The Existence and Uniqueness of an Entropy Solution to Unilateral Orlicz Anisotropic Equations in an Unbounded Domain

: The purpose of this work is to prove the existence and uniqueness of a class of nonlinear unilateral elliptic problem ( P ) in an arbitrary domain, managed by a low-order term and non-polynomial growth described by an N -uplet of N -function satisfying the ∆ 2 -condition. The source term is merely integrable.


Introduction
Let Ω be an arbitrary domain of R N , (N ≥ 2). In this paper, we investigate the existence and uniqueness solution of the following problem: ( a i (x, u, ∇u) ) x i is a Leray-Lions operator defined onW 1 B (Ω) (defined as the adherence space C ∞ 0 (Ω)) into its dual; B(t) = (B 1 (t), · · · , B N (t)) are N-uplet Orlicz functions that satisfy ∆ 2 -condition; the obstacle ψ is a measurable function that belongs to L ∞ (Ω) ∩W 1 B (Ω); and for i = 1, · · · , N, b i (x, s, ξ) : Ω × R × R N −→ R are Carathéodory functions (measurable with respect to x in Ω for every (s, ξ) in R × R N , and continuous with respect to (s, ξ) in R × R N for almost every x in Ω) that do not satisfy any sign condition and the growth described by the vector N-function B(t). Take f ∈ L 1 (Ω) too. Statement of the problems: Suppose they have non-negative measurable functions φ, ϕ ∈ L 1 (Ω); andā,ã are two constants, positive, such that for ξ = (ξ 1 , · · · , ξ N ) ∈ R N and ξ = (ξ 1 , · · · , ξ N ) ∈ R N , we have and withB(t) being the complementary function of B(t), h ∈ L 1 (Ω) and l : R −→ R + being a positive continuous function such that l ∈ L 1 (Ω) ∩ L ∞ (Ω). We recall that in the last few decades, tremendous popularity has been achieved by the investigation of a class of nonlinear unilateral elliptic problem due to their fundamental role in describing several phenomena, such as the study of fluid filtration in porous media, constrained heating, elastoplasticity, optimal control, financial mathematics and others; for those studies, there are large numbers of mathematical articles; see [1][2][3][4] for more details.
When Ω is a bounded open set of R N , we refer to the celebrated paper by Bénilan [5], who presented the idea of entropy solutions adjusted to Boltzmann conditions. For more outcomes concerning the existence of solutions of this class in the Lebesgue Sobolev spaces (to be specific B(t) = |t| p ), we cite [6,7]. We cite [4,8,9] for the Sobolev space with variable exponent. In the case of Orlicz spaces, we have some difficulties due to the non-homogeneity of the N-functions B(t) and a rather indirect definition of the norm. It is generally difficult to move essentially L p techniques to Orlicz spaces. For more work within this framework, we quote [10][11][12][13].
On the other hand, when Ω is an unbounded domain, namely, without expecting any assumptions on the behavior when | x | −→ +∞, Domanska in [14] investigated the well-posedness of nonlinear elliptic systems of equations generalizing the model equation with corresponding indices of nonlinearity p i > 1 ( i = 0, n ). In [15] Bendahmann et al. the problem ( P ) with b(x, u, ∇u) = div(g(u)) and g(u) a polynomial growth like u q in L p -spaces was solved. For more results we refer the reader to the work [16]. We mention [17][18][19], for the Sobolev space with variable exponent, and [20][21][22][23][24][25][26] for the classical anisotropic space. The oddity of our present paper is to continue in this direction and to show the existence and uniqueness of entropy solution for equations (P ) governed with growth and described by an N-uplet of N-functions satisfying the ∆ 2 -condition, within the fulfilling of anisotropic Orlicz spaces. Besides, we address the challenges that come about due to the absence of some topological properties, such as the densities of bounded or smooth functions.
The outline of this work is as follows. In Section 2, we recall some definitions and properties of N-functions and the space of Sobolev-Orlicz anisotropic solutions. In Section 3, we prove the Theorem of the existence of the solutions in an unbounded domain with the help of some propositions; to be demonstrated later. In Section 4, we show the uniqueness of the solution to this problem, which is expected for strictly monotonic operators at least for a broad class of lower-order terms. Finally, there is Appendix A.

Mathematical Background and Auxiliary Results
In this section, we introduce the notation, recall some standard definitions and collect necessary propositions and facts that are used to establish our main result. A comprehensive presentation of Sobolev-Orlicz anisotropic space can be found in the books of M.A Krasnoselskii and Ja. B. Rutickii [23] and in [20,25].
This N-function B admits the following representation: is an increasing function on the right, with b(0) = 0 in the case z > 0 and b(z) −→ ∞ when z −→ ∞.
Its conjugate is noted byB(z) = | z | 0 q(t) dt with q also satisfying all the properties already quoted The Young's inequality is given as follows: This definition is equivalent to, ∀k > 1, ∃ c(k) > 0 such that Definition 3. The N-function B(z) satisfies the ∆ 2 -condition as long as there exist positive numbers c > 1 and z 0 ≥ 0 such that for | z | ≥ z 0 we have Additionally, each N-function B(z)satisfies the inequality We consider the Orlicz space L B (Ω) provided with the norm of Luxemburg given by According with [23] we obtain the inequalities and Moreover, the Hölder's inequality holds and we have for all u ∈ L B (Ω) and v ∈ LB(Ω) In [23,25], if P(z) and B(z) are two N-functions such that P(z) B(z) and meas Ω < ∞, then L B (Ω) ⊂ L P (Ω); furthermore, Additionally, for all N- We define for all N-functions B 1 (z), · · · , B N (z) the space of Sobolev-Orlicz anisotropicW 1 B (Ω) as the adherence space C ∞ 0 (Ω) under the norm Remark 1. Since B satisfies the ∆ 2 -condition, the modular convergence coincides with the norm convergence.

Remark 2.
If the doubling condition is imposed on the modular function, but not on the conjugate, then the space for the solutions to exist is non-reflexive in general. For this reason we will assume in the remainder of this article that B satisfies the both conditions; the ∆ 2 -condition and ∇ 2 -condition, so the Propositions 1 and 2 will remain true.
with B being the right derivative of the N-function B(z) .
Proof. By (6), we take y = B (z); then we obtain and by Ch. I [23], we get the result.
In the following we will assume that for each N-function B i (z) = with b > 1 checks the ∆ 2 -condition and (22).

The Existence of an Entropy Solution
This section is devoted to the proofs of our main results which will be split into different steps. For m ∈ N * , we define the truncation at height m, T m (u) : R −→ R by

Definition 6.
A measurable function u is said to be an entropy solution for the problem (P ), if u ∈W 1 B (Ω) such that u ≥ ψ a.e. in Ω and in Ω }, and sg m (s) = T m (s) m . We and for all v ∈W 1 B (Ω), we consider the following approximate problem: Theorem 1. Assume that conditions (1)-(4) and (22) hold true, then there exists at least one solution of the approximate problem (P m ).
and we assume that 1 0 h(t) t dt converge, so we consider the N-functions B * (z) defined by Step 1. A priori estimate of { u m }: and for a small enough η we deduce that v ≥ ψ. Thus v is an admissible test function in (P m ) and we get for by (2) and (4), we obtain where c is a constant such that 0 < c < 1, and since h, f m , φ ∈ L 1 (Ω) we deduce that (Ω), and by (8), (3) and (6) and the fact that exp(G(±∞)) ≤ exp where c 2 (k) is a positive constant which depends only on k.
Step 2. Almost everywhere convergence of { u m }: Firstly, we prove that meas{ x ∈ Ω : | u m | ≥ k } → 0. According to Lemma 2, we have with c being a positive constant and (k) → 0 when k → ∞. By (31) we obtain Thus, we deduce that Hence Secondly we show that for all {u m } measurable function on Ω such that In the beginning with α → g(α, k) is a decreasing map; then and according to (34) and (35) we have like [28] we obtain lim k→∞ g(0, k) = 0. Hence We have now to demonstrate that the almost everywhere convergence of { u m : } (Ω(R + 1)), and by embedding Theorem, for an N-function P with P B we have and since η R = 1 in Ω(R), we have: in Ω.

Lemma 3 ([29])
. Let an N-functionB(t) satisfy the ∆ 2 -condition and u m , m ≥ 1 and u be two functions of Then, u m u weakly in L B (Ω) as m → ∞. Hence, Step 3. Weak convergence of the gradient: implies the local convergence in measure and, therefore, the local Cauchy property of u m in measure Proving that ∇u m −→ ∇u locally in measure as m → ∞.
For that, we borrow ideas from Evans [13], Demangel-Hebey [12] and Koznikova L. M. [21,22]. Let δ > 0 be given. By Egoroff's Theorem, there exists E δ,k,α ⊂⊂ Ω such that Then, by Lemma 3 and (33) we obtain that According to (1) and the fact that a continuous function on a compact set achieves the lowest value, there exists a function θ(x) > 0 almost everywhere in Ω, such that, for holds. Writing (P m ) twice for { u m } and { u n }, and by subtracting the second relation from the first and according to (23), (27), (29) and (36) we obtain Consider the following test function: Further on, by applying (40), we get Since B(u) satisfies the ∆ 2 -condition, by (14) we have According to Lemma 3, we get and Additionally, using (14) and (3) we have Hence, , and according to (42), (43), (44) and (15) we obtain that Then, For any arbitrary δ > 0 for fixed m and α, by choosing k from (45) we establish the following inequality By applying Lemma 1, for any > 0, we find In addition, according to (37), we have By combining (39), (46) and (47) we deduce the inequality Hence, the sequence { ∇u m } is fundamental in measure on the set Ω(R) for any R > 0. This implies (38) and the selective convergence, Then, we obtain for any fixed k > 0 Applying Lemma 3, we have the following weak convergence

Proposition 4. Suppose that Conditions
(1)-(4) are satisfied and let (u m ) m∈N be a sequence inW 1 B (Ω(R)) such that Proof. Let > 0 fixed, and η > ; then from (1) we have using the condition (c) we get proceeding as in [28], and we obtain ∇u m −→ ∇u; by letting −→ ∞ we get ∇u m χ −→ ∇u, from (2), and the vitali's Theorem, we get Consequently, by Lemma 2.6 in [11] and (48), we get thanks to lemma 1 (see [20]) and (48), we have Step 4. Strong convergence of the gradient: In this step we consider again the following test function: by (2) and (4) we get we then obtain By (2) we get According to (27), (29); and T k (u m ) Then, By Lebesgue dominated convergence theorem, we have T k (u m ) −→ T k (u) strongly inW 1 B,loc (Ω) and ∇T k (u m ) ∇T k (u) weakly inW 1 B (Ω); then the terms on the right hand side of (50) go to zeros as k, j, m tend to infinity, which gives By Proposition 4 and the diagonal process, we deduce for k −→ ∞ that Hence, we obtain for a subsequence ∇u m −→ ∇u a.e. in Ω. (53) Step 5. The equi-integrability of b m i (x, u m , ∇u m ) : In this step we will show that Therefore, it is enough to show that b m i (x, u m , ∇u m ) is uniformly equi-integrable. We take the following We have By (2) and (4) we get Since a m i (x, u m , ∇u m ) is bounded inW 1 B (Ω), and η j (|u m |) ≥ 0 then by (27), (29) we obtain LetV(Ω(R)) be an arbitrary bounded subset for Ω; then, for any measurable set E ⊂V(Ω(R)) we have We conclude that ∀E ⊂V(Ω(R)) with meas(E) < β( ), and Finally, by combining the last formulas we obtain giving the assumed results.
Step 6. Passing to the limit: Let ϕ ∈W 1 B (Ω) ∩ L ∞ (Ω); we take the following test function: By Fatou's Lemma we get and the fact that weakly inW 1 B (Ω). Additionally, since ψ k T k (u m − ϕ) ψ k T k (u − ϕ) weakly inW 1 B (Ω), and by (53) we obtain and and so we get now passing to the limit to infinity in k, we obtain the entropy solution of the problem.

Uniqueness of the Entropy Solution
Theorem 3. Suppose that conditions (1)-(3) are true, and b i (x, u, ∇u) : Ω × R × R N −→ R are strictly monotonic operators, at least for a broad class of lower order terms. Then, the problem (P ) has a unique solution.
Proof. Let u andū belong to K ψ ∩ L ∞ (Ω) being two solutions of problem (P ) with u =ū.
Author Contributions: All authors performed all the steps of the ideas and proofs in this research. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.

Conflicts of Interest:
The authors declare no conflict of interest.