Existence of Bounded Trajectories Via Upper and Lower Solutions

The paper deals with the boundary value problem (on the whole line) u''-f(u,u')+g(u)=0, u(-∞)=0, u(+∞)=1, where g is a continuous non-negative function with support [0, 1], and f 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 the problem 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'). Applications of these results in the field of front-type solutions for reaction diffusion equations can be found in L. Malaguti, C. Marcelli, Math. Nachr. 242 (2002), 148—164