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Konaté, Ibrahime; Ouédraogo, Arouna

doi:https://doi.org/10.1155/2023/9454714

International Journal of Differential Equations

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Research Article | Open Access

Volume 2023 | Article ID 9454714 | https://doi.org/10.1155/2023/9454714

Existence and Uniqueness of Renormalized Solution to Nonlinear Anisotropic Elliptic Problems with Variable Exponent and -Data

Ibrahime Konaté¹and Arouna Ouédraogo²

Academic Editor: Davood D. Ganji

Received26 Sept 2022

Revised18 Feb 2023

Accepted25 Mar 2023

Published10 Apr 2023

Abstract

Nonlinear partial differential equations are considered as an essential tool for describing the behavior of many natural phenomena. The modeling of some phenomena requires to work in Sobolev spaces with constant exponent. But for others, such as electrorheological fluids, the properties of classical spaces are not sufficient to have precision. To overcome this difficulty, we work in the appropriate spaces called Lebesgue and Sobolev spaces with variable exponent. In recent works, researchers are attracted by the study of mathematical problems in the context of variable exponent. This great interest is motivated by their applications in many fields such as elastic mechanics, fluid dynamics, and image restoration. In this paper, we combine the technic of monotone operators in Banach spaces and approximation methods to prove the existence of renormalized solutions of a class of nonlinear anisotropic problem involving Leray–Lions operator, a graph, and data. In particular, we establish the uniqueness of the solution when the graph data are considered a strictly increasing function.

1. Introduction

Partial differential equations (PDEs for short) are considered a fundamental tool for modeling and thus understanding many real-world phenomena. These partial differential equations make it possible to take into account many parameters related to the course of phenomena and the role of these parameters. They also make it possible to predict, sometimes extremely accurately, how the phenomenon evolves over time. This prediction may exist in the very special case of linear PDEs, but when the phenomenon is modeled by a nonlinear PDE, prediction becomes almost impossible.

Nonlinear PDEs appear in many fields including chemistry, physics, and engineering science (see [1–5]). For example, in [6], Gandji et al. used nonlinear equations to study the three-dimensional Bödewadt hybrid nanofluid flow where fluids are composed of water and hexanol. Note that some nonhomogeneous materials such as aluminum oxide or alumina have the ability to change state very quickly (in a few milliseconds) physically when an electric field of very small intensity is applied to them. To model the behavior of these materials, classical Lebesgue and Sobolev spaces with constant exponent are not efficient enough to have accuracy. To this end, we commonly work in the Lebesgue and Sobolev spaces with variable exponent. The properties of these nonhomogeneous materials are widely exploited in many technological applications such as shock absorbers and equipment rehabilitation.

The study of PDEs with variable exponent has increased intensively in recent years. The importance of studying such problems is due to the discovery of their applications in the modeling of behavior of certain nonhomogeneous devices in physics, mechanical process, electrorheological fluids, and stationary thermo-rheological viscous flows of non-Newtonian fluids (see [7–11] for more details). They are also used in modeling the propagation of epidemic diseases (see [12]) and image processing ([13]).

In this paper, we are interested in the existence and uniqueness of renormalized solution of the following anisotropic problem:where is an open bounded domain of with Lipschitz boundary , , is locally Lipschitz continuous, and is a set-valued maximal monotone mapping such that . We allow the term to be multivalued, not necessarily defined in the whole of .

Under our assumptions, problem is generally not well posed in the framework of weak solution because may not belong to ( is just continuous on ). To overcome this difficulty, we use the framework of renormalized solutions which requires low regularity than the weak one. This concept of solution first appeared in the work of Lions and Diperna [14] and used later by Lions and Murat to tackle elliptic equation with low summability data (i.e. when the data are or a measure).

Analysis of problems involving graph data, Sobolev space with constant exponent, and generalized Orlicz spaces is already a classical topic investigated since [15–17]. Then, differential inclusion problems have been extended to variable exponent setting in [18–21] and the references therein. In [22], Akdim and Allalou ensured the existence of renormalized solution of a problem close to but in the framework of weighted space.

In the literature, special cases of problem have been explored in the framework of anisotropic Sobolev space (see [23–25]) and have concerned the problem below:where is a bounded Radon measure or -function.

Let us recall that the first elliptic problems studied in anisotropic Sobolev space with variable exponent were the works of Mihailescu et al. [26, 27].

For the case where , Koné et al. [28, 29] used the minimization technics to prove the existence of weak solutions of problem (see also [30, 31]). The case in which and are continuous nondecreasing functions from to is studied in [23]. In [24], Konaté and Ouaro have proved the existence and uniqueness of an entropy solution of problem when is a Radon measure and is a maximal monotone graph.

When the components of the vector are constants, the authors in [32] studied the problem and established the existence and uniqueness of renormalized solution in the anisotropic Sobolev space with constant components of the vector . Since the components of the vector are able to vary, the Leray–Lions operator which appears in the left-hand side of problem is more general than the one which appears in [32].

To our knowledge, all the previous studies dealing with similar problem in the framework of variable exponent spaces are focused on particular cases.

In this paper, we extend the recent works [21, 24, 25, 32] by using the ideas developed in [21, 32]. More precisely, we used the technic of monotone operators in Banach spaces and approximation methods to prove the existence and uniqueness of a renormalized solution of problem in the context of anisotropic space involving variable exponents . As the novelty of this study, the components of the vector are able to vary and the diffusion convection term is not null. The main difficulty we encounter is how to establish the a priori estimates and convergence results.

Our main results rely on the following assumptions.

Throughout this paper, is a vector such that the components are continuous functions (for any ) satisfyingand we set

For any , let be a Carathéodory function verifying the following assumptions.

There exists positive constants , , such that(i)For a.e. and for every , where is a nonnegative function in , with .(ii)For with and for every ,(iii)For and a.e. ,

We assume thatwhere .

This paper is structured as follows. In Section 2, we recall some fundamental preliminaries which are useful in this work and we give our main results. In Section 3 we study the case where . In Section 4, we study the existence and uniqueness of a renormalized solution when . Finally, in Section 5, we give an example for illustrating our abstract result.

2. Preliminary and Main Results

This section is devoted to some definitions and basic properties of anisotropic Lebesgue with Sobolev spaces and variable exponents. Set

For any , the variable exponent Lebesgue space is defined byendowed with the so-called Luxemburg norm

The -modular of the space is the mapping defined by

For any , we have (see [33, 34])

For any and , with for any , we have the Hölder type inequality

If is bounded and such that for any , then the embedding is continuous (see [35], Theorem 2.8).

We defined the anisotropic Sobolev space with variable exponent as follows:which is a separable and reflexive Banach space (see [26]) under the norm

We have the following embedding results.

Theorem 1 (see [33], Corollary 2.1). Let be a bounded open set and for all a.e. . Then, for any with a.e. such thatwe have the compact embedding

We defined the numbers

Theorem 2 (see [36]). Let ; and

Then, there exists a constant depending on if and also on q and if such thatwhere and .

The Marcinkiewicz space is introduced as the set of measurable function for which the distributionsatisfies the following:

We will use the following pseudonorm in .

We defined the truncation function , by

We observe that and .

For any , we use instead of for the trace of on .

Set as the set of the measurable functions such that for any , .

Lemma 1 (see [30]). Let be a nonnegative function in . Assume and there exists a constant such thatfor every .
Then, there exists a constant , depending on such thatwhere .
We introduce some useful functions as follows.
For , let and be the function defined byLet be defined by for each .
For , we define byand byNow, we give our main results.

Theorem 3. Under assumptions (3)–(9) and , there exists at least one renormalized solution to problem in the sense that(i), , for a.e. .(ii)For all and ,(iii) as .

Theorem 4. Let and be two renormalized solutions of problem . Then,

3. Existence Result for -Data

Theorem 5. Assuming that (3)–(9) hold, , then the problem admits at least one renormalized solution.

Proof. We demonstrate Theorem 5 in five steps.

Step 1. Approximate problem.
Let be the Yosida regularization of (see [37]), defined by such that .
We consider the approximate problem

Lemma 2. The problem has at least one weak solution in the sense thatwhere and denotes the duality pairing between and .

Proof. We define the operators , , and , acting from into its dual as follows:whereReasoning as in [25] (see also [21, 32]), we can prove that the operator is pseudomonotone, coercive, and bounded. Then, we deduce from [38] (Theorem 2.7) that is surjective. Since , it follows that the problem admits at least one solution .
Taking into account the monotonicity of and and following the same lines as in [21, 32], we establish the following comparison principle which will be essential in the proof of uniqueness of the solution.

Proposition 1. Let and such that is a solution of and is a solution of . Then, the following comparison principle holds:

Remark 1 (see [21, 32]). By assuming that a.e. in , an immediate consequence of the proposition above is the inequality . In addition, we have a.e. in .

Step 2. A apriori estimates.

Lemma 3 (see [25]). If is a solution of problem , thenwhere .

Lemma 4 (see [23, 30]). There exist some constants such that(i).(ii).

Remark 2 (see [25]). There exist such that

Step 3. Convergence results.

Lemma 5 (see [23, 25]). Assume that and is a solution of . Then, there exist and such that for a non-relabelled subsequence of as ,Moreover, for any ,

Step 4. Passing to limit.
Let and . We apply the test function in (35) to getwhereWe first observe that . Then, letting and in (43), we obtainwhere and (for the convergence result, see [25]).

Step 5. Subdifferential argument.
To end the proof, we have (see [25, 39])(i), a.e. .(ii) as .

Remark 3 (see [21]). If is a renormalized solution of for second member , then . Moreover, is a weak solution of .

4. The Case of -Data

4.1. Proof of Theorem 1

This proof is made in several steps.

4.1.1. Step 1: Approximate Problem

The first step consists in approximating the second member by bounded function. For and , we define by

Note that , a.e. in and in as . We have also

According to Theorem 5, the problem admits a renormalized solution . That is,where and .

Our goal is to show that these approximated solutions tend, as go to , to a couple of functions which are renormalized solutions of problem . We begin by giving some useful a priori estimates.

4.1.2. Step 2: A Priori Estimates

Lemma 6. If is a renormalized solution of problem , thenfor and .

Proof. For any , by applying as test function in (48), we obtainUsing the same arguments as used in the proof of Lemma 2, we get (49).
For the proof of (50), see [21, 32, 39].

Remark 4 (see [21, 32, 39]). Letting go to , we get the following convergences.where is a measurable function.
The next lemma will be used to show that is finite a.e. in .

Lemma 7. If is a renormalized solution of , then there exists a constant , not depending on , such thatfor all .

Proof. Using Remark 2, we getwhere and is a constant coming from Sobolev embedding in (19). From (49) and (54), we deduce (53).

Remark 5. is finite a.e. in and a.e. in . Indeed, the proof relies on Lemma 4.6 and subdifferential argument (see [21, 32, 39] for details).

Remark 6. If is a renormalized solution of , choosing as a test function in (48), discarding positive terms, and letting go to , we obtainfor any . Now using (53) in (55), we getwhere , , and .

4.1.3. Step 3: Basic Convergence

Lemma 8 (see [25], Lemma 3). For and , if is a renormalized solution of , then there exists a subsequence such that posing , there exists such that a.e. in and the convergences below hold:More over, for any ,

4.1.4. Step 4: Strong Convergence

Remark 7. Arguing as in [25] (Lemma 1), we obtain equality (iii), namely,To complete the proof of Theorem 3, it remains to verify (ii). To this end, we choose as test function in (48) to obtainwhereLetting go to and using Lemma 7, we obtainBy rewriting as follows:whereand reasoning as in [25], we obtainFrom (56), we deduce thatwhere and .
Split , wherePassing to limit as goes to , we getFor all and , letting in (60), we obtainwhereLet such that . Replacing by and passing to limit, as goes to , in each term of (69), we getThanks to (72)–(76), we pass to the limit in (69) as , to get (32).

4.2. Proof of Theorem 2

We highlight that the uniqueness is more delicate and it is necessary to have additional hypothesis.

Here we assume that is strictly increasing, and we prove a uniqueness result for renormalized solution of the problem where .

Proposition 2. Let and and be renormalized solutions of and , respectively. Then, the following comparison principle holds:

Proof. Let , be the Lipschitz approximation of the function.
The fact that , are renormalized solutions implies that for . Hence, is an admissible test function.
Taking and writing the renormalized equalities corresponding to solutions and , respectively, with test function and adding up both results, we getwherewith . Reasoning as in [21], i.e., neglecting the positive part of and using the fact that is locally Lipschitz continuous, we can pass to the limit as .
Using the condition of Theorem 3, we pass to limit as to obtain (77).
To end the proof, we assume . Then, following the same lines as in [40], we obtain

5. Example

An example that is covered by our assumption is the following anisotropic -harmonic problem: set

Then, we have the problem

are Carathéodory functions satisfying the growth condition (4), the coercivity (6), and the monotonicity condition (5).

Since all the hypothesis of Theorem 2 are fulfilled, problem (82) has at least one solution for all .

6. Conclusion

In the present study, we investigated the existence and uniqueness of solution of a class of anisotropic nonlinear elliptic problems defined with inclusion equation and Dirichlet boundary condition. Governing equations are solved by using the technic of monotone operators in Banach spaces and approximation methods. The novelty of this work relies on transposing nonlinear PDEs from classical (Lebesgue and Sobolev) spaces into generalized Lebesgue and Sobolev spaces with variable exponents. The main conclusions of this work are given as follows:(1)When the components of the vector are able to vary, the problem admits an unique renormalized solution in the anisotropic Sobolev space . Moreover if the graph is a nondecreasing function, the solution is unique.(2)The main result of this study extends the previous works in [25, 32] in the context of anisotropic space with variable exponent.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2023 Ibrahime Konaté and Arouna Ouédraogo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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