  We show the existence and multiplicity of concentrating solutions to a pure
Neumann slightly supercritical problem in a ball. This is the first existence
result for this kind of problems in the supercritical regime. Since the
solutions must satisfy a compatibility condition of zero average, all of them
have to change sign. Our proofs are based on a Lyapunov-Schmidt reduction
argument which incorporates the zero-average condition using suitable
symmetries. Our approach also guarantees the existence and multiplicity of
solutions to subcritical Neumann problems in annuli. More general symmetric
domains (e.g. ellipsoids) are also discussed.
