Abstract
PurposeIn this paper, the authors give a new version of the sub-super solution method and prove the existence of positive solution for a (p, q)-Laplacian system under weak assumptions than usually made in such systems. In particular, nonlinearities need not be monotone or positive.Design/methodology/approachThe authors prove that the sub-super solution method can be proved by the Shcauder fixed-point theorem and use the method to prove the existence of a positive solution in elliptic systems, which appear in some problems of population dynamics.FindingsThe results complement and generalize some results already published for similar problems.Originality/valueThe result is completely new and does not appear elsewhere and will be a reference for this line of research.
Highlights
Consider the following (8p1, p2)-Laplacian system, < ÀΔp1u11⁄4 μ1F1ðx; u1; u2Þ in Ω : ÀΔp2 u1 1⁄4 u2 u21⁄4 μ2F2ðx; u1; u2Þ in Ω 1⁄40 on ∂Ω (1)Ω is an open bounded domain of RN with smooth boundary ∂Ω
Our main contribution in this article is, in first, to give a new version of the sub-super solution method based on Schauder’s famous fixed-point theorem and, in second, use the method to prove the existence of a positive solution of problem (1) under the continuity assumptions on functions F and G
We use Theorem 3.1 and solve some elliptic (p1, p2)-Laplacian systems studied in some published articles see [7]
Summary
Ω is an open bounded domain of RN with smooth boundary ∂Ω. For i 5 1, 2, Δpi Fi : 1⁄4 Ω divðj∇uijpi−2∇uiÞ is the pi-Laplacian operator, 3 R 3 R → R is a continuous function. The subsuper solution method, given in [5] by using a monotony argument, is the principal tool used to prove the existence of solution of the problem (1) in [1, 3, 4]. AJMS method of the sub-super solution is given to prove the existence of solution for the (p(x), q(x))Laplacian systems by using the Schaefer’s fixed-point theorem [6]. Our main contribution in this article is, in first, to give a new version of the sub-super solution method based on Schauder’s famous fixed-point theorem and, in second, use the method to prove the existence of a positive solution of problem (1) under the continuity assumptions on functions F and G.
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