Multiplicity of solutions for a nonlinear nonlocal problem with variable exponent

This work deals with a class of value problem involving the $p(x)$-biharmonic elliptic equation {equation*} {gathered} M_1(∫Ω1/p(x)|Δ u|ᵖ⁽ˣ⁾dx)Δ^2ₚ₍ₓ₎u-M_2(∫Ω1/p(x)|∇ u|ᵖ⁽ˣ⁾dx)Δₚ₍ₓ₎u=λ f(x,u)+μ g(x,u) Ω,\\ u=Δ u=0, x∈ ∂ Ω, {gathered} {equation*} where Ω is a bounded domain in RN,~N≥1. with smooth boundary ∂ Ω. Our technical method is based on a theorem obtained by B. Ricceri.