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Δp u “ div `|∇u|p´2 ∇u˘, arises in non-Newtonian fluid flows, turbulent filtration in porous media,plasticity theory, rheology, glaciology, and in many other application areas;see, e.g., Esteban and V´ azquez [48] and Padial, Tak´
c, and Tello [90]. Prob-
lems involving the p-Laplacian have been studied extensively in the literature
during the last fifty years. However, only a few papers have used Morse the-
oretic methods to study such problems; see, e.g., Vannella [130], Cingolani
and Vannella [29, 31], Dancer and Perera [40], Liu and Su [74], Jiu and
Su [58], Perera [98, 99, 100], Bartsch and Liu [15], Jiang [57], Liu and Li
[75], Ayoujil and El Amrouss [10, 11, 12], Cingolani and Degiovanni [30],
Guo and Liu [55], Liu and Liu [73], Degiovanni and Lancelotti [43, 44], Liu
and Geng [70], Tanaka [129], and Fang and Liu [50]. The purpose of this
monograph is to fill this gap in the literature by presenting a Morse theo-
retic study of a very general class of homogeneous operators that includes
the p-Laplacian as a special case.
Infinite dimensional Morse theory has been used extensively in the lit- erature to study semilinear problems (see, e.g., Chang [28] or Mawhin and
Willem [81]). In this theory the behavior of a C1-functional defined on
a Banach space near one of its isolated critical points is described by its
critical groups, and there are standard tools for computing these groups
for the variational functional associated with a semilinear problem. They
include the Morse and splitting lemmas, the shifting theorem, and various
linking and local linking theorems based on eigenspaces that give critical
points with nontrivial critical groups. Unfortunately, none of them apply to
quasilinear problems where the Euler functional is no longer defined on a
Hilbert space or is C2 and there are no eigenspaces to work with. We will
systematically develop alternative tools, such as nonlinear linking and local
linking theories, in order to effectively apply Morse theory to such problems.
A complete description of the spectrum of a quasilinear operator such as the p-Laplacian is in general not available. Unbounded sequences of eigen-values can be constructed using various minimax schemes, but it is generallynot known whether they give a full list, and it is often unclear whether dif-ferent schemes give the same eigenvalues. The standard eigenvalue sequencebased on the Krasnoselskii genus is not useful for obtaining nontrivial critical 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 ofeigenvalues is not a serious obstacle to effectively applying critical point the-ory. Indeed, our sequence of eigenvalues is sufficient to adapt many of thestandard variational methods for solving semilinear problems to the quasi-linear case. In particular, we will obtain nontrivial critical groups and usethe stability and piercing properties of the cohomological index to constructnew linking sets that are readily applicable to quasilinear problems. Ofcourse, such constructions cannot be based on linear subspaces since we nolonger have eigenspaces. We will instead use nonlinear splittings based oncertain sub- and superlevel sets whose cohomological indices can be preciselycalculated. We will also introduce a new notion of local linking based onthese 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 apreliminary survey chapter on Morse theoretic methods used in variationalproblems in order to set up the history and context.

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