A minimax inequality for a class of functionals and applications to the existence of solutions for two-point boundary-value problems

Abstract

In this paper, we establish an equivalent statement to minimax inequality for a special class of functionals. As an application, we prove the existence of three solutions to the Dirichlet problem -u″(x) + m(x)u(x) = λƒ(x, u(x)), x ∈ (α, b), u(α) = u(b) = 0, where λ > 0, ƒ : [α, b] x ℝ → ℝ is a continuous function which changes sign on [α, b] x ℝ and m(x) ∈ C ([α, b]) is a positive function.