Abstract:

In this paper, we consider the existence of the solution for the biharmonic
eigenvalue problem under Navier boundary condition
Δ²u=λa(x)u+f(x, u), x∈ Ω,
u=Δu=0, x∈ Ω,

where Ω is a bounded domain in R^N(N ≥ 5), Δ² is the biharmonic operator, and the weight function a(x) ∈L^r(Ω)(r ≥ N/4) with a(x)> 0 a.e. in Ω. By variational method, we obtain the second eigenvalue of this problem when f(x, u)=0 and study the structure of it, and discuss the existence of the nonzero solutions under resonance and nonresonance conditions.

Key words: Biharmonic operator, Eigenvalue, A variant of the Mountain-pass Lemma, Variational methods.