Positive, unbounded and monotone solutions of the singular second Painlevé equation on the half-line

Abstract

In this paper we study singular boundary value problems on the half-line and we prove the existence of a global, monotone, positive and unbounded solution. The latter satisfies a Neumann condition at the origin and has prescribed asymptotic behavior at infinity. Our approach is based on a generalization of the Kneser's property (continuum) of the cross-sections of the solutions funnel, i.e. on the properties of the so-called consequent mapping and on properties of the associated vector field on the face space. Two applications, one on the well-known second Painlevé-type equation (which is related to superconductivity theory) and a second in the theory of colloids, clarify our results.