2021 UNC Greensboro PDE Conference. Electron. J. Diff. Eqns., Conference 26 (2022), pp. 151-169.

On solutions arising from radial spatial dynamics of some semilinear elliptic equations

Dario A. Valdebenito

Abstract:
We consider the semilinear elliptic equation

where $x\in\mathbb{R}^N\setminus\{0\}$, N≥2, and f satisfies certain smoothness and structural assumptions. We construct solutions of the form $u(r,\phi)=r^{(2-N)/2} \tilde{u}(\log r,\phi)$, where r=|x|>0, $\phi\in\mathbb{S}^{N-1}$, and $\tilde{u}$ is quasiperiodic in its first argument with two nonresonant frequencies. These solutions are found using some recent developments in the theory of spatial dynamics, in which the radial variable r takes the role of time, combined with classical results from dynamical systems and the KAM theory.

Published August 25, 2022.
Math Subject Classifications: 35B08, 35B15, 35J61, 37J40.
Key Words: Semilinear elliptic equations; quasiperiodic solutions; center manifold theorem; radial spatial dynamics.
DOI: https://doi.org/10.58997/ejde.conf.26.v1

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Darío A. Valdebenito
Department of Mathematics
Ave Maria University
5050 Ave Maria Blvd
Ave Maria, FL 34142, USA
email: dario.valdebenito@avemaria.edu

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