i) ∀k > 0, N∑ i=1 ∫ Ω ∣∣∣∣ ∂∂xiTk(u�) ∣∣∣∣pi(x) dx + N∑ i=1 ∫ Ω̃\Ω ( 1 � ∣∣∣∣ ∂∂xiTk(u�) ∣∣∣∣ )pi(x) dx ≤ k(‖d̃‖L1(Γ̃Ne) + |µ|(Ω)); 38 A. Kaboré & S. Ouaro CUBO 23, 1 (2021) (ii) ∫ Ω |u�|pM (x)−1dx + ∫ Γ̃Ne |ρ̃(u�)|dx ≤ (‖d̃‖L1(Γ̃Ne) + |µ|(Ω)); (iii) ∀k > 0, N∑ i=1 ∫ Ω̃ ∣∣∣∣ ∂∂xiTk(u�) ∣∣∣∣pi(x) dx ≤ k(‖d̃‖L1(Γ̃Ne) + |µ|(Ω)). Proof. For any k > 0, we set ξ̃ = Tk(u�) in (3.19), to get  N∑ i=1 ∫ Ω ( ai ( x, ∂ ∂xi u� ) ∂ ∂xi Tk(u�) ) dx + N∑ i=1 ∫ Ω̃\Ω ( 1 �pi(x) ∣∣∣∣ ∂∂xiu� ∣∣∣∣pi(x)−2 ∂∂xiu� ∂∂xiTk(u�) ) dx∫ Ω |u�|pM (x)−2u�Tk(u�)dx = ∫ Ω Tk(u�)dµ� + ∫ Γ̃Ne (d̃� − ρ̃(u�))Tk(u�)dσ. (3.20) (i) Obviously, we have N∑ i=1 ∫ Ω̃\Ω ( 1 �pi(x) ∣∣∣∣ ∂∂xiu� ∣∣∣∣pi(x)−2 ∂∂xiu� ∂∂xiTk(u�) ) dx = N∑ i=1 ∫ Ω̃\Ω ( 1 �pi(x) ∣∣∣∣ ∂∂xiTk(u�) ∣∣∣∣pi(x) ) dx ≥ 0,∫ Γ̃Ne ρ̃(u�)Tk(u�)dσ ≥ 0 and ∫ Ω |u�|pM (x)−2u�Tk(u�)dx ≥ 0. Moreover,   ∫ Ω Tk(u�)dµ� + ∫ Γ̃Ne d̃�Tk(u�)dσ ≤ k ∫ Ω dµ� + k ∫ Γ̃Ne |d̃�|dσ ≤ k ( |µ|(Ω) + ∫ Γ̃Ne |d̃|dσ ) . (3.21) Using the inequalities above and (1.7), it follows that N∑ i=1 ∫ Ω ∣∣∣∣∂Tk(u�)∂xi ∣∣∣∣pi(x) dx ≤ k ( |µ|(Ω) + ∫ Γ̃Ne |d̃|dσ ) . (3.22) As N∑ i=1 ∫ Ω ( ai ( x, ∂ ∂xi u� ) ∂ ∂xi Tk(u�) ) dx ≥ 0, ∫ Γ̃Ne ρ̃(u�)Tk(u�)dσ ≥ 0 and∫ Ω |u�|pM (x)−2u�Tk(u�)dx ≥ 0, therefore, we get from (3.20), N∑ i=1 ∫ Ω̃\Ω ( 1 �pi(x) | ∂ ∂xi Tk(u�)|pi(x) ) dx ≤ k ( |µ|(Ω) + ∫ Γ̃Ne |d̃|dσ ) (3.23) Adding (3.22) and (3.23), we obtain (i). (ii) The first two terms in (3.20) are non-negative and using (3.21), we have from (3.20) the following ∫ Γ̃Ne ρ̃(u�)Tk(u�)dσ + ∫ Ω |u�|pM (x)−2u�Tk(u�)dx ≤ k ( |µ|(Ω) + ∫ Γ̃Ne |d̃|dσ ) . We divide the above inequality by k > 0 and let k go to zero, to get∫ Γ̃Ne ρ̃(u�)sign(u�)dσ + ∫ Ω |u�|pM (x)−2u�sign(u�)dx = ∫ Γ̃Ne |ρ̃(u�)|dσ + ∫ Ω |u�|pM (x)−1dx ≤ ( |µ|(Ω) + ∫ Γ̃Ne |d̃|dσ ) . CUBO 23, 1 (2021) Anisotropic Problem with Non-local Boundary Conditions and Measure Data 39 (iii) For all k > 0, we have N∑ i=1 ∫ Ω̃ ∣∣∣∣ ∂∂xiTk(u�) ∣∣∣∣pi(x) dx ≤ N∑ i=1 ∫ Ω ∣∣∣∣ ∂∂xiTk(u�) ∣∣∣∣pi(x) dx + N∑ i=1 ∫ Ω̃\Ω ∣∣∣∣1� ∂∂xiTk(u�) ∣∣∣∣pi(x) dx, for any 0 < � < 1. According to (i ), we deduce that N∑ i=1 ∫ Ω̃ ∣∣∣∣ ∂∂xiTk(u�) ∣∣∣∣pi(x) dx ≤ k ( |µ|(Ω) + ∫ Γ̃Ne |d̃|dσ ) . Lemma 3.7. There is a positive constant D such that meas{|u�| > k}≤ Dp − m (1 + k) kp − m−1 , ∀k > 0. Proof. Let k > 0; by using Proposition 3.6-(iii), we have N∑ i=1 ∫ Ω̃ ∣∣∣∣∂Tk(u�)∂xi ∣∣∣∣p − m(x) dx ≤ N∑ i=1 ∫  ∣∣∣∣∣∣ ∂Tk(u�) ∂xi ∣∣∣∣∣∣>1   ∣∣∣∣∂Tk(u�)∂xi ∣∣∣∣p − m(x) dx + Nmeas(Ω̃) ≤ N∑ i=1 ∫ Ω̃ ∣∣∣∣∂Tk(u�)∂xi ∣∣∣∣pi(x) dx + Nmeas(Ω̃) ≤ k ( |µ|(Ω) + ∫ Γ̃Ne |d̃|dσ ) + Nmeas(Ω̃) ≤ C′(k + 1), with C′ = max (( |µ|(Ω) + ∫ Γ̃Ne |d̃|dσ ) ; Nmeas(Ω̃) ) . We can write the above inequality as N∑ i=1 ∥∥∥∥∂Tk(u�)∂xi ∥∥∥∥p − m p − m ≤ C′(1 + k) or ‖Tk(u�)‖ W 1,p − m D (Ω̃) ≤ [C′(1 + k)] 1 p − m . By the Poincaré inequality in constant exponent, we obtain ‖Tk(u�)‖Lp−m(Ω̃) ≤ D(1 + k) 1 p − m . The above inequality implies that∫ Ω̃ |Tk(u�)|p − mdx ≤ Dp − m(1 + k), from which we obtain meas{|u�| > k}≤ Dp − m (1 + k) kp − m , since ∫ Ω̃ |Tk(u�)|p − mdx = ∫ {|u�|>k} |Tk(u�)|p − mdx + ∫ {|u�|≤k} |Tk(u�)|p − mdx, 40 A. Kaboré & S. Ouaro CUBO 23, 1 (2021) we get ∫ {|u�|>k} |Tk(u�)|p − mdx ≤ ∫ Ω̃ |Tk(u�)|p − mdx and kp − mmeas{|u�| > k}≤ ∫ Ω̃ |Tk(u�)|p − mdx ≤ Dp − m(1 + k) Lemma 3.8. There is a positive constant C such that N∑ i=1 ∫ Ω̃ (∣∣∣∣ ∂∂xiTk(u�) ∣∣∣∣p − i ) dx ≤ C(k + 1), ∀k > 0. (3.24) Proof. Let k > 0, we set Ω1 = { |u| ≤ k; ∣∣∣∣ ∂∂xiu� ∣∣∣∣ ≤ 1 } and Ω2 = { |u| ≤ k; ∣∣∣∣ ∂∂xiu� ∣∣∣∣ > 1 } ; using Proposition 3.6-(iii), we have N∑ i=1 ∫ Ω̃ (∣∣∣∣ ∂∂xiTk(u�) ∣∣∣∣p − i ) dx = N∑ i=1 ∫ Ω1 (∣∣∣∣ ∂∂xiTk(u�) ∣∣∣∣p − i ) dx + N∑ i=1 ∫ Ω2 (∣∣∣∣ ∂∂xiTk(u�) ∣∣∣∣p − i ) dx ≤ Nmeas(Ω̃) + N∑ i=1 ∫ Ω̃ (∣∣∣∣ ∂∂xiTk(u�) ∣∣∣∣pi(x) ) dx ≤ Nmeas(Ω̃) + k ( |µ|(Ω) + ‖d̃‖L1(Γ̃Ne) ) ≤ C(k + 1), with C = max { Nmeas(Ω̃); ( |µ|(Ω) + ‖d̃‖L1(Γ̃Ne) )} . Lemma 3.9. For all k > 0, there is two constants C1 and C2 such that (i) ‖u�‖Mq∗(Ω̃) ≤ C1; (ii) ∣∣∣∣ ∣∣∣∣ ∂∂xiu� ∣∣∣∣ ∣∣∣∣ Mp − i q/p (Ω̃) ≤ C2. Proof. (i) By Lemma 3.8, we have N∑ i=1 ∫ Ω̃ ∣∣∣∣ ∂∂xiTk(u�) ∣∣∣∣p − i dx ≤ C(1 + k), ∀k > 0 and i = 1, ...,N. • If k > 1, we have N∑ i=1 ∫ Ω̃ ∣∣∣∣ ∂∂xiTk(u�) ∣∣∣∣p − i dx ≤ C′k, which means Tk(u�) ∈ W 1,(p − 1 ,...,p − N )(Ω̃). Using relation (2.8), we deduce that ‖Tk(u�)‖L(p̄)∗(Ω̃ ≤ C1 N∏ i=1 ∥∥∥∥ ∂∂xiTk(u�) ∥∥∥∥ 1 N L p − i (Ω̃) . CUBO 23, 1 (2021) Anisotropic Problem with Non-local Boundary Conditions and Measure Data 41 So, ∫ Ω̃ |Tk(u�)| (p̄)∗ dx ≤ C   N∏ i=1 (∫ Ω̃ ∣∣∣∣ ∂∂xiTk(u�) ∣∣∣∣p − i dx ) 1 Np−i   (p̄)∗ ≤ C′′   N∏ i=1 (k) 1 Np−i   (p̄)∗ ≤ C′′  k N∑ i=1 1 Np−i   (p̄)∗ ≤ C′′k (p̄)∗ p̄ . Thus, ∫ {|u�|>k} |Tk(u�)| (p̄)∗ dx ≤ ∫ Ω̃ |Tk(u�)| (p̄)∗ dx ≤ C′k (p̄)∗ p̄ and so, (k)(p̄) ∗ meas{x ∈ Ω̃ : |u�| > k} ≤ C′k (p̄)∗ p̄ ; which means that λu�(k) ≤ C ′k (p̄)∗( 1 p̄ −1) = C′k−q ∗ , ∀k ≥ 1. • If 0 < k < 1, we have λu�(k) = meas { x ∈ Ω̃ : |u�| > k } ≤ meas(Ω̃) ≤ meas(Ω̃)k−q ∗ . So, λu�(k) ≤ (C ′ + meas(Ω̃))k−q ∗ = C1k −q∗. Therefore, ‖u�‖Mq∗(Ω̃) ≤ C1. 42 A. Kaboré & S. Ouaro CUBO 23, 1 (2021) (ii) • Let α ≥ 1. For all k ≥ 1, we have λ∂u� ∂xi (α) = meas ({∣∣∣∣∂u�∂xi ∣∣∣∣ > α }) = meas ({∣∣∣∣∂u�∂xi ∣∣∣∣ > α; |u�| ≤ k }) + meas ({∣∣∣∣∂u�∂xi ∣∣∣∣ > α; ; |u�| > k }) ≤ ∫  ∣∣∣∣∣∣ ∂u� ∂xi ∣∣∣∣∣∣>α;|u�|≤k   dx + λu�(k) ≤ ∫ {|u�|≤k} ( 1 α ∣∣∣∣∂u�∂xi ∣∣∣∣ )p− i dx + λu�(k) ≤ α−p − i C′k + Ck−q ∗ ≤ B ( α−p − i k + k−q ∗ ) , with B = max(C′; C). Let g : [1;∞) → R, x 7→ g(x) = x αp − i + x−q ∗ . We have g′(x) = 0 with x = ( q∗αp − ) 1 q∗ + 1 . We set k = ( q∗αp − i ) 1 q∗ + 1 ≥ 1 in the above inequality to get, λ∂u� ∂xi (α) ≤ B  α−p−i ×(q∗αp−i ) 1 q∗ + 1 + ( q∗αp − i ) −q∗ q∗ + 1   ≤ B  (q∗) 1 q∗ + 1 ×α −p− i ( 1− 1 q∗ + 1 ) + (q∗) −q∗ q∗ + 1 ×α −p−i q ∗ q∗ + 1   ≤ B  (q∗) 1 q∗ + 1 ×α −p− i   q∗ q∗ + 1   + (q∗) −q∗ q∗ + 1 ×α −p−i q ∗ q∗ + 1   ≤ Mα −p− i q∗ q∗ + 1 ≤ Mα −p− i q p̄ , where M = B × max  (q∗) 1 q∗ + 1 ; (q∗) −q∗ q∗ + 1   and as q∗ = N(p̄− 1) N − p̄ , q = N(p̄− 1) N − 1 . CUBO 23, 1 (2021) Anisotropic Problem with Non-local Boundary Conditions and Measure Data 43 So, q∗ q∗ + 1 = q∗(N − p̄) N(p̄− 1) + N − p̄ = q∗(N − p̄) Np̄− p̄ = N(p̄− 1) (N − 1)p̄ = q p̄ . • If 0 ≤ α < 1, we have. λ∂u� ∂xi (α) = meas ({ x ∈ Ω̃ : ∣∣∣∣∂u�∂xi ∣∣∣∣ > α }) ≤ meas(Ω̃)α −p− i q p̄ . Therefore, λ∂u� ∂xi (α) ≤ ( M + meas(Ω̃) ) α −p− i q p̄ , ∀ α ≥ 0. So, ∥∥∥∥∂u�∂xi ∥∥∥∥ H ≤ C2, where H = M(Ω̃) p−i q p̄ Proposition 3.10. Let u� be a solution of the problem P(ρ̃, µ̃�, d̃�). Then, (i) u� → u in measure, a.e. in Ω and a.e. on Γ̃N ; (ii) For all i = 1, ...N, ∂Tk(u�) ∂xi ⇀ ∂Tk(u) ∂xi = 0 in Lp − i (Ω̃ \ Ω). Proof. (i) By Proposition 3.6 (i), we deduce that (Tk(u�))�>0 is bounded in W 1,~p(.) D (Ω̃) ↪→ Lpm(.)(Ω̃) ↪→ Lp − m(Ω̃) (with compact embedding). Therefore, up to a subsequence, we can assume that as � → 0, (Tk(u�))�>0 converges strongly to some function σk in Lp − m(Ω̃), a.e. in Ω̃ and a.e. on Γ̃Ne. Let us see that the sequence (u�)�>0 is Cauchy in measure. Indeed, let s > 0 and define: E1 = [|u�1| > k], E2 = [|u�2| > k] and E3 = [|Tk(u�1 ) −Tk(u�2 )| > s], where k > 0 is fixed. We note that [|u�1 −u�2| > s] ⊂ E1 ∪E2 ∪E3; 44 A. Kaboré & S. Ouaro CUBO 23, 1 (2021) hence, meas([|u�1 −u�2| > s]) ≤ 3∑ i=1 meas(Ei). (3.25) Let θ > 0, using Lemma 3.7, we choose k = k(θ) such that meas(E1) ≤ θ 3 and meas(E2) ≤ θ 3 . (3.26) Since (Tk(u�))�>0 converges strongly in L p−m(Ω̃), then, it is a Cauchy sequence in Lp − m(Ω̃). Thus, meas(E3) ≤ 1 sp − m ∫ Ω |Tk(u�1 ) −Tk(u�2 )| p−mdx ≤ θ 3 , (3.27) for all �1,�2 ≥ n0(s,θ). Finally, from (3.25), (3.26) and (3.27), we obtain meas([|u�1 −u�2| > s]) ≤ θ for all �1,�2 ≥ n0(s,θ); (3.28) which means that the sequence (u�)�>0 is Cauchy in measure, so u� → u in measure and up to a subsequence, we have u� → u a.e. in Ω̃. Hence, σk = Tk(u) a.e. in Ω̃ and so, u ∈T 1,~p(.) D (Ω). (ii) According to the proof of (i), we have Tk(u�) ⇀ Tk(u) in W 1,~p(.) D (Ω̃) ↪→ W 1,~p− D (Ω̃) which implies on one hand that for all i = 1, ...N, ∂Tk(u�) ∂xi ⇀ ∂Tk(u) ∂xi in Lpi(.)(Ω̃) and on the other hand that for all i = 1, ...N, ∂Tk(u�) ∂xi ⇀ ∂Tk(u) ∂xi in Lpi(.)(Ω̃) and then for all i = 1, ...N, ∂Tk(u�) ∂xi ⇀ ∂Tk(u) ∂xi in Lp − i (Ω̃ \ Ω). Let i = 1, ...,N, by Proposition 3.6-(i), we can assert that ( 1 � ∂Tk(u�) ∂xi ) �>0 is bounded in Lp − i (Ω̃ \ Ω). Indeed, let k > 0, we set Ω1 = { x ∈ Ω̃ \ Ω; |u(x)| ≤ k; ∣∣∣∣ ∂∂xiu�(x) ∣∣∣∣ ≤ � } and Ω2 = { x ∈ Ω̃ \ Ω; |u| ≤ k; ∣∣∣∣ ∂∂xiu�(x) ∣∣∣∣ > � } ; using Proposition 3.6-(i), we have N∑ i=1 ∫ Ω̃\Ω ( 1 � ∣∣∣∣∂Tk(u�)∂xi ∣∣∣∣p − i ) dx = N∑ i=1 ∫ Ω1 ( 1 � ∣∣∣∣∂Tk(u�)∂xi ∣∣∣∣p − i ) dx + N∑ i=1 ∫ Ω2 ( 1 � ∣∣∣∣ ∂∂xiTk(u�) ∣∣∣∣p − i ) dx ≤ Nmeas(Ω̃ \ Ω) + N∑ i=1 ∫ Ω̃\Ω ( 1 � ∣∣∣∣ ∂∂xiTk(u�) ∣∣∣∣pi(x) ) dx ≤ Nmeas(Ω̃ \ Ω) + k ( |µ|(Ω) + ‖d̃‖L1(Γ̃Ne) ) ≤ C′(k + 1), with C′ = max { Nmeas(Ω̃ \ Ω); ( |µ|(Ω) + ‖d̃‖L1(Γ̃Ne) )} . To end, we have ∫ Ω̃\Ω ( 1 � ∣∣∣∣∂Tk(u�)∂xi ∣∣∣∣p − i ) dx ≤ N∑ i=1 ∫ Ω̃\Ω ( 1 � ∣∣∣∣∂Tk(u�)∂xi ∣∣∣∣p − i ) dx, for anyi = 1, . . . ,N. CUBO 23, 1 (2021) Anisotropic Problem with Non-local Boundary Conditions and Measure Data 45 Therefore, there exists Θk ∈ Lp − i (Ω̃ \ Ω) such that 1 � ∂Tk(u�) ∂xi ⇀ Θk in L p − i (Ω̃ \ Ω) as � → 0. For any ψ ∈ L(p ′ i) − (Ω̃ \ Ω), we have∫ Ω̃\Ω ∂Tk(u�) ∂xi ψdx = ∫ Ω̃\Ω ( 1 � ∂Tk(u�) ∂xi − Θk ) (�ψ)dx + � ∫ Ω̃\Ω Θkψdx. (3.29) As (�ψ)�>0 converges strongly to zero in L (p′i) − (Ω̃\Ω), we pass to the limit as � → 0 in (3.29), to get ∂Tk(u�) ∂xi ⇀ 0 in Lp − i (Ω̃ \ Ω). Hence, one has ∂Tk(u�) ∂xi ⇀ ∂Tk(u) ∂xi = 0 in Lp − i (Ω̃ \ Ω), for any i = 1, ...,N. Lemma 3.11. b(u) ∈ L1(Ω) and ρ̃(u) ∈ L1(Γ̃Ne). Proof. Having in mind that by Proposition 3.6-(ii),∫ Ω |b(u�)|dx + ∫ Γ̃Ne |ρ̃(u�)|dσ ≤ (|µ|(Ω) + ‖d̃‖L1(Γ̃Ne)), we deduce that ∫ Ω |b(u�)|dx ≤ (|µ|(Ω) + ‖d̃‖L1(Γ̃Ne)) (3.30) and ∫ Γ̃Ne |ρ̃(u�)|dσ ≤ (|µ|(Ω) + ‖d̃‖L1(Γ̃Ne)). (3.31) By Fatou’s lemma, the continuity of b, ρ̃ and using Proposition 3.10, we have lim inf �→0 ∫ Ω |b(u�)|dx ≥ ∫ Ω |b(u)|dx (3.32) and lim inf �→0 ∫ Γ̃Ne |ρ̃(u�)|dσ ≥ ∫ Γ̃Ne |ρ̃(u)|dσ. (3.33) Using (3.30)-(3.33), we deduce that∫ Ω |b(u)|dx ≤ (|µ|(Ω) + ‖d̃‖L1(Γ̃Ne)) and ∫ Γ̃Ne |ρ̃(u)|dσ ≤ (|µ|(Ω) + ‖d̃‖L1(Γ̃Ne)). Therefore, b(u) ∈ L1(Ω) and ρ̃(u) ∈ L1(Γ̃Ne). 46 A. Kaboré & S. Ouaro CUBO 23, 1 (2021) Lemma 3.12. Assume (1.4)-(1.8) hold and u� be a weak solution of the problem P(ρ,µ̃�, d̃�). Then, (i) ∂ ∂xi u� converges in measure to ∂ ∂xi u . (ii) ai ( x, ∂Tk(u�) ∂xi ) → ai(x, ∂Tk(u) ∂xi ) strongly in L1(Ω) and weakly in Lp ′ i(.)(Ω), for all i=1,...,N. In order to give the proof of Lemma 3.12, we need the following lemmas. Lemma 3.13 (Cf [6]). Let u ∈ T 1,~p(.)(Ω). Then, there exists a unique measurable function νi : Ω → R such that νiχ{|u| 0 and i = 1, ...,N; where χA denotes the characteristic function of a measurable set A. The functions νi are denoted ∂ ∂xi u. Moreover, if u belongs to W 1,~p(.)(Ω), then νi ∈ Lpi(.)(Ω) and coincides with the standard distributional gradient of u i.e. νi = ∂ ∂xi u. Lemma 3.14 (Cf [37], lemma 5.4). Let (vn)n∈N be a sequence of measurable functions. If vn converges in measure to v and is uniformly bounded in Lp(.)(Ω) for some 1 << p(.) ∈ L∞(Ω), then vn → v strongly in L1(Ω). The third technical lemma is a standard fact in measure theory (Cf [16]). Lemma 3.15. Let (X,M,µ) be a measurable space such that µ(X) < ∞. Consider a measurable function γ : X → [0;∞] such that µ({x ∈ X : γ(x) = 0}) = 0. Then, for every � > 0, there exists δ such that µ(A) < �, for all A ∈M with ∫ A γdx < δ. Proof of Lemma 3.12. (i) We claim that ( ∂ ∂xi u� ) �∈N is Cauchy in measure. Indeed, let s > 0, consider An,m := {∣∣∣∣ ∂∂xiun ∣∣∣∣ > h } ∪ {∣∣∣∣ ∂∂xium ∣∣∣∣ > h } , Bn,m := {|un −um| > k} and Cn,m := {∣∣∣∣ ∂∂xiun ∣∣∣∣ ≤ h, ∣∣∣∣ ∂∂xium ∣∣∣∣ ≤ h, |un −um| ≤ k, ∣∣∣∣ ∂∂xiun − ∂∂xium > s ∣∣∣∣ } , where h and k will be chosen later. One has{∣∣∣∣ ∂∂xiun − ∂∂xium ∣∣∣∣ > s } ⊂ An,m ∪Bn,m ∪Cn,m. (3.34) Let ϑ > 0. By Lemma 3.9, we can choose h = h(ϑ) large enough such that meas(An,m) ≤ ϑ 3 for all n,m ≥ 0. On the other hand, by Proposition 3.10, we have that meas(Bn,m) ≤ ϑ 3 CUBO 23, 1 (2021) Anisotropic Problem with Non-local Boundary Conditions and Measure Data 47 for all n,m ≥ n0(k,ϑ). Moreover, by assumption (H3), there exists a real valued function γ : Ω → [0,∞] such that meas{x ∈ Ω : γ(x) = 0} = 0 and (ai(x,ξ) −ai(x,ξ′)).(ξ − ξ′) ≥ γ(x), (3.35) for all i = 1, ...,N, |ξ|, |ξ′| ≤ h, |ξ − ξ′| ≥ s, for a.e. x ∈ Ω. Indeed, let’s set K = {(ξ,η) ∈ R×R : |ξ| ≤ h, |η| ≤ h, |ξ −η| ≥ s}. We have K ⊂ B(0,h) ×B(0,h) and so K is a compact set because it is closed in a compact set. For all x ∈ Ω and for all i = 1, ...,N, let us define ψ : K → [0;∞] such that ψ(ξ,η) = (ai(x,ξ) −ai(x,η)).(ξ −η). As for a.e. x ∈ Ω, ai(x,.) is continuous on R, ψ is continuous on the compact K, by Weier- strass theorem, there exists (ξ0,η0) ∈ K such that ∀(ξ,η) ∈ K, ψ(ξ,η) ≥ ψ(ξ0,η0). Now let us define γ on Ω as follows. γ(x) = ψi(ξ0,η0) = (ai(x,ξ0) −ai(x,η)).(ξ −η0). Since s > 0, the function γ is such that meas ({x ∈ Ω : γ(x) = 0}) = 0. Let δ = δ(�) be given by Lemma 3.15, replacing � and A by � 3 and Cn,m respectively. Taking respectively ξ̃ = Tk(un −um) and ξ̃ = Tk(um −un) for the weak solutions un and um in (3.19) and after adding the two relations, we have  N∑ i=1 ∫ {|un−um| s }) ≤ ϑ, (3.36) for all n,m ≥ n0(s,ϑ), and then the claim is proved. As consequence, ( ∂ ∂xi u� ) �∈N converges in measure to some measurable function νi. In order to end the proof of Lemma 3.12, we need the following lemma. Lemma 3.16. (a) For a.e. k ∈ R, ∂ ∂xi Tk(u�) converges in measure to νiχ{|u| 0, { |χ{|u�| δ } ⊂ { |χ{|u�| δ }) ≤ meas ({|u| = k}) + meas ({u� < k < u}) +meas ({u < k < u�}) +meas ({u� < −k < u}) +meas ({u < −k < u�}) . (3.37) Note that meas ({|u| = k}) ≤ meas ({k −h < u < k + h}) + meas ({−k −h < u < −k + h}) → 0 as h → 0 for a.e. k > 0, since u is fixed function. Next, meas ({u� < k < u}) ≤ meas ({k < u < k + h}) + meas ({|u� −u| > h}) , for all CUBO 23, 1 (2021) Anisotropic Problem with Non-local Boundary Conditions and Measure Data 49 h > 0. Due to Proposition 3.10, we have for all fixed h > 0, meas ({|u� −u| > h}) → 0 as � → 0. Since meas ({k < u < k + h}) → 0 as h → 0, for all ϑ > 0, one can find N such that for all n > N, meas ({|u| = k}) < ϑ 2 + ϑ 2 = ϑ by choosing h and then N. Each of the other terms on the right-hand side of (3.37) can be treated in the same way as for meas ({u� < k < u}) . Thus, meas ({ |χ{|u�| δ} }) → 0 as � → 0. Finally, since ∂ ∂xi Tk(u�) = ∂ ∂xi u�χ{|u�| 0, i = 1, ...,N, ai ( x, ∂ ∂xi Tk(u�) ) con- verges to ai ( x, ∂ ∂xi Tk(u) ) in L1(Ω) strongly. Indeed, let s,k > 0, consider E4 = {∣∣∣∣∂un∂xi − ∂um∂xi ∣∣∣∣ > s, |un| ≤ k, |um| ≤ k } , E5 = {∣∣∣∣∂um∂xi ∣∣∣∣ > s, |un| > k, |um| ≤ k } , E6 ={∣∣∣∣∂un∂xi ∣∣∣∣ > s, |un| ≤ k, |um| > k } . We have {∣∣∣∣∂Tk(un)∂xi − ∂Tk(um)∂xi ∣∣∣∣ > s } ⊂ E4 ∪E5 ∪E6. (3.38) 50 A. Kaboré & S. Ouaro CUBO 23, 1 (2021) ∀ϑ > 0 , by Lemma 3.7, there exists k(ϑ) such that meas(E5) ≤ ϑ 3 and meas(E6) ≤ ϑ 3 . (3.39) Using (3.36)-(3.39), we get meas ({∣∣∣∣ ∂∂xiTk(un) − ∂∂xiTk(um) ∣∣∣∣ > s }) ≤ ϑ, (3.40) for all n,m ≥ n1(s,ϑ). Therefore, ∂Tk(u�) ∂xi converges in measure to ∂Tk(u) ∂xi . Using lem- mas 3.8 and 3.14, we deduce that ∂Tk(u�) ∂xi converges to ∂Tk(u) ∂xi in L1(Ω). So, after pass- ing to a suitable subsequence of ( ∂Tk(u�) ∂xi ) �>0 , we can assume that ∂Tk(u�) ∂xi converges to ∂Tk(u) ∂xi a.e. in Ω. By the continuity of ai(x,.), we deduce that ai ( x, ∂Tk(u�) ∂xi ) converges to ai ( x, ∂Tk(u) ∂xi ) a.e. in Ω. As Ω is bounded, this convergence is in measure. Using lem- mas 3.14 and 3.16, we deduce that for all k > 0, i = 1, ...,N, ai ( x, ∂ ∂xi Tk(u�) ) converges to ai ( x, ∂ ∂xi Tk(u) ) in L1(Ω) strongly and ai ( x, ∂ ∂xi Tk(u�) ) converges to χk ∈ Lp ′ i(.)(Ω) weakly in Lp ′ i(.)(Ω). Since each of the convergences implies the weak L1-convergence, χk can be identified with ai ( x, ∂ ∂xi Tk(u) ) ; thus, ai ( x, ∂ ∂xi Tk(u) ) ∈ Lp ′ i(.)(Ω) By using Lebesgue generalized convergence theorem and above results, we deduce the following result. Proposition 3.17. For any k > 0 and any i = 1, ...,N , as � tends to 0, we have (i) ∂Tk(u�) ∂xi → ∂Tk(u) ∂xi a.e. in Ω, (ii) ai ( x, ∂Tk(u�) ∂xi ) ∂Tk(u�) ∂xi → ai ( x, ∂Tk(u) ∂xi ) ∂Tk(u) ∂xi a.e. in Ω and strongly in L1(Ω), (iii) ∂Tk(u�) ∂xi → ∂Tk(u) ∂xi strongly in Lpi(x)(Ω). 4 Existence and uniqueness of solution to P(ρ,µ,d) We are now able to prove Theorem 2.6. Proof of Theorem 2.6 Thanks to the Proposition 3.10 and as ∀k > 0, ∀i = 1, ...,N, ∂Tk(u) ∂xi = 0 in Lp − i (Ω̃ \ Ω), then, ∀k > 0, Tk(u) = constant a.e. on Ω̃ \ Ω. Hence, we conclude that u ∈T 1,~p(.) Ne (Ω). CUBO 23, 1 (2021) Anisotropic Problem with Non-local Boundary Conditions and Measure Data 51 We already state that b(u) ∈ L1(Ω). To show that u is an entropy solution of P(ρ,µ,d), we only have to prove the inequality in (2.9). Let ϕ ∈ W 1,~p(.)D (Ω) ∩L ∞(Ω). We consider the function ϕ1 ∈ W 1,~p(.) D (Ω̃) ∩L ∞(Ω) such that ϕ1 = ϕχΩ + ϕNχΩ̃\Ω. We set ξ̃ = Tk(u� −ϕ1) in (3.19) to get  N∑ i=1 ∫ Ω ( ai ( x, ∂ ∂xi u� ) . ∂ ∂xi Tk(u� −ϕ) ) dx + N∑ i=1 ∫ Ω̃\Ω ( 1 �pi(x) ∣∣∣∣ ∂∂xiu� ∣∣∣∣pi(x)−2 ∂∂xiu�. ∂∂xiTk(u� −ϕN ) ) dx∫ Ω b(u�)Tk(u� −ϕ)dx = ∫ Ω Tk(u� −ϕ)dµ� + ∫ Γ̃Ne (d̃� − ρ̃(u�))Tk(u� −ϕN )dσ. (4.1) The following convergence result hold true. Lemma 4.1. For any k > 0, for all i = 1, ...,N, as � → 0, ∂ ∂xi Tk(u� −ϕ) → ∂ ∂xi Tk(u−ϕ) strongly in Lpi(.)(Ω). Proof. Let k > 0, i = 1, ...,N. We have∫ Ω ∣∣∣∣ ∂∂xiTk(u� −ϕ) − ∂∂xiTk(u−ϕ) ∣∣∣∣pi(x) dx = ∫ Ω∩[|u�−ϕ|≤k,|u−ϕ|≤k] ∣∣∣∣ ∂∂xiu� − ∂∂xiu ∣∣∣∣pi(x) dx ≤ ∫ Ω∩[|u�|≤l,|u|≤l] ∣∣∣∣∂u�∂xi − ∂u∂xi ∣∣∣∣pi(x) dx, with l = k + ‖ϕ‖∞ = ∫ Ω ∣∣∣∣ ∂∂xiTl(u�) − ∂∂xiTl(u) ∣∣∣∣pi(x) dx → 0 as � → 0 by Proposition 3.17 − (iii). We need to pass to the limit in (4.1) as � → 0. We have N∑ i=1 ∫ Ω ( ai ( x, ∂ ∂xi u� ) ∂ ∂xi Tk(u� −ϕ) ) dx = N∑ i=1 ∫ Ω ( ai ( x, ∂Tl(u�) ∂xi ) ∂ ∂xi Tk(u� −ϕ) ) dx, with l = k + ‖ϕ‖∞, then, by Lemma 3.12- (ii) and Lemma 4.1, we have lim �→0 N∑ i=1 ∫ Ω ( ai ( x, ∂Tl(u�) ∂xi ) ∂ ∂xi Tk(u� −ϕ) ) dx = N∑ i=1 ∫ Ω ( ai ( x, ∂Tl(u) ∂xi ) ∂ ∂xi Tk(u−ϕ) ) dx; that is lim �→0 N∑ i=1 ∫ Ω ( ai ( x, ∂ ∂xi u� ) ∂ ∂xi Tk(u� −ϕ) ) dx = N∑ i=1 ∫ Ω ( ai ( x, ∂Tl(u) ∂xi ) ∂ ∂xi Tk(u−ϕ) ) dx. (4.2) 52 A. Kaboré & S. Ouaro CUBO 23, 1 (2021) For the second term in the left hand side of (4.1), we have lim sup �→0 N∑ i=1 ∫ Ω̃\Ω ( 1 �pi(x) | ∂ ∂xi u�|pi(x)−2 ∂ ∂xi u� ∂ ∂xi Tk(u� −ϕN ) ) dx ≥ 0. (4.3) Indeed   N∑ i=1 ∫ Ω̃\Ω ( 1 �pi(x) ∣∣∣∣ ∂∂xiu� ∣∣∣∣pi(x)−2 ∂∂xiu� ∂∂xiTk(u� −ϕN ) ) dx = N∑ i=1 ∫ Ω̃\Ω∩[|u�−ϕ|≤k] ( 1 �pi(x) | ∂ ∂xi u�|pi(x)−2 ∂ ∂xi u� ∂ ∂xi (u� −ϕN ) ) dx = N∑ i=1 ∫ Ω̃\Ω∩[|u�−ϕ|≤k] ( 1 �pi(x) | ∂ ∂xi u�|pi(x) ) dx ≥ 0. Hence, we get (4.3). Let us examine the last term in the left hand side of (4.1). we have ∫ Ω b(u�)Tk(u� −ϕ)dx = ∫ Ω (b(u�) − b(ϕ))Tk(u� −ϕ)dx + ∫ Ω b(ϕ)Tk(u� −ϕ)dx. As b is non-decreasing, (b(u�) − b(ϕ))Tk(u� −ϕ) ≥ 0 a.e. in Ω and we get by Fatou’s lemma that lim inf �→0 ∫ Ω (b(u�) − b(ϕ))Tk(u� −ϕ)dx ≥ ∫ Ω (b(u) − b(ϕ))Tk(u−ϕ)dx. As ϕ ∈ L∞(Ω), we obtain b(ϕ) ∈ L∞(Ω) and so b(ϕ) ∈ L1(Ω) (as Ω is bounded) and by Lebesgue dominated convergence theorem, we deduce that lim �→0 ∫ Ω b(ϕ)Tk(u� −ϕ)dx = ∫ Ω b(ϕ)Tk(u−ϕ)dx. Consequently, lim sup �→0 ∫ Ω b(u�)Tk(u� −ϕ)dx ≥ ∫ Ω b(u)Tk(u−ϕ)dx. (4.4) As f� → f strongly in L1(Ω) and Tk(u�−v) ⇀∗ Tk(u−v) in L∞(Ω), using the Lebesgue generalized convergence theorem we have  lim �→0 ∫ Ω f�Tk(u� −ϕ)dx = ∫ Ω Tk(u−ϕ)dx, lim �→0 ∫ Γ̃Ne d̃�Tk(u� −ϕN )dσ = ∫ Ω d̃Tk(u−ϕN )dσ. (4.5) Since ∇Tk(u� −ϕ) ⇀ ∇Tk(u−ϕ) in (Lpm(.)(Ω))N and F ∈ (Lp ′ m(.)(Ω))N, lim �→0 ∫ Ω F.∇Tk(u� −ϕ)dx = ∫ Ω F.∇Tk(u−ϕ)dx. (4.6) We know that ∀k > 0, Tk(u) = constant on Ω̃ \ Ω, then, it yields that u = constant on Ω̃ \ Ω. So, one has lim �→0 ∫ Γ̃Ne d̃�Tk(u� −ϕ)dx = dTk(uN −ϕN ). (4.7) CUBO 23, 1 (2021) Anisotropic Problem with Non-local Boundary Conditions and Measure Data 53 At last, we have ∫ Γ̃Ne ρ̃(u�)Tk(u� −ϕN )dσ = ∫ Γ̃Ne (ρ̃(u�) − ρ̃(ϕN ))Tk(u� −ϕN )dσ + ∫ Γ̃Ne ρ̃(ϕN )Tk(u� −ϕN )dσ. As ρ̃ is non-decreasing, (ρ̃(u�) − ρ̃(ϕN ))Tk(u� −ϕN ) ≥ 0 a.e. in Γ̃Ne and we get by Fatou’s lemma that lim inf �→0 ∫ Γ̃Ne (ρ̃(u�) − ρ̃(ϕN ))Tk(u� −ϕN )dσ ≥ ∫ Γ̃Ne (ρ̃(uN ) − ρ̃(ϕN ))Tk(uN −ϕN )dσ = (ρ(uN ) −ρ(ϕN ))Tk(uN −ϕN ). As ϕN ∈ L∞(Γ̃Ne), we obtain ρ̃(ϕN ) ∈ L∞(Γ̃Ne) and so ρ̃(ϕN ) ∈ L1(Γ̃Ne) (as Γ̃Ne is bounded) and by the Lebesgue dominated convergence theorem, we deduce that lim �→0 ∫ Γ̃Ne ρ̃(ϕN )Tk(u� −ϕN )dσ = ∫ Γ̃Ne ρ̃(ϕN )Tk(uN −ϕN )dσ = ρ(ϕN )Tk(uN −ϕN ). Hence, lim sup �→0 ∫ Γ̃Ne ρ̃(u�)Tk(u� −ϕN )dσ ≥ ρ(ϕN )Tk(uN −ϕN ). (4.8) Passing to the limit as � → 0 in (4.1) and using (4.2)-(4.8), we see that u is an entropy solution of P(ρ,µ,d). We now prove the uniqueness part of Theorem 2.6. Let u and v be two entropy solutions of P(ρ,µ,d). Let h > 0. For u, we take ξ = Th(v) as test function and for v, we take ξ = Th(u) as test function in (2.9), to get for any k > 0 with k < h,   ∫ Ω ( N∑ i=1 ai ( x, ∂ ∂xi u ) ∂ ∂xi Tk(u−Th(v)) ) dx + ∫ Ω b(u)Tk(u−Th(v))dx ≤∫ Ω fTk(u−Th(v))dx + ∫ Ω F.∇Tk(u−Th(v))dx + (d−ρ(uNe))Tk(uNe −Th(v)) (4.9) and   ∫ Ω ( N∑ i=1 ai ( x, ∂ ∂xi v ) ∂ ∂xi Tk(v −Th(u)) ) dx + ∫ Ω b(v)Tk(v −Th(u))dx ≤∫ Ω fTk(v −Th(u))dx + ∫ Ω F.∇Tk(v −Th(u))dx + (d−ρ(vNe))Tk(vNe −Th(u)). (4.10) 54 A. Kaboré & S. Ouaro CUBO 23, 1 (2021) By adding (4.9) and (4.10), we obtain  ∫ Ω ( N∑ i=1 ai ( x, ∂ ∂xi u ) ∂ ∂xi Tk(u−Th(v)) ) dx + ∫ Ω ( N∑ i=1 ai ( x, ∂ ∂xi v ) ∂ ∂xi Tk(v −Th(u)) ) dx := A(h,k) + ∫ Ω b(u)Tk(u−Th(v))dx + ∫ Ω b(v)Tk(v −Th(u))dx := B(h,k) +ρ(uNe)Tk(uNe −Th(v)) + ρ(vNe)Tk(vNe −Th(u)) := C(h,k) ≤ ∫ Ω fTk(u−Th(v))dx + ∫ Ω fTk(v −Th(u))dx := D(h,k) + ∫ Ω F.∇Tk(u−Th(v))dx + ∫ Ω F.∇Tk(v −Th(u))dx := T(h,k) +dTk(uNe −Th(v)) + dTk(vNe −Th(u)) := E(h,k). (4.11) Let us introduce the following subsets of Ω. A0 := [|u−v| < k, |u| < h, |v| < h] A1 := [|u−Th(v)| < k, |v| ≥ h] A′1 := [|v −Th(u)| < k, |u| ≥ h] A2 := [|u−Th(v)| < k, |u| ≥ h, |v| < h] A′2 := [|v −Th(u)| < k, |v| ≥ h, |u| < h]. We have the following assertion (see [22] for the proof). Assertion 4.2. If u is an entropy solution of P(ρ,µ,d), then A2 ⊂ Fh,k and A1 ⊂ Fh−k,2k, where Fh,k = {h ≤ |u| < h + k,h > 0,k > 0}. Assertion 4.3. Let u be an entropy solution of P(ρ,µ,d). On A2 (and on A1) we have according to Hölder inequality. (1) ∫ A2 F.∇udx ≤ (∫ A2 |F|(p ′ m) − dx ) 1 (p′m) − (∫ A2 |∇u|p − m ) 1 p − m dx, (4.12) with lim h→∞ (∫ A2 |F|(p ′ m) − dx ) 1 (p′m) − (∫ A2 |∇u|p − mdx ) 1 p − m = 0. (2) ∫ A1 F.∇udx ≤ (∫ A1 |F |(p ′ m) − dx ) 1 (p′m) − (∫ A1 |∇u|p − mdx ) 1 p − m , (4.13) with lim h→∞ (∫ A1 |F|(p ′ m) − dx ) 1 (p′m) − (∫ A1 |∇u|p − mdx ) 1 p − m = 0. CUBO 23, 1 (2021) Anisotropic Problem with Non-local Boundary Conditions and Measure Data 55 Proof. (1) lim h→∞ (∫ A2 |F |(p ′ m) − dx ) 1 (p′m) − = 0 (see [22]). Now, it remains to prove that (∫ A2 |∇u|p − mdx ) 1 p − m is bounded with respect to h. We make the following notations: I = { i ∈{1, ...,N} : {∣∣∣∣ ∂∂xiu ∣∣∣∣ } ≤ 1 } and J = { i ∈{1, ...,N} : {∣∣∣∣ ∂∂xiu ∣∣∣∣ } > 1 } . We have N∑ i=1 ∫ Fh,k ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x) dx = ∑ i∈I (∫ Fh,k ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x) dx ) + ∑ i∈J (∫ Fh,k ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x) dx ) ≥ ∑ i∈J (∫ Fh,k ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x) dx ) ≥ ∑ i∈J (∫ Fh,k ∣∣∣∣ ∂∂xiu ∣∣∣∣p − m dx ) ≥ N∑ i=1 (∫ Fh,k ∣∣∣∣ ∂∂xiu ∣∣∣∣p − m dx ) − ∑ i∈I (∫ Fh,k ∣∣∣∣ ∂∂xiu ∣∣∣∣p − m dx ) ≥ N∑ i=1 (∫ Fh,k ∣∣∣∣ ∂∂xiu ∣∣∣∣p − m ) −Nmeas(Ω) ≥ N∑ i=1 ∣∣∣∣ ∣∣∣∣ ∂∂xiu ∣∣∣∣ ∣∣∣∣p − m (Lp − m(Fh,k))N −Nmeas(Ω) ≥ C‖∇u‖p − m (Lp − m(Fh,k))N −Nmeas(Ω). We deduce that N∑ i=1 ∫ Fh,k ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x) dx ≥ C ∫ Fh,k |∇u|p − mdx−Nmeas(Ω). (4.14) Choosing Th(u) as test function in (2.9), we get  ∫ Ω ( N∑ i=1 ai ( x, ∂ ∂xi u ) ) ∂ ∂xi Tk(u−Th(u)) ) dx + ∫ Ω |u|pM (x)−2uTk(u−Th(u))dx ≤∫ Ω fTk(u−Th(u))dx + ∫ Ω F.∇Tk(u−Th(u))dx + (d−ρ(uNe))Tk(uNe −Th(uNe)). (4.15) According to the fact that ∇Tk(u−Th(u)) = ∇u on {h ≤ |u| < h + k} and zero elsewhere,∫ Ω |u|pM (x)−2uTk(u−Th(u))dx ≥ 0 and ρ(uNe)Tk(uNe−Th(uNe)) ≥ 0, we deduce from (4.15) that   ∫ Fh,k ( N∑ i=1 ai ( x, ∂ ∂xi u ) ∂ ∂xi Tk(u−Th(u)) ) dx ≤ k ∫ |u|≥h |f|dx + ∫ Fh,k ∣∣∣∣∣∣∣ ( 2 Cp−m ) 1 p−m F ∣∣∣∣∣∣∣ ∣∣∣∣∣∣∣ ( Cp−m 2 ) 1 p−m ∇u ∣∣∣∣∣∣∣dx + k|d|. (4.16) 56 A. Kaboré & S. Ouaro CUBO 23, 1 (2021) Using (1.7) (in the left hand side of (4.16)), Young inequality (in the right hand side of(4.16)) and setting c = ( 2 Cp−m )(p′m)− p−m p − m − 1 p−m , we obtain   N∑ i=1 ∫ Fh,k ∣∣∣∣ ∂∂xiu ∣∣∣∣pi(x) dx ≤ k ∫ |u|≥h |f|dx + c ∫ Fh,k |F|(p ′)−mdx + C 2 ∫ Fh,k |∇u|p − mdx + k|d|. (4.17) From (4.14) and (4.17), we deduce  C ∫ Fh,k |∇u|p − mdx ≤ k ∫ |u|≥h |f|dx + c ∫ Fh,k |F|(p ′)−mdx + C 2 ∫ Fh,k |∇u|p − mdx + k|d| + Nmeas(Ω). Therefore,   C 2 ∫ Fh,k |∇u|p − mdx ≤ k ∫ {|u|≥h} |f|dx + c ∫ Fh,k |F|(p ′)−mdx + k|d| + Nmeas(Ω). (4.18) Since A2 ⊂ Fh,k , we deduce from (4.18) that ∫ A2 |∇u|p − mdx is bounded. (2) lim h→∞ (∫ A1 |F|(p ′ m) − dx ) 1 (p′m) − = 0 (see [22]). Now, it remains to prove that (∫ A1 |∇u|p − mdx ) 1 p − m is bounded with respect to h. Since A1 ⊂ Fh−k,2k , we deduce from (4.18) that ∫ A2 |∇u|p − mdx is bounded. Remark 4.4. Similarly, we prove that if v is an entropy solution of P(ρ,f,d), then lim h→∞ ∫ A′2 F.∇vdx ≤ 0 and lim h→∞ ∫ A′1 F.∇vdx ≤ 0. Now, we have  A(h,k) = ∫ A0 ( N∑ i=1 ( ai ( x, ∂ ∂xi u ) −ai ( x, ∂ ∂xi v )) ∂ ∂xi (u−v) ) dx := I0(h,k) + ∫ A1 ( N∑ i=1 ai ( x, ∂ ∂xi u ) ∂ ∂xi u ) dx + ∫ A′1 ( N∑ i=1 ai ( x, ∂ ∂xi v ) ∂ ∂xi v ) dx := I1(h,k) + ∫ A2 ( N∑ i=1 ai ( x, ∂ ∂xi u ) ∂ ∂xi (u−v) ) dx + ∫ A′2 ( N∑ i=1 ai ( x, ∂ ∂xi v ) ∂ ∂xi (v −u) ) dx := I2(h,k). CUBO 23, 1 (2021) Anisotropic Problem with Non-local Boundary Conditions and Measure Data 57 The term I1(h,k) is non-negative since each term in I1(h,k) is non-negative. For the term I2(h,k), as I2(h,k) + ∫ A2 ( N∑ i=1 ai ( x, ∂ ∂xi u ) ∂ ∂xi v ) dx + ∫ A′2 ( N∑ i=1 ai ( x, ∂ ∂xi v ) ∂ ∂xi u ) dx = I1(h,k), so, I2(h,k) ≥− (∫ A2 ( N∑ i=1 ai ( x, ∂ ∂xi u ) ∂ ∂xi v ) dx + ∫ A′2 ( N∑ i=1 ai ( x, ∂ ∂xi v ) ∂ ∂xi u ) dx ) . Let us show that − (∫ A2 ( N∑ i=1 ai ( x, ∂ ∂xi u ) ∂ ∂xi v ) dx ) goes to 0 as h →∞. We have   ∣∣∣∣∣ ∫ A2 ( N∑ i=1 ai ( x, ∂ ∂xi u ) ∂ ∂xi (v) ) dx ∣∣∣∣∣ ≤ C N∑ i=1 ( |ji|p′ i (.) + ∣∣∣∣ ∂u∂xi ∣∣∣∣pi(x)−1 Lpi(.)({h<|u|≤h+k}) )∣∣∣∣ ∂v∂xi ∣∣∣∣ Lpi(.)({h−k<|v|≤h}) . For all i = 1, ...N, the quantity ( |ji|p′ i (.) + ∣∣∣∣ ∂u∂xi ∣∣∣∣pi(x)−1 Lpi(.)({h<|u|≤h+k}) ) is finite since u = Th+k(u) ∈T 1,~p(.) Ne (Ω) and ji ∈ L p′i(.)(Ω); then by Lemma 3.8, the last expression converges to zero as h tends to infinity. Similarly we can show that − (∫ A2 ( N∑ i=1 ai ( x, ∂ ∂xi v ) ∂ ∂xi (u) ) dx ) goes to 0 as h → ∞, hence, we obtain lim sup h→∞ A(h,k) ≥ ∫ [|u−v| 0, ∫ [|u−v|