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

Abstract

In this paper we investigate the boundary eigenvalue problem $$displaylines{ x''-eta(c,t,x)x'+g(t,x)=0 cr x(-infty)=0, quad x(+infty)=1 }$$ depending on the real parameter $c$. We take $eta$ continuous and positive and assume that $g$ is bounded and becomes active and positive only when $x$ exceeds a threshold value $heta in ]0,1[$. At the point $heta$ 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 $eta$ 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.

Keywords

• Boundary eigenvalue problems
• positive bounded solutions
• shooting method