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 the previous work [2] for a Painlevé II equation are extended. It is proved that the uniqueness or multiplicity of solutions depend on the interaction between the mapping ∂g ∂u (·, 0) and the first eigenvalue of the associated linear operator. Furthermore, two open problems posed in [2] regarding, on the one hand, the existence of sign-changing solutions and, on the other hand, exact multiplicity are solved.