Existence of Positive Weak Solutions for Quasi-Linear Kirchhoff Elliptic Systems via Sub-Supersolutions Concept

Abstract

By using the sub- and supersolutions concept (Schmitt, 2007), we prove in this paper the existence of positive solutions of quasi-linear Kirchhoff elliptic systems in bounded smooth domains. This work is an extension of the recent work of Boulaaras et al., 2020.

1. Introduction

The scope of nonlinear partial differential equations is quite wide. One of the main advances in the development of nonlinear PDEs has been the study of wave propagation, then comes the equations related to chemical and biological phenomena, and later, the equations related to solid mechanics, fluid dynamics, acoustics, nonlinear optics, plasma physics, quantum field theory, and engineering.

Studying these equations is a daunting task because there are no general methods for solving them. Each problem requires an appropriate approach depending on the type of linearity ([1–10]).

The -Laplacian operator is a model of quasi-linear elliptic operators which makes it possible to model physical phenomena such as the flow of non-Newtonian aids, reaction flow systems, nonlinear elasticity, the extraction of petroleum, astronomy, through porous media, and glaciology. Several authors in this field obtained many results of existence (see, for example, [1, 3, 5, 11, 12]).

In this work, we consider the following quasi-linear elliptic system:where is a bounded domain and its boundary . Also, and are two continuous functions on , and the parameters , , , and satisfy the following conditions:

Within previous studies [13–15], some nonlocal elliptical problems of the Kirchhoff type of the following model were extensively studied:where is a bounded open domain of with a smooth boundary and the right hand side is defined for some exceptional functions similar to those in [13–16]. In addition, is a defined and continuous function on with values in . In recent years, various Kirchhoff or -Kirchhoff-type problems have been widely studied by many authors due to their theoretical and practical importance. Such problems are often referred to as nonlocal due to the presence of a full term on or in . It is well known that this problem is analogous to the stationary problem of a model introduced by Kirchhoff [17].

More specifically, Kirchhoff proposed this model as an extension of the wave equation of the Alembert classic by considering the effects of variations in the length of the strings during vibration. The parameters of the above equation have the following meanings: is Young’s modulus of the material, is the mass density, is the length of the chain, is the section area, and is the initial tension.

In recent work in [18], we have discussed the existence of the weak positive solution for the following Kirchhoff elliptic systems:where , and are positive parameters, , and .

Motivated by the recent work in [13, 14, 18, 19] and by using the sub- and supersolution method which is defined in [20], existence of positive solutions of quasi-linear Kirchhoff elliptic systems is shown in bounded smooth domains.

The paper outline is as follows: some assumptions and definitions related to problem (1) are given in Section 2. Finally, our main result is given in Section 3.

2. Preliminaries and Assumptions

We assume the following hypothesis: : we assume that is a nonincreasing and continuous function which satisfies where , and there exists such that : and for all . , , , and are on and increasing functions, where  such that

Lemma 1 (see [14]). Under assumption (),we suppose further that function is increasing on .
We assume that and are couple nonnegative functions, whereand then . in .

Lemma 2 (see [1]). If verifies the conditions of Lemma 1, then for each , there exists a unique solution to the -linear problem:

Lemma 3 (see [1]). Let solve . If , then for any , so particularly, is continuous in .

Definition 1. Let , and is said a weak solution of (1) if it satisfiesfor all .

Definition 2. We call the following nonnegative functions , respectively; in are a weak subsolution (respectively, upersolution) of (1) if they verify and in :for all .
Before proving our main result, we need to prove the existence of weak supersolution and subsolution in the following section.

3. Weak Existence Results

3.1. Existence of Weak Supersolution

The existence of a positive weak supersolution for system (1) is established such that each component belongs to , for .

Lemma 4. Suppose that holds, , and . Then, system (1) possesses a positive weak supersolutionfor and .

Proof. Let , for , , be the solution of the following problem:Then, by the strong maximum principle, we get in , .
We definewhere and are positive constants which we will fix them later.
Let , with .
Then, we obtainand similarly,Ifand holds, it is easy to prove that there exist positive constants and such thatThus, from (21), we obtain for all Therefore, by using (18), (19), and (22), we conclude thatHence, is a positive weak supersolution of system (1).

3.2. Existence of Weak Subsolution

Existence of a positive weak subsolution for system (1) is proved such that each component belongs to .

Lemma 5. We assume that holds:

Therefore, system (1) possesses a positive weak subsolution , for all .

Proof. We assume that is the first eigenvalue of with Dirichlet condition with which is its corresponding eigenfunction and belongs to , in and on , for some positive constants , and .
We definewhich belongs to , with to be fixed later, andbecause , , and . Then, for all , with , , we haveSimilarly,Since and on , there exists such that, for every , we haveThen, for each , we getfor all , , andfor all and .
Now, as in and is continuous, then there exists such that for all . Therefore, from (26), we obtain such that the following inequalities hold:for each .
Then,By (33), we haveAnd similarly, from (32), we havein and each .
Therefore,Hence, from (30), (31), (37), and (38), it follows thatThen, by (39) and (40), is a positive weak subsolution of system (1), for each .

4. Main Result

In this section, we give the result of the existence of the positive weak solution to quasi-linear elliptic system (1) by using the sub- and supersolution method which has been already used for some classical elliptic equations by known authors (see [1, 4, 11, 19, 21]).

Theorem 1. Suppose that holds, , and as well as under the results of Lemma 4 and 5. Then, system (1) possesses a weak solution , where each component is positive and belongs to for some , , and each .

Proof 3. In order to obtain a weak solution of problem (1), we shall use the arguments by Azzouz and Bensedik [13]. For this purpose, we define a sequence as follows: , and is the unique solution of the systemProblem (41) is linear in the sense that ifis given, the right-hand sides of (41) are independent of .
SetThen, sinceAccording to the result in [1], we can deduce that system (41) admits a unique solutionBy using (41) and the fact that is a supersolution of (1), we haveAlso, by using Lemma 1, and , and since , , and the monotonicity of and , one hasfrom which, according to Lemma 1, and . For , we writeand then and . Similarly, and becauseRepeating this argument, we get a bounded monotone sequence satisfyingUsing the continuity of the functions and and the definition of the sequence , there exist constants , , independent of such thatFrom (52), we multiply the first equation of (41) by ; in addition, by using the Holder inequality combined with Sobolev embedding, we havewhere is a constant independent of . Similarly, there exists independent of such thatFrom (53) and (54), we deduce that the couple converges weakly in to the couple with and .
By using a standard regularity argument, converges to . Thus, when in (41), we can see that is a positive solution of system (1).
The proof is completed.

5. Conclusion

As a conclusion of this contribution, we have proved the existence of positive solutions of quasi-linear Kirchhoff elliptic systems in bounded smooth domains by using the sub- and super-solution method [20], which is an extension of our recent works of Boulaaras et al. in [18]. In the next work, some other methods such as variational and Galerkin methods (see, for example, [15]) will be used for this problem, and some numerical examples will also be given [9, 22].

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this manuscript.

Authors’ Contributions

The authors contributed equally to this article. They have all read and approved the final manuscript.