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

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 Painlevé 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.