Note on the Uniqueness of a Global Positive Solution to the Second Painleve Equation

Abstract

The purpose of this note is to study the uniqueness of solutions to u'' - u3 + (t - c)u = 0, for t ∈ (0, + ∞) with Neumann condition at 0. Assuming a certain condition at infinity, Helfer and Weissler [6] have found a unique solution. We show that, without any assumptions at infinity, this problem has exactly one global positive solution. Moreover, the solution behaves like as √t as t approaches infinity.