Solutions of second-order and fourth-order ODEs on the half-line

We start by studying the existence of positive solutions for the differential equation u″=a(x)u−g(u),u″=a(x)u−g(u), with u′(0)=u(+∞)=0u′(0)=u(+∞)=0, where aa is a positive function, and gg is a power or a bounded function. In other words, we are concerned with even positive homoclinics of the differential equation. The main motivation is to check that some well-known results concerning the existence of homoclinics for the autonomous case (where aa is constant) are also true for the non-autonomous equation.This also motivates us to study the analogous fourth-order boundary value problem {u(4)−cu″+a(x)u=|u|p−1uu′(0)=u‴(0)=0,u(+∞)=u′(+∞)=0 for which we also find nontrivial (and, in some instances, positive) solutions.