H\'enon type equations and concentration on spheres

TL;DR

This paper investigates the concentration behavior of symmetric solutions to semilinear elliptic equations, showing they blow up on spheres or points depending on symmetry, with implications for astrophysics and diffusion models.

Contribution

It proves that symmetric solutions in high-dimensional balls concentrate and blow up on spheres or points, extending understanding of solution profiles in elliptic problems.

Findings

Solutions concentrate on (m-1)-spheres as parameters grow

Axially symmetric solutions blow up at antipodal points

Results apply to models in astrophysics and diffusion phenomena

Abstract

In this paper we study the concentration profile of various kind of symmetric solutions of some semilinear elliptic problems arising in astrophysics and in diffusion phenomena. Using a reduction method we prove that doubly symmetric positive solutions in a $2m$-dimensional ball must concentrate and blow up on $(m-1)$-spheres as the concentration parameter tends to infinity. We also consider axially symmetric positive solutions in a ball in $\mathbb{R}^N$, $N \geq 3$, and show that concentration and blow up occur on two antipodal points, as the concentration parameter tends to infinity.