Nontrivial solutions on a kind of fourth-order Neumann boundary value problems

Nontrivial solutions for the fourth-order Neumann boundary value problem u(4)(t) − 2u″(t) + u(t) = f(t, u(t)) for all t ∈ (0, 1) and subject to u′(0) = u′(1) = u‴(0) = u‴(1) = 0 are obtained via Morse theory. To compute the critical groups at infinity of the relevant functional, we propose an approach by combining the homotopy and reduction methods.