  We use Morse theory to study impulsive problems. First we consider
asymptotically piecewise linear problems with superlinear impulses, and prove a
new existence result for this class of problems using the saddle point theorem.
Next we compute the critical groups at zero when the impulses are
asymptotically linear near zero, in particular, we identify an important
resonance set for this problem. As an application, we finally obtain a
nontrivial solution for asymptotically piecewise linear problems with impulses
that are asymptotically linear at zero and superlinear at infinity. Our results
here are based on the simple observation that the underlying Sobolev space
naturally splits into a certain finite dimensional subspace where all the
impulses take place and its orthogonal complement that is free of impulsive
effects.
