Positive homoclinic solutions for a class of second order differential equations

We study 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 epsilon R, where the coefficient functions (alpha(x), beta(x), and gamma(x) are continuous and satisfy 0 < a less than or equal to alpha(x), 0 less than or equal to b less than or equal to beta(x) less than or equal to B, 0 < c less than or equal to gamma(x) less than or equal to C. Assuming that the coefficient functions are 2 pi-periodic, we prove the existence of a nontrivial positive homoclinic solution of Eq. (1) whenever B-2 - b(2) < 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. (C) 1999 Academic Press.