Multiple Solutions To Eigenvalue Problem Of Higher Order Nonlinear Elliptic Equation

In this paper, we study the existence of multiple weak solutions to eigenvalue problem of higher order nonlinear elliptic equation in W0m,p(Ω), where p>1, Ω is a bounded, convex, open domain, λ∈R,v is the outer normal derivative of (?)Ω,α、β∈Nn, m∈N, n≥mp. Suppose that A=A(x, ζ0,ζ1,ζm-1,η)is a continuous function on Ω×R×Rn×…×Rnm. we denote ζ=(ζ0,…,ζm-1)∈R×…×Rnm-1=Rnm-1/n-1. Assume that A=A(x,ζ,η) has continuous partial derivatives with respect to variables ζ,η,DηA(x,ζ,η)=a(x,ζ,η), A and f satisfy certain structural conditions and A(x,ζ,η) is strictly convex in77, namely, for all η1,η2∈Rnm,η1≠η2, which is weaker than the p-uniformly convexity assumed in [3] for second order equations. Applying a Three Critical Points Theorem in [7], we prove the existence of at least three solutions to (pλ) in W0m,p(Ω) for λ in some open interval (?)(?)R. This result extends the main result of [10] to the higher order equation and extends partially the main result of [8].