Hénon type equations and concentration on spheres

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 ${\mathbb{R}}^N$, $N\ge 3$, and show that concentration and blow up occur on two antipodal points, as the concentration parameter tends to infinity.