CHAPTER 2

Modifications of the potential and of

one-dimensional solutions

We now construct some barriers, which will be of use in the proof of the main

results. Such barriers will be obtained by appropriate modifications on the potential

ho, which induce corresponding modifications on one-dimensional solutions.

Here and below, we fix Co 0, to be conveniently chosen in the following

(actually, during the proof of Proposition 2.13 here below). We will also fix R,

to be assumed suitably large (with respect to Co and some universal constants).

The first function needed in our construction is the following modification of the

potential ho in the interval [-3/4,3/4]:

DEFINITION 2.1. Fix |s

0

| 1/4. For any |s| 3/4, we define1

h0(s)Rp

(2.1)

tPs0As)

= ^(

5

) :=

R-Cois-soX^hois))*

(2.2)

Note that, by construction,

1

1 Co

R

so) •

Roughly speaking, for large R, if is close to ho: this is the reason for which we

consider (^asa modified potential in [—3/4, 3/4]. We now consider some properties

enjoyed by ip. First of all, we estimate (p in terms of ho in [—3/4, —1/2] U [1/2, 3/4]:

LEMMA 2.2. The following inequalities hold:

3 1'

(2.3)

and

(2.4)

p(s) h0{s) - - ^

ip{s) h0(s) +

2Co

R

if se

if se

1 3

2' 4

provided that R and

CQ/CQ

are suitably large. Also,

CQ

may be taken large if so is

C0.

PROOF. TO prove (2.3), note that for s G [ - f , - | ] we have C0{s - s0) G

[—Co, — ^ ] . Also, from (1.1), there exists k 0 such that

(2.5)

0 k inf

C T G [ - 3 / 4 , - 1 / 2 ] U [ 1 / 2 , 3 / 4 ] \p

(^°M):

1

Notice that R - C0(s - s0)(-^h0(s)) P 0 for any \s\ 3/4 and \s0\ 1/4, if R is large

enough, thus the definition of ip is well posed.

7