  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.
