CHAPTER 1

Introduction

Given a domain Q C RN, we define the following functional on W1,P(Q):

;Fn(u) = /

| W ( X ) | P

+ ho(u(x)) dx .

Here above and in the sequel, we suppose that 1 p oo and that ho G

C°([—1,1])

nCrl'1((—1,1))

can be extended to a function which is C

1

in a neigh-

borhood of [—1,1]. We will also assume that, for some 0 c 1 C and some

(9* G (0,1), we have

for any 0 G [0,1], c0p h0(-l + 0) C6P and

(1.1) c0p ho(l-0) C0p,

for any 0 G [0,0*), ti0(-l + 0)

cdp~Y

and

(1.2) K{\-6)

-c0p-\

We also assume that h0 is monotone increasing in ( — 1,-1 + 6*) U (1 — 0*, 1).

Quantities depending only on the constants above will be referred to as "universal

constants". As a model example for a potential ho satisfying the conditions stated

here above, one may consider

/i

0

(C):=(l-C 2 ) P -

In the literature, ho is often referred to as a "double-well" potential, and its deriv-

ative as a "bi-stable nonlinearity".

In the light of (1.1) and (1.2), we have that, with no loss of generality, possibly

reducing the size of 0*, we may and do assume that

for any Ce [-1 + 0\ 1 - 0%

(1-3) MC) - max[-i,-i+0*]u[i-0*,i] ho,

l