In this paper, the author establishes existence and multiplicity results for elliptic resonant problems
where is an eigenvalue of , and
for an appropriate constant , where is the eigenspace associated to . Let and , and consider the conditions
For and 2, under the conditions or with or , and suitable values of (with extra assumptions), the author obtain multiplicity results. The proofs rely on the Morse theory and new observations on the critical groups of degenerate critical points of a local linking type.