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

Laboratory of Analysis and Control of Differential Equations “ACED”, Fac. MISM, Department of Mathematics, Faculty of MISM Guelma University, P.O. Box 401, Guelma 24000, Algeria Department of Mathematics, College of Sciences and Arts, Al-Rass, Qassim University, Saudi Arabia Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Ahmed Benbella, Algeria Department of Mathematics, Faculty of Exact Sciences, University Tebessa, Tebessa, Algeria


Introduction
e 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][2][3][4][5][6][7][8][9][10]). e p-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 Ω ⊂ R N (N ≥ 3) is a bounded domain and its boundary zΩ. Also, A and B are two continuous functions on R + , and the parameters α, β, δ, and c satisfy the following conditions: Within previous studies [13][14][15], some nonlocal elliptical problems of the Kirchhoff type of the following model were extensively studied: where Ω is a bounded open domain of R n with a smooth boundary zΩ and h(x, u) the right hand side is defined for some exceptional functions similar to those in [13][14][15][16]. In addition, M is a defined and continuous function on R + with values in R * + . In recent years, various Kirchhoff or p(x)-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 Ω or in R n . 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. e parameters of the above equation have the following meanings: E is Young's modulus of the material, ρ is the mass density, L is the length of the chain, h is the section area, and P 0 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 λ 1 , μ 1 ′ , λ 2 ′ , and μ 2 ′ are positive parameters, α + c < 1, and β + d < 1. 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. e 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.

Preliminaries and Assumptions
We assume the following hypothesis: (H1): we assume that M: R + ⟶ R + is a nonincreasing and continuous function which satisfies where m 0 > 0, and there exists (H2): and α, β ∈ C(Ω), for all x ∈ Ω.
(H3): f, g, h, and τ are C 1 on (0, +∞) and increasing functions, where Lemma 1 (see [14]). Under assumption (H1),we suppose further that function We assume that u and v are couple nonnegative functions, where and then u ≥ v a.e. in Ω.
Proof. Let e i ∈ C 0,ρ i (Ω), for i � 1, 2, ρ i > 0, be the solution of the following problem: en, by the strong maximum principle, we get e i > 0 in Ω, i � 1, 2.
We define where C 1 and C 2 are positive constants which we will fix them later.

Existence of Weak Subsolution.
Existence of a positive weak subsolution for system (1) is proved such that each component belongs to C 0 (Ω).

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]). 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 (u n , v n ) ⊂ (H 1 0 (Ω) × H 1 0 (Ω)) as follows: u 0 : � u, v 0 � v, and (u n , v n ) is the unique solution of the system 4 Mathematical Problems in Engineering is given, the right-hand sides of (41) are independent of u n , v n . Set en, since According to the result in [1], we can deduce that system (41) admits a unique solution By using (41) and the fact that (u 0 , v 0 ) is a supersolution of (1), we have Also, by using Lemma 1, u 0 ≥ u 1 and v 0 ≥ v 1 , and since u 0 ≥ u, v 0 ≥ v, and the monotonicity of from which, according to Lemma 1, and then u 1 ≥ u 2 and v 1 ≥ v 2 . Similarly, Repeating this argument, we get a bounded monotone Using the continuity of the functions f and g and the definition of the sequence u n , v n , there exist constants From (52), we multiply the first equation of (41) by u n ; in addition, by using the Holder inequality combined with Sobolev embedding, we have where C 3 > 0 is a constant independent of n. Similarly, there exists C 2 > 0 independent of n such that v n � � � � � � � � H 1 0 (Ω) ≤ C 4 , ∀n.
By using a standard regularity argument, (u n , v n ) converges to (u, v). us, when n ⟶ +∞ in (41), we can see that (u, v) is a positive solution of system (1). e proof is completed.

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].
Mathematical Problems in Engineering 5