Existence and Nonexistence of Positive Solutions for a Weighted Quasilinear Elliptic System. (29th March 2023)

• Record Type: Journal Article
• Authors: Hamzaoui, Yamina; Matallah, Atika; Ould El Mokhtar, Mohammed El Mokhtar
• Other Names: Nagy A. M. Academic Editor.
• Is Part Of: Journal of mathematics. Volume 2023(2023)
• Publication Date: 2023-03-29
• Subjects: Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1155/2023/8629590
• Languages: English
• ISSNs: 2314-4629

Abstract

Abstract : This paper deals with the existence and nonexistence of solutions for the following weighted quasilinear elliptic system, S μ 1, μ 2 q 1, q 2 − div q 1 x ∇ u p − 2 ∇ u = μ 1 u p − 2 u + α + 1 u α − 1 u v β + 1 in Ω − div q 2 x ∇ v p − 2 ∇ v = μ 2 v p − 2 v + β + 1 u α + 1 v β − 1 v in Ω u > 0, v > 0 in Ω u = v = 0 on ∂ Ω , where Ω ⊂ ℜ N N ≥ 3, 2 ≤ p < N, q 1, q 2 ∈ W 1, p Ω ∩ C Ω ¯, α, β ≥ 0, μ 1, μ 2, ≥ 0 and α, β > 0 satisfy α + β = p ∗ − 2 with p ∗ = p N / N − p is the critical Sobolev exponent. By means of variational methods we prove the existence of positive solutions which depends on the behavior of the weights q 1, q 2 near their minima and the dimension N . Moreover, we use the well known Pohozaev identity for prove the nonexistence result.