Infinitely many homoclinic solutions for fourth-order differential equations with locally defined potentials

Abstract

In this paper, we are concerned with the existence of infinitely many homoclinic solutions for a class of fourth-order differential equations u (4) (x) + ωu (x) + a(x)u(x) = f (x, u(x)), ∀x ∈ R, where the function a ∈ C(R, R) may be negative on a bounded interval and the potential F(x, u) = u 0 f (x,t)dt is only locally defined near the origin with respect to the second variable. Some recent results in the literature are generalized and improved.