A supercritical elliptic equation in the annulus

By a combination of variational and topological techniques in the presence of invariant cones, we detect a new type of positive axially symmetric solutions of the Dirichlet problem for the elliptic equation -Δu + u = a(x)|u|^p-2u in an annulus A ⊂ ℝ^N (N ≥ 3). Here p > 2 is allowed to be supercritical and a(x) is an axially symmetric but possibly nonradial function with additional symmetry and monotonicity properties, which are shared by the solution u we construct. In the case where a equals a positive constant, we detect conditions, only depending on the exponent p and on the inner radius of the annulus, that ensure that the solution is nonradial.