Small coupling limit and multiple solutions to the Dirichlet Problem for Yang-Mills connections in 4 dimensions - Part II

Takeshi Isobe, Antonella Marini

Published 2010-05-28Version 1

In this paper we complete the proof of the existence of multiple solutions (and, in particular, non minimal ones), to the epsilon-Dirichlet problem obtained as a variational problem for the SU(2)-epsilon-Yang Mills functional. This is equivalent to proving the existence of multiple solutions to the Dirichlet problem for the SU(2)-Yang Mills functional with small boundary data. In the first paper of this series this non-compact variational problem is transformed into the finite dimensional problem of finding the critical points of the function J(q), which is essentially the Yang Mills functional evaluated on the approximate solutions, constructed via a gluing technique. In the present paper, we establish a Morse theory for this function, by means of Ljusternik-Schnirelmann theory, thus complete the proofs of the existence theorems (Theorems 1-3).