Existence and Nonexistence for Boundary Problem Involving the p-Biharmonic Operator and Singular Nonlinearities. (13th March 2023)

• Record Type: Journal Article
• Authors: Ould El Mokhtar, Mohammed El Mokhtar
• Other Names: Verbitsky Igor E. Academic Editor.
• Is Part Of: Journal of function spaces. Volume 2023(2023)
• Publication Date: 2023-03-13
• Subjects: Function spaces -- Periodicals; 515.7305
• DOI: 10.1155/2023/7311332
• Languages: English
• ISSNs: 2314-8896

Abstract

Abstract : This article concerns the existence and the nonexistence of solution for the following boundary problem involving the p-biharmonic operator and singular nonlinearities, Δ p 2 u = u γ − 1 u + μ u − 1 − α / x β u in Ω and u = ∂ u / ∂ n = 0 on ∂ Ω, where 4 < 2 p < N, 0 ∈ Ω, − ∞ < μ < μ ∗, = N − 2 p 1 − α / p N, p < γ < p ∗ = p N / N − 2 p is the critical Sobolev exponent, 0 ≤ β < N γ + α / γ + 1, 0 < α < 1 . Under some sufficient conditions on coefficients, we prove the existence of at least one nontrivial solutions in E by using variational methods. By using the Pohozaev identity type, we show the nonexistence of positive solution when Ω ⊂ ℝ N be a bounded, smoothandstrictlystar-shapeddomain, β = 0 and γ ≥ γ ∗, = pN 1 − α / N − 2 p 1 − α − μ Np > p ∗ = pN / N − 2 p .