Multiplicity of solutions to the generalized extensible beam equations with critical growth

Abstract In this paper by variational methods, we study the multiplicity of solutions to the following fourth-order elliptic equations of Kirchhoff type with critical nonlinearity in R N : Δ 2 u − M ∫ R N | ∇ u | 2 d x Δ u + V ( x ) u = k ( x ) | u | q − 2 u + λ | u | 2 ∗ ∗ − 2 u , x ∈ R N , where Δ 2 u = Δ ( Δ u ) is the biharmonic operator, M : R + → R + is a continuous function, λ > 0 is a parameter, V : R N → R + is a potential function, and k ( x ) is a nonnegative continuous real valued function satisfying some conditions. The compactness condition is proved by the Lions’ second Concentration-compactness principle and Concentration-compactness principle at infinity.