Electron. J. Diff. Eqns., Vol. 2003(2003), No. 118, pp. 1-21.

Existence and multiplicity of heteroclinic solutions for a non-autonomous boundary eigenvalue problem

Luisa Malaguti & Cristina Marcelli

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 April 15, 2003. Published November 28, 2003.
Math Subject Classifications: 34B40, 34B18, 34C37.
Key Words: Boundary eigenvalue problems, positive bounded solutions, shooting method

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Luisa Malaguti
Department of Engineering Sciences and Methods
University of Modena and Reggio Emilia
via Fogliani 1 - 42100 Reggio Emilia, Italy
email: malaguti.luisa@unimore.it
Cristina Marcelli
Department of Mathematical Sciences
Polytechnic University of Marche
via Brecce Bianche - 60131 Ancona, Italy
email: marcelli@dipmat.univpm.it

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