Two positive solutions for nonlinear fourth-order elastic beam equations

The aim of this paper is to study the existence of at least two non-trivial solutions to a boundary value problem for fourth-order elastic beam equations given by u (4) + Au00 + Bu = λ f(x, u) in [0, 1], u(0) = u(1) = 0, u 00(0) = u 00(1) = 0, under suitable conditions on the nonlinear term on the right hand side. Our approach is based on variational methods, and in particular, on an abstract two critical points theorem given for differentiable functionals defined on a real Banach space