For a class of p(x)-biharmonic operators with weights

We study the fourth order nonlinear problem with a p ( x )-biharmonic operators Δ ( | Δ u | p 1 ( x ) - 2 Δ u ) + Δ ( | Δ u | p 2 ( x ) - 2 Δ u ) = λ V 1 ( x ) | u | q ( x ) - 2 u - μ V 2 ( x ) | u | α ( x ) - 2 u , x ∈ Ω u = Δ u = 0 , x ∈ ∂ Ω where Ω ∈ R N with N ≥ 2 is a bounded domain with smooth boundary, λ , μ are positive real numbers, p 1 , p 2 , q and α are continuous functions on Ω ¯ , V 1 and V 2 are weight functions in a generalized Lebesgue spaces L s 1 ( x ) ( Ω ) and L s 2 ( x ) ( Ω ) respectively such that V 1 may change sign in Ω and V 2 ≥ 0 on Ω . We established an existence results using variational approaches and Ekeland’s variational principle.