## Topological Methods in Nonlinear Analysis

- Topol. Methods Nonlinear Anal.
- Volume 54, Number 1 (2019), 233-246.

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

Pablo Amster and Mariel P. Kuna

#### 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 $\frac {\partial g}{\partial u}(\cdot,0)$ 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.

#### Article information

**Source**

Topol. Methods Nonlinear Anal., Volume 54, Number 1 (2019), 233-246.

**Dates**

First available in Project Euclid: 16 July 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.tmna/1563242561

**Digital Object Identifier**

doi:10.12775/TMNA.2019.039

**Mathematical Reviews number (MathSciNet)**

MR4018278

**Zentralblatt MATH identifier**

07131282

#### Citation

Amster, Pablo; Kuna, Mariel P. On exact multiplicity for a second order equation with radiation boundary conditions. Topol. Methods Nonlinear Anal. 54 (2019), no. 1, 233--246. doi:10.12775/TMNA.2019.039. https://projecteuclid.org/euclid.tmna/1563242561