## Topological Methods in Nonlinear Analysis

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

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

#### References

• P. Amster and M. P. Kuna, Multiple solutions for a second order equation with radiation boundary conditions, Electron. J. Qual. Theory Differ. Equ. 2017 (2017), no. 37, 1–11.
• P. Amster, M. K. Kwong and C. Rogers, A Painlevé \romII model in two-ion electrodiffusion with radiation boundary conditions, Nonlinear Anal. 16 (2013), 120–131.
• L. Bass, Electric structures of interfaces in steady electrolysis, Transf. Faraday. Soc. 60 (1964), 1656–1663.
• A. Castro, J. Cossio and J. M. Neuberger, A sign changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math. 27 (1997), 1041–1053.
• B. Grafov and A. Chernenko, Theory of the passage of a constant current through a solution of a binary electrolyte, Dokl. Akad. Nauk SSR 146 (1962), 135–138.
• Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc. 95 (1960), 101–123.