Pseudo-radial solutions of semi-linear elliptic equations on symmetric domains

TL;DR

This paper studies the existence and properties of non-radial, separable solutions to semi-linear elliptic equations on symmetric 2D domains, using phase plane analysis to characterize solutions on spherical and hyperbolic geometries.

Contribution

It provides a comprehensive description of pseudo-radial solutions for specific semi-linear elliptic equations on symmetric domains, extending analysis to spherical and hyperbolic cases.

Findings

Full characterization of pseudo-radial solutions for $ riangle u = ext{sign} imes a^2(|x|) u|u|^{q-1}$.

Analysis of solutions on spherical and hyperbolic symmetric domains.

Reduction of PDE problem to phase plane analysis of a dynamical system.

Abstract

In this paper we investigate existence and characterization of non-radial pseudo-radial (or separable) solutions of some semi-linear elliptic equations on symmetric 2-dimensional domains. The problem reduces to the phase plane analysis of a dynamical system. In particular, we give a full description of the set of pseudo-radial solutions of equations of the form $\Delta u = \pm a^2(|x|) u|u|^{q-1}$, with $q>0$, $q\neq 1$. We also study such equations over spherical or hyperbolic symmetric domains.