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 of
front-type solutions for reaction diffusion equations.
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