On a p ( x ) ‐Kirchhoff equation with critical exponent and an additional nonlocal term via truncation argument

We study existence and multiplicity of solutions of the following nonlocal p ( x ) ‐Kirchhoff equation with critical exponent, via truncation argument on the Sobolev space with variable exponent,− M ∫ Ω 1 p ( x )| ∇ u | p ( x )Δ p ( x ) u=λ f ( x , u )∫ Ω F ( x , u )r + | u | q ( x ) − 2 u ,inΩ ,u =0 ,on∂ Ω ,where Ω is a bounded smooth domain ofR N , p , q ∈ C ( Ω ¯ ) , M , f are continuous functions, 1 < p ( x ) < N , F ( x , u ) = ∫ 0 u f ( x , ξ )d ξ and λ , r > 0 are real parameter.