Homoclinic and Periodic Solutions for Some Classes of Second Order Differential Equations

Abstract

In this paper we state the existence of positive homoclinic solutions of the second order equation

$$ u - \alpha (x)u + \beta (x)u^2 + \gamma (x)u^3 = 0, x \in \mathbb{R}, $$

(I)

where the coefficient functions a(x)ß(x) and y(x) are continuous, positive and 27-periodic. We obtain, in some sense, generalizations of results contained in [10] and [6], where ß(x) is assumed identically zero. The homoclinic solution u of equation (I) is obtained as the limit of 2n7-periodic solutions of (I). We establish the fact that the quadratic form associated to the linear operator is positive definite and the particular type of the nonlinearity considered introduces simplicity and clearness in the proof, namely when we use the mountain pass lemma to study some periodic approximating problems. We present only the main ideas and sketch the proofs briefly. In Section 2, we study equation (I). The approximating procedure used in the proofs appears in several papers concerning the existence of homoclinics, namely in the case of Hamiltonian systems. We refer to Rabinowitz [10], Ambrosetti and Bertotti [1], Korman and Lazer [6], Arioli and Szulkin [2]. However those results do not apply to equation (I). Essentially, not only does the nonlinearity we consider not satisfy the hypotheses assumed there, but also [1], [10] and [2] do not concern positive solutions. For further details concerning Section 2, see[5].