Existence of entropy solutions for nonlinear elliptic equations in Musielak framework with L1 data

Bol. Soc. Paran. Mat. c SPM –ISSN-2175-1188 on line SPM: www.spm.uem.br/bspm (3s.) v. 36 1 (2018): 125–150. ISSN-00378712 in press doi:10.5269/bspm.v36i1.29440 Existence of entropy solutions for nonlinear elliptic equations in Musielak framework with L1 data M.S.B. Elemine Vall, A. Ahmed, A. Touzani and A. Benkirane abstract: We prove existence of solutions for strongly nonlinear elliptic equations of the form  A(u) + g(x, u, ∇u) = f + div(φ(u)) u ≡ 0 ∂Ω. in Ω, Where A(u) = −div(a(x, u, ∇u)) be a Leray-Lions operator defined in D(A) ⊂ W01 Lϕ (Ω) → W −1 Lψ (Ω), the right hand side belongs in L1 (Ω), and φ ∈ C 0 (R, RN ), without assuming the ∆2 -condition on the Musielak function. Key Words: Musielak Orlicz spaces, elliptic problem, Musielak Orlicz function. Contents 1 Introduction 125 2 Preliminary 126 3 Essential assumptions 129 4 Some technical Lemmas 130 5 Main results 135 1. Introduction Let Ω be a bounded open subset of RN , we consider the following nonlinear boundary problem  −diva(x, u, ∇u) + g(x, u, ∇u) = f − div(φ(u)) in Ω, (1.1) u ≡ 0, ∂Ω, where A(u) = −div(a(x, u, ∇u)) is an operator of Leray-Lions type, g is a nonlinearity with the sign condition but any restriction on its growth, f ∈ L1 (Ω) and φ ∈ C 0 (R, RN ). The notion of entropy solution, used in [14], allows us to give a meaning to a possible solution of (1.1). In the classical Sobolv spaces, Boccardo in [14] has proved the existence and regularity of an entropy solution u of problem (1.1) for 2 − N1 < p < N , in the particular case where g ≡ 0, see also [13,17] for related topics. Submitted October 11, 2015. Published January 22, 2016 125 Typeset by BSP style. M c Soc. Paran. de Mat. 126 M.S.B. Elemine Vall, A. Ahmed, A. Touzani and A. Benkirane In the sitting of Orlicz spaces, A.Benkirane and J.Bennouna in [6] have studied the existence of entropy solution of (1.1) where g ≡ 0, Aharouch and Azroul [1] studied the problem (1.1), where g ≡ 0, for more results see [11,12]. In the Sobolev variable exponent, E.Azroul, H.Hjiaj, and A.Touzani [4] have proved the existence and some regularity result for the problem (1.1), Bendahmane and Wittbold in [5] proved the existence and uniqueness of renormalized solution to the problem (1.1) in the particular case a(x, s, ξ) = |ξ|p(x)−2 ξ, g ≡ 0, φ = 0. In Musielak Orlicz framework, M. Ait Khellou, A. Benkirane, S.M. Douiri (see [3]) have proved the existence of entropic solution of (1.1) in the variational case where φ = 0, M. L. Ahmed Oubeid, A. Benkirane, M. Sidi El Vally in [2] proved the existence of entropy solution of (1.1) where g ≡ 0, φ = 0 and the right hand side is a measure data, recently A.Benkirane, F.Blali and M.Sidi El Vally in [7] have solved (1.1) in the case where the Musielak-orlicz function complementary to ϕ satisfies the ∆2 -condition. For some existing results for strongly nonlinear elliptic equations in Musielak-Orlicz-Sobolev spaces [ 10, 22]. Our purpose is to generalize the result [3] and we prove the existence of entropy solution of (1.1). We first give a proof of a Poincaré-type inequality allowing us to prove our result (Lemma 4.4). This article is organized as follows. In the second section we are going to recall some important definitions and results of Musielak-Orlicz-Sobolev spaces. We introduce in the third section some assumptions on a(x, s, ξ) and g(x, s, ξ) for which our problem has a solution. The fourth section contains some important lemmas useful to prove our main results. The section 5 will be devoted to show the existence of entropy solutions for the problem (1.1). 2. Preliminary N Let Ω be an open set in R and let ϕ be a real-valued function defined in Ω × R+ , and satisfiying the following conditions :  a) ϕ(x, .) is an N-function convex, increasing, continous, ϕ(x, 0) = 0, ϕ(x, t) > 0,  ϕ(x,t) ∀t > 0, ϕ(x,t) −→ 0 as t −→ 0, −→ ∞ as t −→ ∞ . t t b) ϕ(., t) is a measurable function. A function ϕ, which satisfies the conditions a) and b) is called Musielak-Orlicz function. For a Musielak-Orlicz function ϕ we put ϕx (t) = ϕ(x, t) and we associate its nonnegative reciprocal function ϕ−1 x , with respect to t that is −1 ϕ−1 x (ϕ(x, t)) = ϕ(x, ϕx (t)) = t. The Musielak-Orlicz function ϕ is said to satisfy the ∆2 -condition if for some k > 0; and a non negative function h; integrable in Ω we have ϕ(x, 2t) ≤ kϕ(x, t) + h(x) for all x ∈ Ω and t ≥ 0. (2.1) Existence of Entropy Solutions 127 When (2.1) holds only for t ≥ t0 > 0; then ϕ said satisfies ∆2 near infinity. Let ϕ and γ be two Musielak-orlicz functions, we say that ϕ dominate γ, and we write γ ≺ ϕ, near infinity (resp. globally) if there exist two positive constants c and t0 such that for almost all x ∈ Ω γ(x, t) ≤ ϕ(x, ct) for all t ≥ t0 , ( resp. for all t ≥ 0 i.e. t0 = 0). We say that γ grows essentially less rapidly than ϕ at 0 (resp. near infinity), and we write γ ≺≺ ϕ, If for every positive constant c we have     γ(x, ct) γ(x, ct) sup = 0, (resp. lim = 0). lim sup t−→∞ t−→0 x∈Ω ϕ(x, t) x∈Ω ϕ(x, t) Remark 2.1. [8] If γ ≺≺ ϕ near infinity, then ∀ε > 0 there exist k(ε) > 0 such that for almost all x ∈ Ω we have for all t ≥ 0. γ(x, t) ≤ k(ε)ϕ(x, εt), (2.2) We define the functional ρϕ,Ω (u) = Z ϕ(x, |u(x)|)dx. Ω where u : Ω −→ R a Lebesgue measurable function. In the following the measurability of a function u : Ω −→ R means the Lebesgue measurability. The set n . o Kϕ (Ω) = u : Ω −→ R measurable ρϕ,Ω (u) < +∞ . is called the generalized Orlicz class. The Musielak-Orlicz space (the generalized Orlicz spaces) Lϕ (Ω) is the vector space generated by Kϕ (Ω), that is, Lϕ (Ω) is the smallest linear space containing the set Kϕ (Ω). Equivalently   .  |u(x)|  < +∞, for some λ > 0 . Lϕ (Ω) = u : Ω −→ R measurable ρϕ,Ω λ Let ψ(x, s) = sup {st − ϕ(x, t)}. t≥0 that is, ψ is the Musielak-Orlicz function complementary to ϕ in the sens of Young with respect to the variable s. In the space Lϕ (Ω) we define the following two norms :   Z  |u(x)|  kukϕ,Ω = inf λ > 0/ ϕ x, dx ≤ 1 . λ Ω which is called the Luxemburg norm and the so called Orlicz norm by : Z k|u|kϕ,Ω = sup |u(x)v(x)|dx. kvkψ,Ω ≤1 Ω 128 M.S.B. Elemine Vall, A. Ahmed, A. Touzani and A. Benkirane where ψ is the Musielak Orlicz function complementary to ϕ. There two norms are equivalent [21]. The closure in Lϕ (Ω) of the bounded measurable functions with compact support in Ω is denoted by Eϕ (Ω). It is a separable space [21]. We say that sequence of functions un ∈ Lϕ (Ω) is modular convergent to u ∈ Lϕ (Ω) if there exists a constant k > 0 such that u − u n = 0. lim ρϕ,Ω n→∞ k For any fixed nonnegative integer m we define   m α W Lϕ (Ω) = u ∈ Lϕ (Ω) : ∀|α| ≤ m, D u ∈ Lϕ (Ω) . and   α W Eϕ (Ω) = u ∈ Eϕ (Ω) : ∀|α| ≤ m, D u ∈ Eϕ (Ω) . m where α = (α1 , ..., αn ) with nonnegative integers αi , |α| = |α1 | + ... + |αn | and Dα u denote the distributional derivatives. The space W m Lϕ (Ω) is called the Musielak Orlicz Sobolev space. Let o   n u X ≤ 1 . ρϕ,Ω (...truncated)