CHAPTER 0
Morse Theory and Variational Problems
In this preliminary chapter we give a brief survey of Morse theoretic
methods used in variational problems. General references are Milnor [84],
Mawhin and Willem [81], Chang [28], and Benci [19].
We consider a real-valued function Φ defined on a real Banach space
pW, }¨}q. We say that Φ is Fr´ echet differentiable at u P W if there is an
element
Φ1puq
of the dual space pW
˚,
}¨}˚q,
called the Fr´ echet derivative of
Φ at u, such that
Φpu ` vq Φpuq `
Φ1puq,v
` op}v}q as v Ñ 0 in W,
where ¨, ¨ is the duality pairing. The functional Φ is continuously Fr´echet
differentiable on W , or belongs to the class
C1pW,
Rq, if
Φ1
is defined every-
where and the map W Ñ W
˚,
u Þ Ñ
Φ1puq
is continuous. We assume that
Φ P
C1pW,
Rq for the rest of the chapter. We say that u is a critical point
of Φ if
Φ1puq
0. A real number c P ΦpW q is a critical value of Φ if there is
a critical point u with Φpuq c, otherwise it is a regular value. We use the
notations
Φa u P W : Φpuq ě a
(
,
Φb
u P W : Φpuq ď b
(
, Φa
b
Φa X
Φb,
K u P W :
Φ1puq
0
(
,
Ă
W W zK, Ka
b
K X Φa,
b Kc

Kcc
for the various superlevel, sublevel, critical, and regular sets of Φ.
We begin by recalling the compactness condition of Palais and Smale and
its weaker variant given by Cerami in Section 0.1. Then we state the first
and second deformation lemmas under the Cerami’s condition in Section
0.2. In Section 0.3 we define the critical groups of an isolated critical point
and summarize the basic results of Morse theory. These include the Morse
inequalities, Morse lemma and its generalization splitting lemma, shifting
theorem of Gromoll and Meyer, and the handle body theorem. Next we dis-
cuss the minimax principle in Section 0.4. Section 0.5 contains a discussion
of homotopical linking, pairs of critical points with nontrivial critical groups
produced by homological linking, and nonstandard geometries without a fi-
nite dimensional closed loop. We recall the notion of local linking and an
alternative for a critical point produced by a local linking in Section 0.6.
We conclude with a result on nontrivial critical groups associated with the
p-Laplacian in Section 0.7.
1
http://dx.doi.org/10.1090/surv/161/01
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