  We study critical growth elliptic problems with jumping nonlinearities.
Standard linking arguments based on decompositions of $H^1_0(\Omega)$ into
eigenspaces of $- \Delta$ cannot be used to obtain nontrivial solutions to such
problems. We show that the associated variational functional admits certain
linking structures based on splittings of $H^1_0(\Omega)$ into nonlinear
submanifolds. In order to capture these linking geometries, we prove several
generalizations of the classical linking theorem of Rabinowitz that are not
based on linear subspaces. We then use these new linking theorems to obtain
nontrivial solutions of our problems. Our abstract results are of independent
interest and can be used to obtain nontrivial solutions of other types of
problems with jumping nonlinearities as well.
