Existence of nontrivial weak solutions for p-biharmonic Kirchhoff-type equations

We are concerned with the following p -biharmonic equations: Δ p 2 u + M ( ∫ R N Φ 0 ( x , ∇ u ) d x ) div ( φ ( x , ∇ u ) ) + V ( x ) | u | p − 2 u = λ f ( x , u ) in R N , where 2 < 2 p < N , Δ p 2 u = Δ ( | Δ u | p − 2 Δ u ) , the function φ ( x , v ) is of type | v | p − 2 v , φ ( x , v ) = d d v Φ 0 ( x , v ) , the potential function V : R N → ( 0 , ∞ ) is continuous, and f : R N × R → R satisfies the Carathéodory condition. We study the existence of weak solutions for the problem above via mountain pass and fountain theorems.