Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 118, pp. 1-21.
Title: Existence and multiplicity of heteroclinic solutions for a
non-autonomous boundary eigenvalue problem
Authors: Luisa Malaguti (Univ. of Modena and Reggio Emilia, Italy)
Cristina Marcelli (Polytechnic Univ. of Marche, Italy)
Abstract:
In this paper we investigate the boundary eigenvalue problem
$$\displaylines{
x''-\beta(c,t,x)x'+g(t,x)=0 \cr
x(-\infty)=0, \quad x(+\infty)=1
}$$
depending on the real parameter $c$. We take $\beta$ continuous
and positive and assume that $g$ is bounded and becomes active
and positive only when $x$ exceeds a threshold value
$\theta \in ]0,1[$. At the point $\theta$ we allow $g(t, \cdot)$
to have a jump. Additional monotonicity properties are required,
when needed. Our main discussion deals with the non-autonomous case.
In this context we prove the existence of a continuum of values $c$
for which this problem is solvable and we estimate the interval
of such admissible values. In the autonomous case, we show its
solvability for at most one $c^*$. In the special case when
$\beta$ reduces to $c+h(x)$ with $h$ continuous, we also give a
non-existence result, for any real $c$. Our methods combine
comparison-type arguments, both for first and second order
dynamics, with a shooting technique. Some applications of the
obtained results are included.
Submitted Submitted April 15, 2003. Published November 28, 2003.
Math Subject Classifications: 34B40, 34B18, 34C37.
Key Words: Boundary eigenvalue problems; positive bounded solutions;
shooting method