Small coupling limit and multiple solutions to the Dirichlet problem for Yang-Mills connections in four dimensions. II

Abstract

In this paper, we complete the proof of the existence of multiple solutions (and, in particular, non minimal ones), to the ∈-Dirichlet problem obtained as a variational problem for the SU(2)∈-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 J∈(q), by means of Ljusternik-Schnirelmann theory, thus complete the proofs of Theorems 1-3 given by Isobe and Marini ["Small coupling limit and multiple solutions to the Dirichlet Problem for Yang-Mills connections in 4 dimensions - Part I," J. Math. Phys.53, 063706 (2012)]. © 2012 American Institute of Physics.