p(x)-Kirchhoff bi-nonlocal elliptic problem driven by both p(x)-Laplacian and p(x)-Biharmonic operators.

We investigate the existence of non-trivial weak solutions for the following p(x)-Kirchhoff bi-nonlocal elliptic problem driven by both p(x)-Laplacian and p(x)-Biharmonic operators { M (σ) ( Δ p (x) 2 u - Δ p (x) u) = λ ϑ (x) | u | q (x) - 2 u ( ∫ Ω ϑ (x) q (x) | u | q (x) d x ) r in Ω , u ∈ W 2 , p (.) (Ω) ∩ W 0 1 , p (.) (Ω) , under some suitable conditions on the continuous functions p, q, the non-negative function ϑ and M(σ), where σ : = ∫ Ω | Δ u | p (x) p (x) + | ∇ u | p (x) p (x) d x. Our main results is obtained by employing variational techniques and the well-known symmetric mountain pass lemma. [ABSTRACT FROM AUTHOR]