Existence of bounded trajectories via upper and lower solutions

The paper deals with the boundary value problem on the whole line (P): u''-f(u,u')+g(u)=0, u(-∞)=0, u(+∞)=1, where g:R →R is a continuous non-negative function with support [0,1] and f:R^2 →R is a continuous function. By means of a new approach, based on a combination of lower and upper-solutions methods and phase-plane techniques, we prove an existence result for (P) when f is superlinear in u'; by a similar technique, we also get a non-existence one. As an application, we investigate the attractivity of the singular point (0,0) in the phase-plane (u,u'). We refer to a forthcoming paper for a further application in the field of front-type solutions for reaction-diffusion equations