Multiplicity of solutions for a class of fourth-order elliptic equations of p(x)-Kirchhoff type

Abstract

This paper deals with a class of fourth order elliptic equations of Kirchhoff type with variable exponent

Δ²p(x) u − M( ∫_Ω 1/ p(x) |∇u|^p(x)dx) Δ_p(x) u + |u|^p(x)−2u = λ f(x,u) + μ g(x,u)  in Ω,
u = Δu =0 on ∂Ω,

where

p⁻ := inf _x∈\bar{Ω} p(x) > max{1, N/2}, λ>0 and μ ≥ 0 are real numbers, Ω ⊂ ℝ^N (N ≥ 1) is a smooth bounded domain, Δ²_p(x)u=Δ(|Δu|^p(x)−2Δu) is the operator of fourth order called the p(x)-biharmonic operator, Δ_p(x)u = div(|∇u|^p(x)–2∇u) is the p(x)-Laplacian, p : Ω -> ℝ is a log-Hölder continuous function, M : [0, +∞) -> ℝ is a continuous function and f, g : Ω × ℝ -> ℝ are two L¹-Carathéodory functions satisfying some certain conditions. Using two kinds of three critical point theorems, we establish the existence of at least three weak solutions for the problem in an appropriate space of functions.