The paper gives conditions for the existence of non-trivial solutions to the fourth-order boundary value problem
as well as to the sixth-order one
in which , , are constants, and and are even, continuous, positive, -periodic functions. The proof of the results uses the variational method introduced by D. C. Clark [“A variant of the Lusternik-Schnirelman theory”, Indiana Univ. Math. J. 22, 65–74 (1972; Zbl 0228.58006)].