  In the setting of saddle point reduction, we prove that the critical groups
of the original functional and the reduced functional are isomorphic. As
application, we obtain two nontrivial solutions for elliptic gradient systems
which may be resonant both at the origin and at infinity. The difficulty that
the variational functional does not satisfy the Palais-Smale condition is
overcame by taking advantage of saddle point reduction. Our abstract results on
critical groups are crucial.
