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
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