The p-Laplacian operator
Δp u div
, p P p1, 8q
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 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
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
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 generally
not known whether they give a full list, and it is often unclear whether dif-
ferent schemes give the same eigenvalues. The standard eigenvalue sequence
based on the Krasnoselskii genus is not useful for obtaining nontrivial critical
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