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:
\[
H
×
(
−
∞
,
λ
1
+
ε
)
a
m
p
;
→
≈
α
a
m
p
;
R
2
×
E
A
↓
a
m
p
;
a
m
p
;
↓
G
H
′
×
(
−
∞
,
λ
1
+
ε
)
a
m
p
;
→
≈
β
a
m
p
;
R
2
×
E
\begin {array}{*{20}{c}} {H \times ( - \infty ,{\lambda _1} + \varepsilon )} & {\xrightarrow [ \approx ]{\alpha }} & {{\mathbb {R}^2} \times E} \\ {A \downarrow } & {} & { \downarrow G} \\ {H’ \times ( - \infty ,{\lambda _1} + \varepsilon )} & {\xrightarrow [ \approx ]{\beta }} & {{\mathbb {R}^2} \times E} \\ \end {array}
\]
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
].