The structure of a nonlinear elliptic operator
Authors: Church P. T. , Dancer E. N. , Timourian J. G. 
Published: June 2012

Consider the nonlinear Dirichlet problem

(
1
)
−
Δ
u
−
λ
u
+

u
3

=
g

(1) - \Delta u - \lambda u + {u^3} = g

, for

u
:
Ω
→

R

u:\Omega \to \mathbb {R}

,

u

|

∂
Ω
=
0

u|\partial \Omega = 0

, and

Ω
⊂

R

n

\Omega \subset {\mathbb {R}^n}

connected and bounded, and let

λ
i

{\lambda _i}

be the

i
i

th eigenvalue of

−
Δ
u

- \Delta u

on

Ω
\Omega

with

u

|

∂
Ω
=
0

u|\partial \Omega = 0

,

(
i
=
1
,
2
,
…
)

(i = 1,2, \ldots )

. Define a map

A
λ

:
H
→
H
′

{A_\lambda }:H \to H\prime

by

A
λ

(
u
)
=
−
Δ
u
−
λ
u
+

u
3

{A_\lambda }(u) = - \Delta u - \lambda u + {u^3}

, for either the Sobolev space

W
0

1
,
2

(
Ω
)
=
H
=
H
′

W_0^{1,2}(\Omega ) = H = H\prime

(if

n
≤
4
)

n \leq 4)

or the Hölder spaces

C
0

2
,
α

(

Ω
¯

)
=
H

C_0^{2,\alpha }(\bar \Omega ) = H

and

C

0
,
α

(

Ω
¯

)
=
H
′

{C^{0,\alpha }}(\bar \Omega ) = H\prime

(if

∂
Ω

\partial \Omega

is

C

2
,
α

{C^{2,\alpha }}

), and define

A
:
H
×

R

→
H
′
×

R

A:H \times \mathbb {R} \to H\prime \times \mathbb {R}

by

A
(
u
,
λ
)
=
(

A
λ

(
u
)
,
λ
)

A(u,\lambda ) = ({A_\lambda }(u),\lambda )

. Let

G
:

R

2

×
E
→

R

2

×
E

G:{\mathbb {R}^2} \times E \to {\mathbb {R}^2} \times E

be the global cusp map given by

G
(
s
,
t
,
v
)
=
(

s
3

−
t
s
,
t
,
v
)

G(s,t,v) = ({s^3} - ts,t,v)

, and let

F
:

R

×
E
→

R

×
E

F:\mathbb {R} \times E \to \mathbb {R} \times E

be the global fold map given by

F
(
t
,
v
)
=
(

t
2

,
v
)

F(t,v) = ({t^2},v)

, where

E
E

is any Fréchet space. Theorem 1. If

H
=
H
′
=

W
0

1
,
2

(
Ω
)

H = H\prime = W_0^{1,2}(\Omega )

, assume in addition that

n
⩽
3

n \leqslant 3

. There exit

ε
>
0

\varepsilon > 0

and homeomorphisms

α
\alpha

and

β
\beta

such that the following diagram commutes:
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 1st Column upper H times left-parenthesis negative normal infinity comma lamda 1 plus epsilon right-parenthesis 2nd Column a m p semicolon ModifyingBelow right-arrow With almost-equals Overscript alpha Endscripts 3rd Column a m p semicolon double-struck upper R squared times upper E 2nd Row 1st Column upper A down-arrow 2nd Column a m p semicolon 3rd Column a m p semicolon down-arrow upper G 3rd Row 1st Column upper H prime times left-parenthesis negative normal infinity comma lamda 1 plus epsilon right-parenthesis 2nd Column a m p semicolon ModifyingBelow right-arrow With almost-equals Overscript beta Endscripts 3rd Column a m p semicolon double-struck upper R squared times upper E EndLayout">

H
×
(
−
∞
,

λ
1

+
ε
)

a
m
p
;

→

≈

α

a
m
p
;

R

2

×
E

A
↓

a
m
p
;

a
m
p
;

↓
G

H
′

×
(
−
∞
,

λ
1

+
ε
)

a
m
p
;

→

≈

β

a
m
p
;

R

2

×
E

\begin {array}{*{20}{c}} {H \times ( - \infty ,{\lambda _1} + \varepsilon )} & {\xrightarrow [ \approx ]{\alpha }} & {{\mathbb {R}^2} \times E} \\ {A \downarrow } & {} & { \downarrow G} \\ {H’ \times ( - \infty ,{\lambda _1} + \varepsilon )} & {\xrightarrow [ \approx ]{\beta }} & {{\mathbb {R}^2} \times E} \\ \end {array}

\]
The analog for

A
λ

{A_\lambda }

with

λ
1

>
λ
>

λ
1

+
ε

{\lambda _1} > \lambda > {\lambda _1} + \varepsilon

is also given. In a very strong sense this theorem is a perturbation result for the problem (1): As

g
g

(and

λ
\lambda

) are perturbed, it shows how the number of solutions

u
u

of (1) varies; in particular, that number is always

1
1

,

2
2

or

3
3

for

λ
>

λ
1

+
ε

\lambda > {\lambda _1} + \varepsilon

. A point

u
∈
H

u \in H

is a fold point of

A
A

if the germ of

A
A

at

u
u

is

C
0

{C^0}

equivalent to the germ of

F
F

at

(
0
,
0
)

(0,0)

(i.e. under homeomorphic coordinate changes in domain near

u
u

and in range near

A
(
u
)

A(u)

,

A
A

becomes

F
F

), and the singular set

S
A

SA

is the set of points at which

A
A

fails to be a local diffeomorphism. For larger values of

λ
\lambda

our information is limited: Theorem 2. Consider the Sobolev case with

n
⩽
4

n \leqslant 4

and

∂
Ω

C
∞

\partial \Omega \,{C^\infty }

. For all

λ
∈

R

\lambda \in \mathbb {R}

, (i)

int
⁡
(
S
A
)
=
∅

\operatorname {int} (SA) = \emptyset

; (ii) there is a dense subset

Γ
\Gamma

in

S
A

SA

of fold points, and (iii) for

λ
>

λ
2

\lambda > {\lambda _2}

,

S
A

SA

[resp., for

n
⩽
3

n \leqslant 3

and

λ
>

λ
2

\lambda > {\lambda _2}

,

S
A
−
Γ

SA - \Gamma

] is a real analytic submanifold of codimension

1
1

in

H
×

R

H \times \mathbb {R}

[resp.,

S
A

SA

].