Three solutions of a fourth order elliptic problem via variational theorems of mixed type

Abstract

We prove the existence of two nontrivial solutions for the fourth order problem and u = 0 on ∂ω when λ₁≥c>λ_i+1 and either b<λ_k(λ_k-c) and b is close to λ_k(λ_k-c) where 2≤k≤i or b>λ_j(λ_j-c) and b is close to λ_j(λ_j-c) where j≥i+1. (Here (λ_i)i≥1 is the sequence of the eigenvalues of –Δin H¹ ₀(ω)). Moreover if c>λ₁, c is close to λ₁, b>λ_j(λ_j-c) and b is close to λ_j(λ_j-c) where j≥2 we get three non trivial solutions

Keywords:

• Multiple solutions
• variational theorems of mixed type
• aeroelastic oscillations