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.
Highlights
By using the sub- and supersolutions concept (Schmitt, 2007), we prove in this paper the existence of positive solutions of quasilinear Kirchhoff elliptic systems in bounded smooth domains. is work is an extension of the recent work of Boulaaras et al, 2020
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
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]
Summary
(H1): we assume that M: R+ ⟶ R+ is a nonincreasing and continuous function which satisfies lim t⟶0+. If M verifies the conditions of Lemma 1, for each f ∈ L2(Ω), there exists a unique solution u ∈ H10(Ω) to the M-linear problem:. Let (u, v) ∈ (H10(Ω) ∩ L∞(Ω) × H10(Ω) ∩ L∞ (Ω)), and (u, v) is said a weak solution of (1) if it satisfies. We call the following nonnegative functions (u, v), respectively; (u, v) in (H10(Ω) ∩ L∞(Ω)× H10(Ω) ∩ L∞(Ω)) are a weak subsolution (respectively, upersolution) of (1) if they verify (u, v) and (u, v) (0, 0) in zΩ: A ∇ u2dx ∇ u ∇φdx ≤ λ uα vcφdx in Ω, B ∇ v2dx ∇ v ∇ψdx ≤ λ uδ vβ ψdx in Ω,. Before proving our main result, we need to prove the existence of weak supersolution and subsolution
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