Existence and multiplicity results for a new p(x)-Kirchhoff problem

In this work, we study the existence and multiplicity results for the following nonlocal p(x)-Kirchhoff problem: −a−b∫Ω [Formula presented] |∇u|p(x)dxdiv(|∇u|p(x)−2∇u)=λ|u|p(x)−2u+g(x,u) in Ω,u=0, on ∂Ω,where a≥b>0 are constants, Ω⊂RN is a bounded smooth domain, p∈C(Ω¯) with N>p(x)>1, λ is a real parameter and g is a continuous function. The analysis developed in this paper proposes an approach based on the idea of considering a new nonlocal term which presents interesting difficulties.