groups or for constructing linking sets or local linkings. We will work with
a new sequence of eigenvalues introduced by the first author in [98] that
uses the Z2-cohomological index of Fadell and Rabinowitz. The necessary
background material on algebraic topology and the cohomological index will
be given in order to make the text as self-contained as possible.
One of the main points that we would like to make here is that, contrary
to the prevailing sentiment in the literature, the lack of a complete list of
eigenvalues is not a serious obstacle to effectively applying critical point the-
ory. Indeed, our sequence of eigenvalues is sufficient to adapt many of the
standard variational methods for solving semilinear problems to the quasi-
linear case. In particular, we will obtain nontrivial critical groups and use
the stability and piercing properties of the cohomological index to construct
new linking sets that are readily applicable to quasilinear problems. Of
course, such constructions cannot be based on linear subspaces since we no
longer have eigenspaces. We will instead use nonlinear splittings based on
certain sub- and superlevel sets whose cohomological indices can be precisely
calculated. We will also introduce a new notion of local linking based on
these splittings.
We will describe the general setting and give some examples in Chap-
ter 1, but first we give an overview of the theory developed here and a
preliminary survey chapter on Morse theoretic methods used in variational
problems in order to set up the history and context.
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