2 AN OVERVIE W

where here and in future example s Q denotes a bounded domai n i n R

n

whos e

boundary, 50 , i s a smoot h manifold . Suppos e / G C(ll). A funetion u i s a

elassical Solution of (1.1) if u G

C2(ü)

0 C(fl). Fo r such a Solution, multiplying

(1.1) by p G C§° (Q) yields

(1.2) f{Vu-

Ja

Vp-fp)dx = 0

after a n Integratio n b y parts. Le t

WQ

,2(Ü)

denot e the elosure of C§°(Q) with

respeet t o

If u G WQ

'2(fi)

an d satisfies (1.2) for all p G Q)°(n), then it is said to be a weak

Solution of (1.1). By our above remarks, any elassical Solution of (1.1) is a weak

Solution. Unde r slightly stronger hypotheses on / (e.g . / Holde r continuous) the

converse is also true. Choosin g E = Wo'

2(0),

set

(1.3) I(u)= t {\\Vu\* - fu)dx.

Ja

It is not difficult t o verify tha t I i s Frechet differentiable o n E an d

(1.4) I'(u)p= f (Vu-Vp-fp)dx

Ja

for p G E. Thu s u is a critical point of I i f and only if u is a weak Solution of

(i.D.

As was noted earlier, when E = R

n

th e most familiär sort s of critical points

obtained ar e maxim a o r minima . I n thes e lecture s w e will be dealing mainl y

with functionals whic h may not b e bounded from above or below even modulo

finite dimensional subspac^ or submanifolds. Suc h "indefinite " functionals may

not posses s any local maxima or minim a other tha n trivia l ones . Fo r example

let 0 = (0 , TT) C R, E = W^

52([0,

TT]), and

(1.5) I(u)= r ( | K |

2

- | u

4

) d x ,

Jo

where ' = d/dx . I t i s not difficult t o show that / i s differentiable o n E an d has

u = 0 as a local minimum. Fo r any other u G E an d a G R,

I(au) = J*(^\ur-^u*)dx

-0 0

as |a | — o o so i" is no t bounde d from below . Furthermor e fo r eac h k G N,

sin kx G U, and

I(sinkx)

jk2

- j — oo

as k —• oo so I is not bounded from above. Thus it is not obvious that I possesses

any critical points other than the trivial one u = 0. Nevertheless we will see later

as an application of the Mountain Pass Theorem thati* possesses positive critical

values and the same thing is true for higher dimensional versions of (1.5).