Existence of solutions for a p(x)-biharmonic problem under Neumann boundary conditions. Issue 10 (27th July 2021)

• Record Type: Journal Article
• Authors: Hsini, Mounir; Irzi, Nawal; Kefi, Khaled
• Is Part Of: Applicable analysis. Volume 100:Issue 10(2021)
• Page Start: 2188
• Page End: 2199
• Publication Date: 2021-07-27
• Subjects: S. Leonardi; p(x)-biharmonic operator -- Ekeland's variational principle -- generalized Sobolev spaces; 35J65 -- 35D05 -- 35J60 -- 35D30 -- 35J58; Mathematical analysis -- Periodicals; 515
• DOI: 10.1080/00036811.2019.1679788
• Languages: English
• ISSNs: 0003-6811

Abstract

Abstract : In this paper, we consider for a given smooth bounded domain Ω of R N, ( N ≥ 3 ), the following p ( x ) -biharmonic type problem 1 Δ ( ξ ( x ) | Δ u | p ( x ) − 2 Δ u ) + a ( x ) | u | p ( x ) − 2 u = λ ∂ F ∂ u ( x, u ) in Ω ∂ u ∂ n = 0 on ∂ Ω ∂ ∂ n ( ξ ( x ) | Δ u | p ( x ) − 2 Δ u ) = 0 on ∂ Ω, where λ is a positive parameter, p ∈ C + ( Ω ¯ ) with p − := inf x ∈ Ω ¯ p ( x ) > 1, ξ is a function which satisfies the condition 0 < ξ 1 ≤ ξ ( x ) ≤ ξ 2, a ∈ L ∞ ( Ω ) with e s s i n f x ∈ Ω ¯ a ( x ) > 0 and F : Ω ¯ × R ⟶ R is a C 1 function. We prove the existence of a continuous family of eigenvalues in a neighbourhood of the origin, under some suitable conditions by using Ekeland's principle and variational method.