Existence and uniqueness of positive even homoclinic solutions for second order differential equations

This paper is concerned with the existence of positive even homoclinic solutions for the p-Laplacian equation (vertical bar u'vertical bar(p-2)u')' - a(t)vertical bar u vertical bar(p-2)u + f(t,u) = 0, t is an element of R, where p >= 2 and the functions a and f satisfy some reasonable conditions. Using the Mountain Pass Theorem, we obtain the existence of a positive even homoclinic solution. In case p = 2, the solution obtained is unique under a condition of monotonicity on the function u bar right arrow f(t,u)/u. Some known results in the literature are generalized and significantly improved.