  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 $$ -\Delta u + u =
a(x)|u|^{p-2}u $$ in an annulus $A \subset \mathbb R^N$ ($N\ge3$). 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.
