Positive Homoclinic Solutions for a Class of Second Order Differential Equations

We study the existence of positive homoclinic solutions of the second order equation uYya x.uqb x.u2qg x.u3s0, xgR, I. where the coefficient functions a x., b x., and g x. are continuous and satisfy 0-aFa x., 0FbFb x.FB, 0-cFg x.FC. Assuming that the coefficient functions are 2p-periodic, we prove the existence of a nontrivial positive homoclinic solution of Eq. I. whenever B2yb2-4ac. 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