  We consider the semilinear elliptic boundary value problem \[ -\Delta
u=\left\vert u\right\vert ^{p-2}u\text{ in }\Omega,\text{\quad }u=0\text{ on
}\partial\Omega, \] in a bounded smooth domain $\Omega$ of $\mathbb{R}^{N}$ for
supercritical exponents $p>\frac{2N}{N-2}.$
  Until recently, only few existence results were known. An approach which has
been successfully applied to study this problem, consists in reducing it to a
more general critical or subcritical problem, either by considering rotational
symmetries, or by means of maps which preserve the Laplace operator, or by a
combination of both.
  The aim of this paper is to illustrate this approach by presenting a
selection of recent results where it is used to establish existence and
multiplicity or to study the concentration behavior of solutions at
supercritical exponents.
