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

Abstract

We consider the semilinear elliptic equation $$\Delta u+f(x,u)=0, $$ where \(x\in\mathbb{R}^N\setminus\{0\}\), \(N\geq 2,\) 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 \(r=|x|>0|0\), \(\phi\in\mathbb{S}^{N-1}\), 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. For more information see https://ejde.math.txstate.edu/conf-proc/26/v1/abstr.html