Existence of bounded trajectories via upper and lower solutions

Abstract

The paper deals with the boundary value problem on the whole line $u'' - f(u,u') + g(u) = 0 $ $u(\infty,1) = 0$, $ u(+\infty) = 1$ $(P)$ where $g : R \to R$ is a continuous non-negative function with support $[0,1]$, and $f : R^2\to R$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 [13] for a further application in the field offront-type solutions for reaction diffusion equations.