Existence result for a Kirchhoff elliptic system with variable parameters and additive right-hand side via sub- and supersolution method

The paper deals with the study of the existence result for a Kirchhoff elliptic system with additive right-hand side and variable parameters by using the sub-/supersolution method. Our study is a natural extension result of our previous one in (Boulaaras and Guefaifia in Math. Methods Appl. Sci. 41:5203–5210, 2018), where we discussed only the simple case when the parameters are constant.


Introduction
Consider the following system: where Ω ⊂ R N (N ≥ 3) is a bounded smooth domain with C 2 boundary ∂Ω, and A, B : R + → R + are continuous functions with further conditions to be given later, α, β, γ , η ∈ C(Ω). This nonlocal problem originates from the stationary version of Kirchhoff 's work [16] in 1883, namely ρ ∂ 2 u ∂t 2 - 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 [1-7, 9, 11-15, 17-20].
In [2], Alves and Correa proved the validity of sub-/supersolution method for problems of Kirchhoff class involving a single equation and a boundary condition By using a comparison principle that requires M to be nonnegative and nonincreasing 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 was increasing in u for each x ∈ Ω fixed.
For systems involving similar equations, this result cannot 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. In [8], we studied the system inΩ, Using a weak positive supersolution as the first term of a constructed iterative sequence (u n , v n ) in H 1 0 (Ω) × H 1 0 (Ω), and a comparison principle introduced in [2], the authors established the convergence of this sequence to a positive weak solution of the considered problem.
To complement our above work in [8], where we discussed only the simple case when the parameters are constant, in this paper we prove an existence result for problem (1.1) by considering the complicated case when the parameters α, β, γ , and η on the righthand side are variable. We also give a better subsolution providing easier computations compared with the earlier work in [8].
For the supersolution part, consider e the solution of the following problem: We give the supersolution of problem (1.1) by where C > 0 is a large positive real number to be given later. Indeed, for all φ ∈ H 1 0 (Ω) with φ ≥ 0 in Ω, we get from (2.3) and the condition (H1) By (H4 ) and (H5), we can choose C large enough so that Therefore, Also, Using (H4) and (H5) again for C large enough, we get Combining (2.5) and (2.6), we obtain By (2.4) and (2.7), 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 that in [8] in order to obtain a weak solution of our problem. Consider the following sequence (2.8) Since A and B satisfy (H1) and α(x)f (v n-1 ), β(x)g(u n-1 ), γ (x)h(u n-1 ), and η(x)l(v n-1 ) ∈ L 2 (Ω) (in x), we deduce from a result in [2] that system (2.8) has a unique solution (u n , v n ) ∈ (H 1 0 (Ω) × H 1 0 (Ω)). Using (2.8) 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 due to 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 By repeating the same arguments, 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.
From (2.11), multiplying the first equation of (2.8) by u n , integrating, using Hölder inequality and Sobolev embedding, we check that where C 5 > 0 is a constant independent of n. Similarly, there exists C 6 > 0 independent of n such that v n H 1 0 (Ω) ≤ C 6 , ∀n.