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 RN: Δ2u−M∫RN|∇u|2dxΔu+V(x)u=k(x)|u|q−2u+λ|u|2∗∗−2u,x∈RN, where Δ2u=Δ(Δu) is the biharmonic operator, M:R+→R+ is a continuous function, λ>0 is a parameter, V:RN→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.