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Topological Methods in Nonlinear Analysis

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

Authors

  • Pablo Amster https://orcid.org/0000-0003-2829-7072
  • Mariel P. Kuna

Keywords

Second order ODEs, radiation boundary conditions, multiple solutions, electro-diffusion models

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 \cite{AKR3} 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 \cite{AKR3} regarding, on the one hand, the existence of sign-changing solutions and, on the other hand, exact multiplicity are solved.

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 II 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.

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Published

2019-07-13

How to Cite

1.
AMSTER, Pablo and KUNA, Mariel P. On exact multiplicity for a second order equation with radiation boundary conditions. Topological Methods in Nonlinear Analysis. Online. 13 July 2019. Vol. 54, no. 1, pp. 233 - 246. [Accessed 4 July 2025].
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Vol 54, No 1 (September 2019)

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