July  2000, 6(3): 575-590. doi: 10.3934/dcds.2000.6.575

Existence of bounded trajectories via upper and lower solutions

1. 

Dipartimento di Matematica Pura e Applicata, Università di Modena e Reggio Emilia, via Campi 213/B, 41100 Modena, Italy

2. 

Dipartimento di Matematica e Informatica, Università di Perugia, via Vanvitelli 1, 06123 Perugia, Italy

Received  August 1999 Revised  April 2000 Published  April 2000

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.

Citation: Luisa Malaguti, Cristina Marcelli. Existence of bounded trajectories via upper and lower solutions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 575-590. doi: 10.3934/dcds.2000.6.575
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