Positive Homoclinic Solutions for a Class of Second Order Differential Equations

Abstract

We study the existence of positive homoclinic solutions of the second order equationu″−αxu+βxu2+γxu3=0,x∈R, (I)where the coefficient functions α(x), β(x), and γ(x) are continuous and satisfy0<a≤αx,0≤b≤βx≤B,0<c≤γx≤C.Assuming that the coefficient functions are 2π-periodic, we prove the existence of a nontrivial positive homoclinic solution of Eq. (I) wheneverB2−b2<4ac.This homoclinic is derived as the limit of positive solutions of some approximating problems that are obtained by using the mountain pass theorem. Using the same method we also prove under adequate assumptions the existence of positive symmetric homoclinic solutions.