Multiplicity Solutions to a Doubly Eigenvalue Fourth-Order Equation

In this paper, using a three critical points theorem and variational methods, we study the existence of at least three solutions for the following fourth-order boundary value problem on a compact interval [a, b] ⊆ R {u^iv + αu"+βu = λf(x, u) + μg(x, u) x ∈ (a,b), u(a) = u(b) = u"(a) = u"(b) = 0) where α, β are real constants and λ, μ are positive parameters and f, g: (a,b) × R → R are L²-Caratheodory functions.