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


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

Author: Dario A. Valdebenito (Ave Maria Univ., Ave Maria, FL, USA)
 
Abstract: 
We consider the semilinear elliptic equation
$$
\Delta u+f(x,u)=0,
$$
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.