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where p > 1 and a : R → R satisfies: (a1) a ∈ L∞(R), inf a > 0, (a2) a = a∞ + a0, with a∞ T -periodic and a0(t)→ 0 as t→ ±∞. Noting that 0 is a hyperbolic rest point for (1.1), we look for homoclinic orbits to 0, namely non trivial solutions to (1.1) such that u(t) → 0 and u(t) → 0 as t→ ±∞. The homoclinic problem for equation (1.1), possibly with a more general nonlinearity, as well as the analogous subcritical elliptic problem on R, has been successfully studied with variational methods by several authors, for different kinds of behaviour of the coefficient a.

On the existence of homoclinic orbits for the asymptotically periodic Duffing equation