On exact multiplicity for a second order equation with radiation boundary conditions

TL;DR

This paper investigates the solution multiplicity of a second order differential equation with radiation boundary conditions, extending previous results, and clarifies how solution uniqueness depends on the interaction between the nonlinearity and the linear operator's eigenvalues.

Contribution

It extends prior work on Painlevé II equations to more general equations, establishing conditions for solution multiplicity and solving open problems on sign-changing solutions and exact counts.

Findings

Solution multiplicity depends on the interaction between the derivative of g and the first eigenvalue.

Extended previous results to a broader class of equations with radiation boundary conditions.

Solved open problems regarding existence of sign-changing solutions and exact multiplicity.

Abstract

A second order ordinary differential equation with a superlinear term $g(x,u)$ under radiation boundary conditions is studied. Using a shooting argument, all the results obtained in a previous work for a Painlev\'e II equation are extended. It is proved that the uniqueness or multiplicity of solutions depend on the interaction between the mapping $\frac {\partial g}{\partial u}(\cdot,0)$ and the first eigenvalue of the associated linear operator. Furthermore, two open problems regarding, on the one hand, the existence of sign-changing solutions and, on the other hand, exact multiplicity are solved.