Symmetry-breaking and Multiple Solutions For A Neumann Problem In An Exterior Domain

In this paper we prove existence of multiple positive solutions for a Neumann problem in R(N)\B(0, R), R large, with a superquadratic and odd nonlinearity. The proof is based on the fact that in such a situation the minimum of the corresponding energy functional (which is achieved) is not an even function and that there is quite a large gap (for large R) between such a minimum and the minimum of the same functional on even functions. In the set of functions whose energy lies in such a gap, we can apply index theory to prove the desired multiplicity result.