A reduction method for semilinear elliptic equations and solutions concentrating on spheres

We show that any general semilinear elliptic problem with Dirichlet or Neumann boundary conditions in an annulus A⊆R2m, m⩾2, invariant by the action of a certain symmetry group can be reduced to a nonhomogeneous similar problem in an annulus D⊂Rm+1, invariant by another related symmetry. We apply this result to prove the existence of positive and sign changing solutions of a singularly perturbed elliptic problem in A which concentrate on one or two (m−1) dimensional spheres. We also prove that the Morse indices of these solutions tend to infinity as the parameter of concentration tends to infinity.