Existence of solutions for a p(x)-biharmonic problem under Neumann boundary conditions. Issue 10 (27th July 2021) - Hsini, Mounir; Irzi, Nawal; Kefi, Khaled
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.