Existence result for a Kirchhoff elliptic system involving p-Laplacian operator with variable parameters and additive right hand side via sub and super solution methods

The paper deals with the study of the existence result for a Kirchhoff elliptic system with additive right hand side and variable parameters involving p−Laplacian operator by using the sub-super solutions method. Our study is an natural extension result of our previous once in (Math. Methods Appl. Sci. 41 (2018), 5203–5210), where in the latter we discussed only the simple case when the parameters are constant.

This nonlocal problem originates from the stationary version of Kirchhoff's work [15] in 1883.
where Kirchhoff extended the classical d'Alembert's wave equation by considering the effect of the changes in the length of the string during vibrations. The parameters in (1.2) have the following meanings: L is the length of the string, h is the area of the cross-section, E is the Young modulus of the material, ρ is the mass density, and P 0 is the initial tension.
Recently, Kirchhoff elliptic equations have been heavily studied, we refer to [1-21, 23, 24]. In [1], Alves and Correa proved the validity of Sub-super solutions method for problems of Kirchhoff class involving a single equation and a boundary condition with f ∈ C Ω × R . By using a comparison principle that requires M to be non-negative and non-increasing in [0, +∞), with H (t) := M t 2 t increasing and H (R) = R, they managed to prove the existence of positive solutions assuming f increasing in the variable u for each x ∈ Ω fixed.
For systems involving similar class of equations, this result can not be used directly, i.e. the existence of a subsolution and a supersolution does not guarantee the existence of the solution. Therefore, a further construction is needed. As in [22], where we studied the system Using a weak positive supersolution as first term of a constructed iterative sequence (u n , v n ) in W 1,p 0 (Ω) × W 1,p 0 (Ω), and a comparison principle introduced in [1], the authors established the convergence of this sequence to a positive weak solution of the considered problem.
To complement our above works in [22], where we discussed only the simple case when the parameters are constant, we are working in this paper for proving the existence result for problem (1.1) by considering the complicated case when the parameters α, β, γ and η in the right hand side are variable. We also give a better subsolution providing easier computations compared with the last work in [22].

Existence result
Definition 2. Let u, v , (u, v) be a pair of nonnegative functions in W 1,p 0 (Ω) × W 1,p 0 (Ω) , they are called positive weak subsolution and positive weak supersolution (respectively) of (1.1) if they satisfy the following If u, v are two non-negative functions verifying then u ≥ v a.e. in Ω.
Proof. Thanks to [24]. Define the functional J : W 1,p 0 (Ω) → R by the formula It is obvious that the functional J is a continuously Gâteaux differentiable whose Gâteaux derivative at the point u ∈ W 1,p 0 (Ω) is the functional J ∈ W −1,p 0 (Ω) , given by It is obvious that J is continuous and bounded since the function M is continuous. We will show that J is strictly monotone in W 1,p 0 (Ω). Indeed, for any u, v ∈ W 1,p 0 (Ω) , u v, without loss of generality, we may assume that Otherwise, changing the role of u and v in the following proof. Therefore, we have Since M (s) is a monotone function. Using Cauchy's inequality, we have and If |∇v (x)| ≥ |∇u (x)| for all x ∈ Ω, changing the role of u and v in (2.3)-(2.7), we have From (2.6) and (2.7) we have and J is strictly monotone in W 1,p 0 (Ω) . Let u, v be two functions such that (2.2) is verified. Taking ϕ = (u − v) + , the positive part of u − v as a test function of (2.2), we have Relations (2.10) and (2.11) mean that u ≤ v.
Before stating and proving our main result, here are the conditions we need.
(H1) M i : R + → R + , i = 1, 2, are two continuous and increasing functions that satisfy the monotonicity conditions of lemma 2.2 so that we can use the Comparison principle, and assume further that there exists m 1 , m 2 > 0 such that ( Proof of Theorem 1. Consider σ p the first eigenvalue of − p with Dirichlet boundary conditions and φ 1 the corresponding positive eigenfunction with φ 1 = 1 and φ 1 ∈ C ∞ Ω (see [10]).
For the supersolution part, consider e p the solution of the following problem We give the supersolution of problem (2.12) by where C > 0 is a large positive real number to be given later. Indeed, for all φ ∈ W 1,p 0 (Ω) with φ ≥ 0 in Ω, we get from (2.12) and the condition (H1) By (H4) and (H5), we can choose C large enough so that Therefore, (2.14) Using (H4) and (H5) again for C large enough we get (2.15) Combining (2.13) and (2.14), we obtain (2.16) By (2.12) and (2.15), we conclude that (u, v) is a supersolution of problem (1.1). Furthermore, u ≤ u and v ≤ v for C chosen large enough. Now, we use a similar argument to [22] in order to obtain a weak solution of our problem. Consider the following sequence {(u n , v n )} ⊂ W 1,p 0 (Ω) × W 1,p 0 (Ω) , where: u 0 := u, v 0 = v and (u n , v n ) is the unique solution of the system we deduce from a result in [1] that system (2.16) has a unique solution (u n , v n ) ∈ W 1,p 0 (Ω) × W 1,p 0 (Ω) . Using (2.16) and the fact that (u 0 , v 0 ) is a supersolution of (1.1), we get Then by Lemma 1, u 0 ≥ u 1 and v 0 ≥ v 1 . Also, since u 0 ≥ u, v 0 ≥ v and the monotonicity of f, g, h, and l one has According to Lemma 1 again, we obtain u 1 ≥ u, v 1 ≥ v.
Repeating the same argument for u 2 , v 2 , observe that Similarly, we get u 2 ≥ u and v 2 ≥ v from By repeating these implementations we construct a bounded decreasing sequence By continuity of functions f, g, h, and l and the definition of the sequences (u n ) and (v n ) , there exist positive constants C i > 0, i = 1, ..., 4 such that and |l (u n−1 )| ≤ C 4 for all n.

Conclusions
In [22], we discussed only the simple case when the parameters are constant, in this current work, we have proved the existence result for problem (1.1) by considering the complicated case when the parameters α, β, γ and η in the right hand side are variable. We also give a better subsolution providing easier computations compared with the last work in [22]. In the next work, we will try to apply the same techniques in the Hall-MHD equations which is nonlinear partial differential equation that arises in hydrodynamics and some physical applications. It was subsequently applied to problems in the percolation of water in porous subsurface strata (see for example [2,8,9]).