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On a Nonlinear Eigenvalue Problem
Recent Topics in Nonlinear P D E I I I , Tokyo, 1986 Lecture Notes in Num. Appl. Anal., 9, 185-218 (1987)
On a Nonlinear Eigenvalue Problem Ken’ i c h i NAGASAKI Department of Mathematics, Faculty of Engineering Chiba Institute of Technology
and Takashi SUZUKI Department of Mathematics, Faculty of Science University of Tokyo
61. Introduction. In this paper we are concerned with the nonlinear boundary value groblem Au
(1.1)
(1.2)
(P):
+
Ae‘(x)
=
o
for
x 6
51
=
0
for
x
a51
u(x)
,
.
is a sim$ly connected and bounded domain in R 2 a2 a2 smooth boundary an , x = ( x l , x2) and A = 2+axl a x2
Here
51
deal with classical solutions negative
A,
so that
u
(u,
A )
of
is nonnegative in
with
. \?e
( P ) only for non-
n
by the maximum
principle. Nonlinear problems of this ty2e arise in the theory of thermal self-ignition of a chemically active mixture, in the theory of nonlinear heat generation, in the study of abstract surface with constant Gaussian curvature. 185
Ken'ichi NAGASAKI and Takashi SUZUKI
186
To begin with, we review briefly the results which have been known till now. Generally speaking, when
0
is a disk, radially symmetric
solutions of partial differential equations are reduced to those of ordinary ones, and in this way Gel'fand C71 studied radially symmetric solutions for (1.1) with (1.2) on = t 1x1 < R I .
Q
However, in our case, any nontrivial solution
is positive so that is radially symmetric from Gidas-Ni-
u
Gel ' f and C 7 1
Nirenberg C 8 1 .
there exists a positive if
X > X
*
X
*
has proved consequently that
, a unique solution if
solutions if
0 <
solutions of
(P)
<
X
( P ) has no solution
such that A
= X
*
In fact, for
A*.
and exactly two txlIxl < R),
Q =
the
are given as
X RL
for
A
E (0, X
*
1,
where
X
*
is equal to
-
(On the other
R2'
hand, Joseph and Lundgren C 9 J showed curious phenomena about the number of solutions for the same problem spherical domain
0
The nonlinearity
(P) with a
where 2 < n < 10). e(u) = eu : -+ R --t R -+ is monotone in-
in
R"
creasing and convex, where the monotone iteration method works well.
Keller and Cohen C113 showed that the spectrum of
that is the set of nonnegative solutions, is a closed interval X = X
*
(a) > 0
is determined by
X's
CO, X R.
*
1.
5 w
in
Q
has
Here,
For every
spectrum, there exists the minimal solution the solution satisfying
(P)
for which
PI
3
in the
X =
u
-X
namely
for any solution
w
Nonlinear Eigenvalue Problem
( P ) for the same
of
A .
Fujita [6]
187
and Laetsch [121
proved the nonexistence of the ordered triple of solutions of ( P ) for fixed
X.
( P ) is also
Nonlinear functional analytic approach to feasible (Keener-Keller [lo],
Crandall-Rabinowitz [41).
First, the implicit function theorem indicates that the minimal
(gx,A )
solutions
form a branch
and continues up to
(0,
0)
at
(x , u
*
x
*).
=
i
/B x*
which originates from and that z -f bends back
Secondly, the mountain-pass lemma, a modified
version of the Ljusternik-Schnirelman theory, shows the existence of a second solution for every
in
(0,
*
i )
the other hand, by the theory of Rabinowitz [16],
J
nent in
C(?i)
of the solutions of x
[Of
-).
(P)
containing
/B
([4 j ) .
On
the compois unbounded
Nevertheless nothing is known as to whether
this component contains the second solution in [4] and whether
d
x
has a branch which goes to infinity as
4
0.
A more detailed and delicate approach is possible with the
Liouville integral. z = x1
+ ix2,
z = x1
- ix2 reduces the equation (1.1) into
2 4 - a u + x eU = o ,
(1.3)
where
Note that the change of variables such as
-
azaZ
1 axl a - i-) a 2 a z = -(-
ax2
and
a
a - 1
a- - ~ ( % + i % ) '
a
Owing
to Liouville [131, every real solution of (1.3) is expressed in the form (1.4)
u
=
log
If' I
2
(l+;/f12)2
I
Ken'ichi NAGASAKI and Takashi SUZUKI
188
where
f(z)
is meromorphic with at most simple zeros and a.
simple poles in
With this expression, Schwarz' symmetriza-
tion and isoperimetric inequalities, Bandle [l-21 derived the lower and upper estimates of
A
*
= A
*
(a).
She also gave an a
priori estimate for a certain class of solutions, which will be precisely refered to later as Lemma 2.
Lastly, we look over
the results of Weston [18] and Moseley [14]. D C R2
connected domain
For a simply
satisfying appropriate conditions,
they constructed, by the method of singular perturbations, "large solutions" u blow up as
x
0.
C
*
which exist for positive
x
near
0 and
More exactly,
(1.5)
uniformly in
a
as
x
0
Here ,
g : D = Izl
(1.6)
is a Riemann mapping, that is, one-to-one and conformal mapping having a homeomorphic extension
-
-
g ; D
-+ n.
Moreover,
6 C
D
solves the equation
Thus as
*
u (x) brings about one-point blowing-up at
K
= g(6) E
n
ACO.
The main object of the present paper is to investigate the possibility of the connectedness between the branch of minimal solutions and that of Weston-Moseley's large solutions. We shall show the existence of a solution branch connecting
Nonlinear Eigenvalue Problem
n
them when
189
is close to a disc.
In this connection, De Figueiredo-Lions-Nussbaum [ S ] studied the equation AU
(P1 1
where
Q
c R2
+ xf(U)
= 0
for
x
C
n,
u = o
for
x
C
an,
f
;
is bounded and the nonlinearity
?
-
R
+
is monotone increasing, convex and superlinear (that is, f(tl = lim -
t++m t
and satisfies a growth condition at infinity:
+-)
f(t) = 0, where lim t++- ta
ilof in E
[O,
m)
C
containing
{(u,
but
They showed that the component
x)
J1Ix
€
is unbounded
(0, 0 )
2
E }
is bounded for each
Further they showed that the spectrum of
closed interval A
(P1)
solutions of
C(z) x
> 0.
> 0.
*
(0, X ) ,
[0,
(P1)
x
*
]
for some
A
*
> 0
(P,)
is a
and that for every
has at least two solutions on
J1.In
their proof employing the topological degree argument,essential are a priori estimates in
Lm(n)-sense of solutions.
Those estimates are derived from standard arguements by Sobolev's imbedding and the elliptic estimate.
However it
seems to be difficult to derive such estimates for solutions of
( P ) with the nonlinearity
a component
A
for
U
e
,
and hence to obtain such
( P ) in that way.
Further, even if such
a component can be proved topologically to exist, its relation with Weston-Moseley's solutions would be obscure. 5 2 . Heuristics and Statement of Main Result.
In the beginning of this section, we illustrate the const-
Ken'ichi NAGASAKI and Takashi SUZUKI
190
ruction of Weston-Moseley's large solutions. Substitution
F(z)
for
(+)lI2f ( z )
(1.4) yields
in
where
F(z)
f(z).
Besides, boundary condition (1.2) is reduced to the con-
satisfies the same conditions in
as those of
R
dition (2.2)
At this point, we recall the Riemann mapping
g
:
D cR
of
(1.6) and transform (2.2) into
(2.3)
with
G
=
F
0
g.
First, asymptotic solutions
G
=
G(0
as
At0
for (2.3)
is constructed in the following way (Weston [18]): Putting G
=
1-lI2G0
in (2.3), we have
modulo-0 ( All2 ) , (2.3).
which is the first asymptotic equation for
The n-th order equation will follow from putting
G = A-1/2EAPGp
dure.
in (2.3) and making modulo-0(1n/2) proce'P 0 We have to solve those asymptotic equations in turn.
The first equation (2.4) is reduced to
Nonlinear Eigenvalue Problem
if =
satisfies
A(<) A(<)
IA(r,)(
s h o u l d be t a k e n ,
case when
0
w e may t a k e
=
1
on
5
191
T o see how
c aD.
A
l e t u s c o n s i d e r f o r t h e moment t h e
is a unit disc:
= Ix = ( x l ,
x2)IlxI < 11.
Then
As i s d e s c r i b e d i n 51, t h e s o l u t i o n s f o r
g = id.
(P) are g i v e n as
c,
u -+ ( x ) = 2 l o g
(2.6)
with
C,
=
4
+&%i X
Therefore, we can set (2.7)
in (1.4
G+
.
Hence
5) = F+([)
-
with
h-1’2Go(~)
so t h a t ( 2 . 5 ) h o l d s f o r
A(c)
.
1
=
CL
[18]
Go(<) =
-65
as
X
J.
0,
I n view of t h i s , Weston
imposed i n ( 2 . 5 ) t h a t L
f o r some
6 f D.
The e q u a t i o n ( 1 . 7
follows f o r the f i r s t
e q u a t i o n ( 2 . 5 ) t o be s o l v a b l e . Under c e r t a i n a s s u m p t i o n s f o r
(or
n
g : D
-
a ) , he
showed t h e s o v a b i l i t y o f t h e a s y m p t o t i c e q u a t i o n s o f h i g h e r
.
order, and o b t a f n e d t h e f u n c t i o n (2.9)
and
+ Xe
U
=
o
(in
u
n
= un(x)
a)
on
such t h a t
Ken'ichi NAGASAKI and Takashi SUZUKI
192
un
(2.10) for
n = 1,2,
=
o(x")
...,
(on a a ) ,
1 O(1og T )
I I U ~ I I=
L-
x
as
+ O
Moseley [14] a d o p t e d a n o t h e r e x p r e s s i o n o f
t h e L i o u v i l l e i n t e g r a l (1.4), t h a t i s
v = v(z)
Here,
and
v"(z,)
= 0
i s h o l o m o r p h i c w i t h a t most s i m p l e z e r o s i n
when
v(zo) = 0
H e showed t h a t
a).
(zo €
a
then t h e assumptions f o r t h e s o l v a b i l i t y of asymptotic equat i o n s of any order i s r e d u c e d t o a s i m p l e c o n d i t i o n , t h a t i s Det(6)
#
0,
where
6
€
solves
D
1 . 7 ) and
D e t ( 5 ) = lg'
(2.12)
I t i s known i n t h e complex f u n c t i o n t h e o r y t h a t
least a solution
6 6 D
=
g(6)t a NOW,
has a t
f o r a n y s i m p l y c o n n e c t e d domain
F u r t h e r i n t h e case o f a c o n v e x domain K
(1.7)
i s unique and
a,
6
a.
and hence
E D
D e t ( 6 ) > 0.
t h e genuine l a r g e s o l u t i o n s f o r
by a m o d i f i e d N e w t o n ' s i t e r a t i o n t a k i n g s o l u t i o n as a s t a r t i n g p o i n t .
If
n 2 3
( P ) are c o n s t r u c t e d n-th order asymptotic and
h
> 0
is suffi-
c i e n t l y s m a l l , t h e i t e r a t i o n scheme c o n v e r g e s t o p r o d u c e a genuine s o l u t i o n , e x c e p t f o r a " p a t h o l o g i c a l case".
W e can
show t h a t t h i s p a t h o l o g i c a l case d o e s n o t h o l d when
Det(6) > 0
([15] c . f .
[17]).
T h e s e c o n d i t i o n s c a n b e s t a t e d more s i m p l y
i f we n o t e t h a t a s o l u t i o n 6
= 0
by composing
$Yr)
D
6 =
5+6
h a n d s i d e : g N = g a y : D + 8.
of
to Then,
( 1 . 7 ) i s reduced t o
g = g(5)
from t h e r i g h t -
( 1 . 7 ) reads:
Nonlinear Eigenvalue Problem
(2.13) and
193
gi(0) = 0
Det(6)
$
0
is equivalent to
Thus, we have arrived at the point to illustrate our idea. From the complex analytic viewpoint, the equation (2.2) can be interpreted as follows.
Let
denote the Riemann
K
sphere with unit diameter, tangent to w-plane at the origin, and let
be the point on
w
corresponding to
K
Further, let the linear elements on
Q
and on
ponding points by
do
=
F
be denoted by
F(z). K
Idz( and
at
corresdT,
res-
do holds. In this 1+IFI situation, the equality (2.2) implies that the length of F ( a 0 )
pectively.
on
K
Then the relation
d
=
is equal to
Similarly, from (2.1) the area of
F(n)
on
K
is given by
(2.16) which plays a key role in our theory. When
Q
is a unit disc, we can show that all solutions
of
( P ) is parametrized by
T.
In fact, the value
s
u+(x) is equal to
s,
for u? = 2
s
where
log
varies from
:
C ,T
A 2
1+-c a + 1x1
2
0
to
Ken'ichi NAGASAKI and Takashi SUZUKI
194
(2.17)
s = q 2 1
*
c 1
-
-1.
Inspired by the above example, we conceive to analyze the so ution branch of ( 2 16).
( P ) by the parametrization with w
c R2
aw
and
To state our result, let
ed domain with a smooth boundary
s
in
be a simply connectlet
gl:D,w be a Riemann mapping such that ciently small
gY(0) = 0.
Then, for suffi-
Igl,
becomes a Riemann mapping such that
g N , E ' ' ( 0 ) = 0,
where
RE
- g N f E ( D ) . In fact, the univalentness follows from Dorboux's
theorem. tions,
Then, a criterion for the existence of large solu'I'
'gN,E
( 0 ) / g i r E ( O ) ( < 2 , holds when
IE I
is sufficiently
small. Further, Theorem 1.
The branch of Weston-Moseley's large solutions
connects to that of minimal solutions provided that
(€1
is
sufficiently small. 53. Lemmas and Proofs.
In this section, we present five lemmas and afterwards complete the proof of Theorem 1. to the simple connectedness of
These lemmas are irrelevant
a.
The first lemma concerning a priori estimates of solutions in question is due to C. Bandle [ 3 ] . another proof.
However we will give
Nonlinear Eigenvalue Problem
Lemma 1. 0 < s <
If a s o l u t i o n
where
A,
(3.1)
s
(u,
of
A)
195
(P)
i s t h e r i g h t - h a n d s i d e of
s a t i s f es t h a t (2.16
,
then
S
5 - 2 log(l-;)
IlUII, m
Proof. Ix
For
Qlu(x) = t l
6
r(t)
case
t 2 0,
r(t)
and
6
{x
of
Qlu(x) > t )
i s o b v i o u s l y t h e b o u n d a r y of
argument i n 5 2 , t h e l e n g t h D(t)
Q ( t )d e n o t e t h e s e t s
and
L(t)
F ( n ( t ) ) on t h e s p h e r e
of K
F(
respectively.
n(t).
r ( t ))
In t h i s
From t h e a n d t h e area
are d e f i n e d such as
and
where
F
is t h e function i n (2.1).
From t h e f a c t t h a t
(u, X)
solves
(P),
(3.2) n
b e i n g t h e o u t e r n o r m a l of
Q(t).
Owing t o t h e co-area f o r m u l a [ 3 ] ,
(3.3)
we have t h e r e l a t i o n
Ken'ichi NAGASAKI and Takashi SUZUKI
196
With Schwarz' inequality we get from (3.2) and (3.3) the inequality
The isoperimetric inequality on the sphere
K
with unit
diameter means that (3.5)
and
From (3.4
.
t) I
L(tI2 3.5), we find D'(t)
1
-n-D0)7 Integration from
( J u ( ( ~with respect to
to
0
t
yields
m
the desired estimate. For
p
Co(3i)
6
,
Note that
D(0)
m
The set
{p.(p) )j=l
(--m
3
denotes its eigenvalues. tial operator
- A -
s.
we introduce the eigenvalue problem
y = o
(3.7)
=
p
for < p1(p) < p , ( p )
x c an.
5 u3(p) 5
...+-)
Further, A
denotes the differenP with the zero Dirichlet condi-
on
tion. (P)
We note that the linearized eigenvalue problem of with respect to with
p = xeU
with
U
p = xe
.
p,(pl
u
at
(u,
x)
is reduced to the above problem
It is known about the least eigenvalue that > 0
(respectively, pl(p)
= 0)
P
l(p)
Nonlinear Eigenvalue Problem
for the minimal solution =
*
(u,
x ( a ) , (respectively, x
X)
*
of
197
(P) with
0 <
x
c X*
= x ),
for any non-minimal solution
(u,
For these facts, we
X).
refer to Crandall-Rabinowitz [ 4 ] . The following lemma is also due to C. Bandle [l], from which we find that the solution
(u,
X)
with small
is the
s
minimal one. Lemma 2 .
p = XeU
with
Proof.
If
(u, A )
solves
(P)
and
s <
,
then
. This is an immediate consequence of Proposition A .
In fact, set
q = eU ,
KO =
K M = 4 s 0
and
M =
n
eU dx,
then
< 2 n .
This means that the assumption of Proposition A is satisfied.
Therefore we know that the least eigenvalue
v1
of the
following problem:
- A + - v eU J I = o
for
x 6 n ,
0
for
x c an ,
J,=
is greater than assertion.
2K0
= A.
Obviously this fact proves the
Q
Ken'ichi NAGASAKI and Takashi SUZUKI
198
Lemma 3.
p = Ae
with
(u,
If
s < n,
then
.
U
Denote the eigenfunction of (3.6) with ( 3 . 7 )
Proof.
corresponding to
nl
domains
solves (1.1) and
A)
e2,
p2(p) by n2.
and
then it has two nodal
Either
or s 2 s A J02
holds for
s
=
eiqenvalue of
+
s
-
A
s2,
eU dx c
while
- p on
2
p2(p)
or
Ol
J
O2
coincides with the least under the Dirichlet
boundary condition. Moreover, the corresponding eigenfunction is
4 4 a,
or
9
I
I
vious one.
.
O2
Hence the lemma follows from the pre-
0
The next lemma is useful in parametrizing the solution (u,
A)
of
P
in terms of
implicit function theorem.
for
h = (
U A
) C
X ?
S
X
R
as
s
=
J
sz
eUdx
with the aid of
We introduce the nonlinear function
Nonlinear Eigenvalue Problem
(i
AU
o(h,
(3.8)
S )
=
Here
X = C
2+a -
(n)
n Co(a)
+ xeU
s = 75
and
Y = Ca(n) w i t h (u,
0 < a < 1.
1)
of
(1.1)
eudx.
i n Note t h a t i t s F r e c h e t d e r i v a t i v e
(h, s)
.
z ;
e d x - -
Then i t s z e r o c h a r a c t e r i z e s t h e s o l u t i o n such t h a t
Y
6 f
8f)
n
199
with respect to
dh@ :
f
?
---f
at
i s g i v e n by t h e m a t r i x
h
For t h i s o p e r a t o r , we claim t h e f o l l o w i n g . Lemma 4 .
of
0,
whenever
Proof. f =
(
V p )
(3.9)
dho :
f
f
--p
Supposing t h a t
w e have
AV
+ pv +
pe'
(h, s )
0.
p,(p)
?
is i n v e r t i b l e a t a n y z e r o
=
dh@(h, s)[f] = 0
o
(in
:}
dx = 0 ,
n),
v =
o
with
(on
an)
pl(p)
> 0,
and
i,
(3.10)
where
p = ieU ( h =
v = 2 A-'(p), A P
(r)
p{v
so t h a t
+
).
I n t h e case o f
we have
Ken’ichi NAGASAKI and Takashi SUZUKI
200
Further, A -1 ( p ) > 0 P v = 0.
v1
If
yl
where have
and hence we get
Q),
= 0
p
and
( p ) = 0, we get from (3.9) that
is the first eigenfunction.
> 0
= 0
p
(in
and hence
v = constant
x
p > 0,
Since
9,.
Now
we
v = 0
0
follows from (3.10). From the above proof, we see that lent to
Since
I
q*
=
= 1
0
A-l(p)
P
a(dh@)
h
0 $6 a(Ap),
in the case of
+
0
satisfies
hg*
+
is equiva-
where
pep*
=
o
in
a,
we
have
Here, we note Lemma 5.
In the case of as -
(3.12)
where
ax s
0
8i
o(Ap) , we have
1,
is the right-hand side of (2.16).
the relation Proof. By virtue of 0 L o(Ap)i - -1 A-l ( p ) is obtained by differentiating
X P
in
A.
AU
u =xe
Hence we have
(in a ) ,
u = 0
(on aa)
v
5
a ax
U
Noniinear Eigenvalue Problem
dx = & j Q p { l
+
A-'(p P
20 1
-0 1 I.
ldx =
W e p r o v e Theorem 1 u n d e r t h e s e p r e p a r t i o n s ;
O u t l i n e of P r o o f o f Theorem 1:
qN,€ - gN,E(s) = 5
+
D
:
cq1(5)
R e c a l l t h e Riemann mapping with
7 QE
qN,E1l(0) = 0.
From t h e Weston-Moseley t h e o r y , t h e r e e x i s t p o s i t i v e c o n s t a n t s and
E~
such t h a t t h e l a r g e s o l u t i o n s
6
-
(3.13)
arise for e
=
{ ( u "A,
Er
hu = k e
(in
and
aE)
I E ~<
and
0 < A < 6 A)IO
U
E
u
~
.
*
of
A I E
(on
u = 0
anE)
$:
W e set
< A < 61.
A s i s d e s c r i b e d above,
t h r o u g h t h e mapping
@E
:
( 3 . 1 3 ) i s r e - f o r m u l a t e d as
2
R -+ ?
d e f i n e d as
hU i L e u
L
S) =
for -
h =
U
,
( A )
where
C2+a -
and
(nE)r\C:(RE)
(3.14)
s*(x,
E)
=
2
I
eUdx
= XE x
IR,
YE = C a ( z E ) U
-
0s
PE
= YE
with
IR,
x
X
E
Putting
0 < a < 1.
*
(yAfE), *
e 'IEdx
and
h;,€
=
OE
we have
*
*
@ E ( h A , Es,
(A,
o f t h e diffeomorphism
€1)
-'
gN,€
= 0
( 0 < A < 6,
: RE
-+ no = D,
IEI
<
E
~
)
.
In t e r m s
t h e problem ( 3 . 1 3 )
Ken'ichi NAGASAKI and Takashi SUZUKI
202
on
Then,
go
x
R into
Po
x
R,
which is denoted by
*
*
X , E - ~ N , Eand
=
=
H X r ~
(Po),
As for
Step 1.
I
Let it be
(PE).
will be transformed into another mapping from
OE
* UX,E
no.
is transformed into that on
QE
( ~"'),
Then, setting
FE.
we have
every solution
is
= uf , X
U,
given explicitly as in (2.6) and is parametrized by
s =
n
eUdx
with
0 < s < n.
Let it be ( 0 < s < n).
We have
OO(hsfO,s )
(0 < s < n )
=
O < s < n
0
.
Further, d h ~ o ( h s , O ,S
is invertible by Lemmas 4 and 5, and the express
ion (2.17).
be
s = s+(X), -
where
s+(h)
-
X = X o ( s ) is two-valued.
The inverse mapping of
--+
n
s-(X)
and
Let it
0
as
< cO)
is
X * 0.
Step 2.
as
X
F (0,
Lm-sense, so that
bounded in (Po)
For each
E
+
0
CU:fEII~I
6),
U
*
X f E
converges to a solution of
from the elliptic estimate.
Lm-norm of the limit function goes to it ought to be
u + ~:
-
as
Further, the XtO,
and hence
Nonlinear Eigenvalue Problem
203
Consequently, *
as
* 0.
E
Step 3 .
On the other hand, a branch
of solutions of
OE(h, s )
originating from
-8,
.
-1
gN,E*
*E
( f:::)
-
-
I ( y S I E,
=
=
exists for
(h, s ) = (0, 0 )
is,)
10 < s <
‘0’
let
.
0
TI)
hslE
Then, we have
h =
.
bs
0 < s <
by Lemmas 1 and 3 .
We set
Through the diffeomorphism
be transformed into
FE
TI,
(11s ,E
H
-s
,E
( 0 < s < n).
s) = 0
In the same way as in Step 2, we see that
as
E
* 0
uniform in
(0,
such that Is
-
sol <
W S f Ef
8 )
Take an 6).
i
>O
.
Further, the convergence is
1s - s o l
such that
so
clo-e to
. s- ,\ ,
TI
IE I
5
p1/21
(0, n ) .
such that
-is - invertible - -- - - - - - f -o - -r
from Steps 1 and 3 .
as a solut on set of K
(0, n
Then, there exist constants
d - .F ~ -f’ -Hs , P
6
on each compact interval contained in
s
Step 4.
xo(so) t
s
for each
E~
IF I < I -
I
and F -
-1
pl>
0
and
Consequently, the branch
has a local uniqueness property
FE(H, s )
=
0.
Namely
5 ~,/2, Is1 - s o l
(p1/2,
there exists a FE(H1, s l ) = 0
Ken'ichi NAGASAKI and Takashi SUZUKI
204
and
(IH1-HsOfEI)<~ imply
(H1,S1) C t ( ~ s , E f ~ ) ~ ~ ~ - S O ~ ( ~ 1 / 2 1 .
On t h e o t h e r h a n d , S t e p 2 i m p l i e s t h a t
and
as
E
*
so t h a t
(as
E
IE(
(Hi
0
(s
0
) f
S
Hence
0).
+
*
F u r t h e r we have
0.
-*
*
(Ao(so)f
.d*
*
I
E ) )
f t($s,E~
connects with
s (x0(so),
-
S ) I IS
-8, ,
€ 1 ) = 0,
SoI
5 ~1/21
provided t h a t
i s s u f f i c i e n t l y small.
54.
C o n c l u d i n g Remarks.
1.
If
n
i s star-shaped,
o r i g i n , and i f
(u,
x)
f o r instance, with respect t o t h e
solves
(PI, t h e n R e l l i c h ' s i d e n t i t y
yields that
where
[l]).
e d x , A = In1 a n d B = n W e always have B 2n. I n case
do.
s =
B
5
-
XA,
4n
(Bandle
and
s
n,
we get
1 and hence
(
4
-~ El2 5
(4s
x 5 8n(B - 2n)/AB.
-
El2 5 B
I n o t h e r words,
s < n
holds
205
Nonlinear Eigenvalue Problem
when
and
5 4n
B
-
X > X :8n(B
From Lemmas 1 a n d
2n)/AB.
3 , w e c a n show t h a t
<
x
Theorem 2 .
*
=
i o n s on
x
*
(0).
(X,
*
Suppose t h a t
5
B
2n)/AB
They f o r m a b r a n c h w h i c h c o n n e c t s t o t h a t
o f minimal s o l u t i o n s and bends a t 2.
-
A = 8n(B -
(P) h a s e x a c t l y t w o s o l u t -
Then, t h e p r o b l e m ).
and
4n
.
=
I t i s c o n v e n i e n t t o regard t h e o p e r a t o r
+ xeU
A
(4.2)
'he
=[
J
L 2 (.) as t h a t i n
.
-
8s eU
e U dx
under t h e D i r i c h l e t c o n d i t i o n f o r t h e f i r s t
x
R
X
component, r a t h e r t h a n a s t h a t on
t h e domain
D(dh B) =
H
a positivity-preserving
H1 O(a).
1
2
( 0 ) n V
x
R
into
Y x
R
V x
R
Then, w i t h
i t becomes s e l f - a d j o i n t a n d h a s
X
R
a n d compact r e s o l v e n t ,
The a s s o c i a t e d s y m m e t r i c b i l i n e a r f o r m
d e f i n e d on
.
V
being
,
bL(
)
is
through t h e r e l a t i o n
Actually f o r
V
V
v x
and
g =
(r)~ V x
R
V
,
Ken’ichi NAGASAKI and Takashi SUZUKI
206
with
p = Xe
U
.
V
We identify T
x
IR
V* = tv f
with
being the unit tangent vector on
1 a H (n)l=v a$,
= 0
on
an),
through the isomor-
phism
Then, the identity ( 4 . 3 ) reads (4.5)
where
a(-,
0
)
is the symmetric bilinear form on
a(v, w)
=
SQ
(Vv-Vw- pvw)dx
(v, w C H1(Q)).
Now we can introduce the self-adjoint operator associated with the form
a1 *
*
v
x
=
0
H1(Q) :
A*
P
in
L2(n)
such as
v
Namely,
with D(A*) = {v f H 2 ( a ) I
a v
on
an
and
P
Because of the relation ( 4 . 3 ) , (4.5) and ( 4 . 6 1 , invertibility of
dh
4
in
L2 (n 1 x
R
a
v
= 0).
the
*
is equivalent to that of A. P
Nonlinear Eigenvalue Problem
in
201
L*
-< ....
denotes t h e set of e i g e n v a l u e s of
For t h e c o n s t a n t f u n c t i o n
.a(c, c ) < 0.
ln11/2 1
C =
*
.
P we have
On a c c o u n t o f t h e mini-max p r i n c i p l e , t h i s means
*
that
A
v1 ( p )
Since t h e codimension of
< 0.
V
in
V
*
is one,
we have
*
P,(Pl
1. P1(P)
*
and
! J 3 ( P ) 1. P , ( P )
a l s o f r o m t h e mini-max p r i n c i p l e . s <
*
by Lemma 3.
TI
P,(p)
*
p2(p)
p3(p) > 0
if
A l s o , b y v i r t u e o f Lemma 4 w e h a v e
F u r t h e r , we have
> 0
if
P l ( p ) 1. 0 .
$ o
is equivalent t o
O u t l i n e of Proof:
y*
*
Therefore,
= 1
(p)
I =
do
$
0.
We see t h a t
+
A-’(p) P
E
V*
satisfies
*
a ( y , yj) where
yj
= 0
denotes t h e
that I = a(cp
*
j-th eigenfunction of
, y*)= -
F u r t h e r , t h e e q u i v a l e n c e between follows d i r e c t l y .
... ) ,
( j = 1,2,
I n t h e case o f
A
P
and a l s o
p y*dx. 0 €
5
(dh 0 )
I > 0,
and
I = 0
t h e closed subspace
Ken'ichi NAGASAKI and Takashi SUZUKI
208
Inf
a(v, v ) > 0,
*
v tV1
*
so that
> 0.
p2
*
9, + C 2
v = C1
For this
v,
the case
p i
Appendix.
we have = 0
*
VIC V
any closed subspace element
I < 0,
In the case of
Lp*
on the other hand,
of codimension 1 contains an such that
a(v, v) < 0,
I
holds only if
=
5
(C1,
=
so that 0.
P2
C2) 6 R2\rOl.
5 0.
Hence we have
However, P *2 < 0.
Generalized Schwarz Symmetrization and the estimates
of eigenvalues. ~ , ( p ) in ( 3 . 6 ) and (3.7)
The estimate of the eigenvalue with
p = keU
was derived by
C.
Bandle as an application of
generalized Schwarz Symmetrization.
We review briefly the
theory for reader's conven ence. Definition A.l. Riemannian metric For real constants rized domain
is a disk
{
D
*
Kotb
Let
doL KO
=
D
be a domain in
p x)ds 2 ,
and
where
b(> O ) ,
ds2
such that
=
dxl 2
+ dx2. 2
the generalized symmet-
with Riemannian metric
x 1x1 < R 1
R2 with
0
Nonlinear Eigenvalue Problem
For a closed domain disk
*
(XI
we define
D
*
as a closed
Kotb
1x1 2 R ) .
The area elements of Riemann surfaces
(DKo,b'
do) are denoted by
d T = p(x)dx
and
curvatures of :
D,
209
'log -2p(xI
and
d;
d; = c(r; K O , b)dx.
(D, do) and
*
( DKo f b
and
respectively, that is, We note that Gaussian
d^o) are equal to k(x)
u : D -IR,
For a function
*
*
uK o f b : DKo,b
-+
JR
we define a
as follows:
*
The function
u ~ ~ , which ~ ,is radially symmetric, is
called a generalized symmetrized function of
*
Henceforth abbreviated to
and
respectively.
KO
Definition A.2. function
dr
(D, do)
DKO,bf c(r; K o f b ) and D*, c(r)
and
u*
u.
Uiof,
may be
for simplicity.
Among the many interesting properties of the generalized symmetrized function, we take up two of them, which will be needed later. Lemma A.l.
Let
g(x)
and
h(x)
be continuous in
-
D,
then
ho = min h(x), hl = max h(x) D D = { x 6 D,Ih(x) > t), then we have Proof.
Let
and
D(t)
Ken'ichi NAGASAKI and Takashi SUZUKI
210
and
the inequality (A.1) follows. Lemma
A.2.
Let
be a domain in E2 and
D
be Gaussian curvature of assumption that
(D, d u) = (D,
k(x) 5 K O and
in Definition A.1, and
M =
KOM
dT
holds for a real analytic function in
D
and
u(x) = 0
on
m)ds). ,
4
I, J =
k(x)
D
where
p(x)dx,
u(x)
Under the is that
KO
the inequality
such that
u(x) 1. 0
aD.
Before proving the above lemma, we recall an isoperimetric inequality on a Riemann surface, that is, Bol's inequality: For any subdomain =
J Bp(x)dx
and
L~ =
B
of
D,
let
MB =
JB
dT
=JaBm ds denote the area
211
Nonlinear Eigenvalue Problem
of
B
and the length of
aBl
then
For
t 2 0,
(A.3)
Proof of Lemma A . 2 . a(t)
and
of
D(t), T(t),
a f t ) as follows:
For a real analytic function t(a)
we define
a(t)
decreasing in
u,
the inverse function of
is well-defined because (0, urnax). Moreover,
a(t)
a(t)
is strictly
and
t(a)
differentiable. On account of
a
co-area formula
- da dt
[
-
ds
r(t)
and Schwarz‘s inequality, we have
From an application of Bolls inequality follows (A.
41
(vulds 2 { 4 n
- K,,a(t)l
a(t)
(-
dt
are
) .
Ken’ichi NAGASAKI and Takashi SUZUKI
212
Combining (A.4) with another co-area formula
and integrating in
u 3 max
[0
with respect to
t,
we get
r
In the right-hand side of (A.5), the substitution a
with the relation
a
d; = J 1x1< r
equality
2 nbr 1+-K r 4 0
and
we use the fact that M
=
i,*
d;
=
nbR
lvu
*
I
=
du = - dr
-
ID*
Vu * 1 2 dx ,
dt
da
2nbr b 2) 2 (l+;rKor
2
b 2 . 1+-K R 4 0
The assertion follows from (A.5) and (A.6). We derive some estimates of the minimal eigenvalues v,l
*
v1
of the following eigenvalue problems:
+
for
yields the
,
22nr dr =
where
r
= o
for
x 6 D
= o
for
x E aD
I
,
Nonlinear Eigenvalue Problem
The comparison of
w1
*
with
213
*
for
x E D
for
x E aD
*
,
is given in the follow-
v1
ing . Lemma A . 3 . in Lemma A . 2 ,
Proof.
If
K O M < 477,
where
KO
and
are the same
M
we have
We introduce Rayleigh quotients
and
According to the variational characterization of the minimal eigenvalue, we have w1
=
inf
1 YeHo(D)
R[q]
and
w;
=
inf R*[$] 1 * $€HO(D
.
On account o f the positivity of the eigenfunction corresponding to
v1
and the denseness in
analytic functions, the minimizing sequence can be chosen such that every assumptions in Lemma A . 2 .
4,
(n
=
Ci(E)
of real
[qnt of
1, 2 , . . . )
R[q]
satisfies the
Hence, the application of Lemma A.1
Ken’ichi NAGASAKI and Takashi SUZUKI
214
and A . 2 to
9,
yields the estimate
The assertion follows immediately from ( A . 7 ) . Next we will calculate the minimal eigenvalue of for particular Lemma A . 4 . vl(Bp)
p
and
D.
For a disk
B,
= {XI 1x1 < P I ,
we denote by
the minimal eigenvalue of the eigenvalue problem: 4c
A Y + v
,
(E.P)
2 2 q = 0
(l+r
9 where
(E.P)
r = 1x1
and
C
for
x 6 B,
,
for
x E aBp
,
)
= o
is a positive constant.
Then we have for
P
=
1
and
Proof.
In polar coordinates
(E. P),
can be expressed
as follows.
9
(A.9)
Setting q(r,
8)
=
u(r)O(e)
= O
in ( A . 8 ) ,
for we get
r =
P .
Nonlinear Eigenvalue Problem
@(el
+
= an sin n e
On the other hand, (A.10)
1 F(rur r
(A.11)
u(0
bn cos n 8
,
0, 1, 2,
=
... ) .
is a solution of
u(r)
rn
(n
215
.
u(p) = 0 z =
By the substitution of
r2-1 2
and
r +1
v(z) = u(r),
we
transform (A.lO,ll) into the associated Legendre equation: t(1-z 2 dv
(A.12) d
n2 v + - 1-2
v(-1) <
(A.13)
V(+)
m,
p
=
2v
=
P 2-1) z c (-1,-
for
0
p
2-1
P
In case
Y C
=
0
2+1
.
+1
1, the equation (A.12) is satisfied with
and v(z) = - z . This means that - is an C2 C2 eigenvalue of (E.P)l with the corresponding eigenfunction
n = O ,
$(X)
=
vl(B1)
v = -
2
1-r . Moreover
l+r
-
C2
must be the minimal eigenvalue n
because of the positivity of
Cp(x)
=
1-i-' 7 in
l+r
B1'
The latter part of the assertion is a consequence of the monotonity of the minimal eigenvalue of to the size of the domain
Bp,
that is,
(E.P) with respect P.
Lastly we reach the position to state the main proposition in this Appendix. Proposition A .
If
KO > 0
and
KOM < 2 n
,
where K O
and
Ken'ichi NAGASAKI and Takashi SUZUKI
216
M
are the same as in Lemma A.2., then "1 > 2KOt
being the minimal eigenvalue of
v1
Proof.
Setting
b
equal to
4Ki1
(E.P).
in
v(r: K O , b),
we
can reduce the problem ( E . P ) to the problem ( E . P ) R in g instead of C 2 . In this situation, Lemma A.4 with K -1
R < 1
follows from the assumption
KOM < 2n
.
The application of Lemma A.3 and A.4 provides that v 1 = vl(BR) > 2K0
v1
.
Note :
1) The complex number
tE(e)
=
gk,E(eie)/(gh,E(e ie )
I
indicates the unit normal vector of a n e at ie gN,€(e ) f a n E . Therefore, the univalentness of
(eie E S1 = a D ) the point
g N I E on the boundary
aD
follows if the winding number of the
mapping tE is
:
e i e .s
s1 + + ~ € ( e )
E
s1
+1 and
Both hold for small.
E
=
0
and hence when
/ E /
is sufficiently
Nonlinear Eigenvalue Problem
217
References [l]
Bandle, C., Existence theorems, qualitative results and a priori bounds for a class of a nonlinear Dirichlet problems, Arch. Rat. Mech. Anal., 58 (1975) 219-238.
[2]
Bandle, C., Isoperimetric inequalities for a nonlinear eigenvalue problem, Proc. Amer. Math. SOC., 56 (1976) 243246.
[3]
Bandle, C., Isoperimetric Inequalities and Applications, Pitman, Boston/London/Melbourne, 1980.
[a]
Crandall, M.G., Rabinowitz, P.H., Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rat. Mech. Anal., 58 (1975) 207-218.
[5]
De Figueiredo, D.G., Lions, P.L., Nussbaum, R.D., A priori estimates and existence of positive solutions of nonlinear elliptic equations, J. Math. Pure Appl., 61 (1982) 41-63.
[6]
Fujita, H., On the nonlinear equations
U
Au+e = O
and av/at
V
=AV+e , Bull. Amer. Math. SOC., 75 (1969) 132-135. [7]
Gel'fand, I.M., Some problems in the theory of quasilinear equations, Amer. Math. SOC. Transl., l(2) 29 (1963) 295-381.
[a]
Gidas, B., Ni Wei-Ming, Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979)'209-243.
[9]
Joseph, D.D., Lundgren, T.S., Quasilinear Dirichlet problems driven by positive sources, Arch. Rat. Mech. Anal., 49 (1973) 241-269.
Ken'ichi NAGASAKI and Takashi SUZUKI
218
[lo] Keener, J.P., Keller, H.B., Positive solutions of convex nonlinear eigenvalue problem, J. Diff. Equations, 16 (1974) 103-125. [ll] Keller
H.B., Cohen, D.S., Some positive problems
suggested by nonlinear heat generation, J. Math. Mech., 16 (1967) 1361-1376. [12] Laetsch, T., On the number of solutions of boundary value problems with convex nonlinearities, J. Math. Anal. Appl., 35 (1971) 389-404. [13] Liouville, J., Sur l'equation aux derivees partielles (a2 log x)/auav+2xa 2= O f J. de Math., 18 (1853) 71-72. [14] Moseley, J.L., Asymptotic solutions for a Dirichlet problem with an exponential nonlinearity, SIAM J. Math. Anal., 14 (1983) 719-735. [15] Nagasaki, K., Suzuki, T., in preparation. [16] Rabinowitz, P.H., Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math., 3 (1973) 161-202. [17] Wente, H., Counter example to a conjecture of H, Hopf, Pacific J. Math., 121 (1986) 193-244. [l8] Weston, V.H., On the asymptotic solution of a partial differential equation with an exponential nonlinearity, SIAM J. Math. Anal., 9 (1978) 1030-1053.
On a Nonlinear Eigenvalue Problem Ken’ i c h i NAGASAKI Department of Mathematics, Faculty of Engineering Chiba Institute of Technology
and Takashi SUZUKI Department of Mathematics, Faculty of Science University of Tokyo
61. Introduction. In this paper we are concerned with the nonlinear boundary value groblem Au
(1.1)
(1.2)
(P):
+
Ae‘(x)
=
o
for
x 6
51
=
0
for
x
a51
u(x)
,
.
is a sim$ly connected and bounded domain in R 2 a2 a2 smooth boundary an , x = ( x l , x2) and A = 2+axl a x2
Here
51
deal with classical solutions negative
A,
so that
u
(u,
A )
of
is nonnegative in
with
. \?e
( P ) only for non-
n
by the maximum
principle. Nonlinear problems of this ty2e arise in the theory of thermal self-ignition of a chemically active mixture, in the theory of nonlinear heat generation, in the study of abstract surface with constant Gaussian curvature. 185
Ken'ichi NAGASAKI and Takashi SUZUKI
186
To begin with, we review briefly the results which have been known till now. Generally speaking, when
0
is a disk, radially symmetric
solutions of partial differential equations are reduced to those of ordinary ones, and in this way Gel'fand C71 studied radially symmetric solutions for (1.1) with (1.2) on = t 1x1 < R I .
Q
However, in our case, any nontrivial solution
is positive so that is radially symmetric from Gidas-Ni-
u
Gel ' f and C 7 1
Nirenberg C 8 1 .
there exists a positive if
X > X
*
X
*
has proved consequently that
, a unique solution if
solutions if
0 <
solutions of
(P)
<
X
( P ) has no solution
such that A
= X
*
In fact, for
A*.
and exactly two txlIxl < R),
Q =
the
are given as
X RL
for
A
E (0, X
*
1,
where
X
*
is equal to
-
(On the other
R2'
hand, Joseph and Lundgren C 9 J showed curious phenomena about the number of solutions for the same problem spherical domain
0
The nonlinearity
(P) with a
where 2 < n < 10). e(u) = eu : -+ R --t R -+ is monotone in-
in
R"
creasing and convex, where the monotone iteration method works well.
Keller and Cohen C113 showed that the spectrum of
that is the set of nonnegative solutions, is a closed interval X = X
*
(a) > 0
is determined by
X's
CO, X R.
*
1.
5 w
in
Q
has
Here,
For every
spectrum, there exists the minimal solution the solution satisfying
(P)
for which
PI
3
in the
X =
u
-X
namely
for any solution
w
Nonlinear Eigenvalue Problem
( P ) for the same
of
A .
Fujita [6]
187
and Laetsch [121
proved the nonexistence of the ordered triple of solutions of ( P ) for fixed
X.
( P ) is also
Nonlinear functional analytic approach to feasible (Keener-Keller [lo],
Crandall-Rabinowitz [41).
First, the implicit function theorem indicates that the minimal
(gx,A )
solutions
form a branch
and continues up to
(0,
0)
at
(x , u
*
x
*).
=
i
/B x*
which originates from and that z -f bends back
Secondly, the mountain-pass lemma, a modified
version of the Ljusternik-Schnirelman theory, shows the existence of a second solution for every
in
(0,
*
i )
the other hand, by the theory of Rabinowitz [16],
J
nent in
C(?i)
of the solutions of x
[Of
-).
(P)
containing
/B
([4 j ) .
On
the compois unbounded
Nevertheless nothing is known as to whether
this component contains the second solution in [4] and whether
d
x
has a branch which goes to infinity as
4
0.
A more detailed and delicate approach is possible with the
Liouville integral. z = x1
+ ix2,
z = x1
- ix2 reduces the equation (1.1) into
2 4 - a u + x eU = o ,
(1.3)
where
Note that the change of variables such as
-
azaZ
1 axl a - i-) a 2 a z = -(-
ax2
and
a
a - 1
a- - ~ ( % + i % ) '
a
Owing
to Liouville [131, every real solution of (1.3) is expressed in the form (1.4)
u
=
log
If' I
2
(l+;/f12)2
I
Ken'ichi NAGASAKI and Takashi SUZUKI
188
where
f(z)
is meromorphic with at most simple zeros and a.
simple poles in
With this expression, Schwarz' symmetriza-
tion and isoperimetric inequalities, Bandle [l-21 derived the lower and upper estimates of
A
*
= A
*
(a).
She also gave an a
priori estimate for a certain class of solutions, which will be precisely refered to later as Lemma 2.
Lastly, we look over
the results of Weston [18] and Moseley [14]. D C R2
connected domain
For a simply
satisfying appropriate conditions,
they constructed, by the method of singular perturbations, "large solutions" u blow up as
x
0.
C
*
which exist for positive
x
near
0 and
More exactly,
(1.5)
uniformly in
a
as
x
0
Here ,
g : D = Izl
(1.6)
is a Riemann mapping, that is, one-to-one and conformal mapping having a homeomorphic extension
-
-
g ; D
-+ n.
Moreover,
6 C
D
solves the equation
Thus as
*
u (x) brings about one-point blowing-up at
K
= g(6) E
n
ACO.
The main object of the present paper is to investigate the possibility of the connectedness between the branch of minimal solutions and that of Weston-Moseley's large solutions. We shall show the existence of a solution branch connecting
Nonlinear Eigenvalue Problem
n
them when
189
is close to a disc.
In this connection, De Figueiredo-Lions-Nussbaum [ S ] studied the equation AU
(P1 1
where
Q
c R2
+ xf(U)
= 0
for
x
C
n,
u = o
for
x
C
an,
f
;
is bounded and the nonlinearity
?
-
R
+
is monotone increasing, convex and superlinear (that is, f(tl = lim -
t++m t
and satisfies a growth condition at infinity:
+-)
f(t) = 0, where lim t++- ta
ilof in E
[O,
m)
C
containing
{(u,
but
They showed that the component
x)
J1Ix
€
is unbounded
(0, 0 )
2
E }
is bounded for each
Further they showed that the spectrum of
closed interval A
(P1)
solutions of
C(z) x
> 0.
> 0.
*
(0, X ) ,
[0,
(P1)
x
*
]
for some
A
*
> 0
(P,)
is a
and that for every
has at least two solutions on
J1.In
their proof employing the topological degree argument,essential are a priori estimates in
Lm(n)-sense of solutions.
Those estimates are derived from standard arguements by Sobolev's imbedding and the elliptic estimate.
However it
seems to be difficult to derive such estimates for solutions of
( P ) with the nonlinearity
a component
A
for
U
e
,
and hence to obtain such
( P ) in that way.
Further, even if such
a component can be proved topologically to exist, its relation with Weston-Moseley's solutions would be obscure. 5 2 . Heuristics and Statement of Main Result.
In the beginning of this section, we illustrate the const-
Ken'ichi NAGASAKI and Takashi SUZUKI
190
ruction of Weston-Moseley's large solutions. Substitution
F(z)
for
(+)lI2f ( z )
(1.4) yields
in
where
F(z)
f(z).
Besides, boundary condition (1.2) is reduced to the con-
satisfies the same conditions in
as those of
R
dition (2.2)
At this point, we recall the Riemann mapping
g
:
D cR
of
(1.6) and transform (2.2) into
(2.3)
with
G
=
F
0
g.
First, asymptotic solutions
G
=
G(0
as
At0
for (2.3)
is constructed in the following way (Weston [18]): Putting G
=
1-lI2G0
in (2.3), we have
modulo-0 ( All2 ) , (2.3).
which is the first asymptotic equation for
The n-th order equation will follow from putting
G = A-1/2EAPGp
dure.
in (2.3) and making modulo-0(1n/2) proce'P 0 We have to solve those asymptotic equations in turn.
The first equation (2.4) is reduced to
Nonlinear Eigenvalue Problem
if =
satisfies
A(<) A(<)
IA(r,)(
s h o u l d be t a k e n ,
case when
0
w e may t a k e
=
1
on
5
191
T o see how
c aD.
A
l e t u s c o n s i d e r f o r t h e moment t h e
is a unit disc:
= Ix = ( x l ,
x2)IlxI < 11.
Then
As i s d e s c r i b e d i n 51, t h e s o l u t i o n s f o r
g = id.
(P) are g i v e n as
c,
u -+ ( x ) = 2 l o g
(2.6)
with
C,
=
4
+&%i X
Therefore, we can set (2.7)
in (1.4
G+
.
Hence
5) = F+([)
-
with
h-1’2Go(~)
so t h a t ( 2 . 5 ) h o l d s f o r
A(c)
.
1
=
CL
[18]
Go(<) =
-65
as
X
J.
0,
I n view of t h i s , Weston
imposed i n ( 2 . 5 ) t h a t L
f o r some
6 f D.
The e q u a t i o n ( 1 . 7
follows f o r the f i r s t
e q u a t i o n ( 2 . 5 ) t o be s o l v a b l e . Under c e r t a i n a s s u m p t i o n s f o r
(or
n
g : D
-
a ) , he
showed t h e s o v a b i l i t y o f t h e a s y m p t o t i c e q u a t i o n s o f h i g h e r
.
order, and o b t a f n e d t h e f u n c t i o n (2.9)
and
+ Xe
U
=
o
(in
u
n
= un(x)
a)
on
such t h a t
Ken'ichi NAGASAKI and Takashi SUZUKI
192
un
(2.10) for
n = 1,2,
=
o(x")
...,
(on a a ) ,
1 O(1og T )
I I U ~ I I=
L-
x
as
+ O
Moseley [14] a d o p t e d a n o t h e r e x p r e s s i o n o f
t h e L i o u v i l l e i n t e g r a l (1.4), t h a t i s
v = v(z)
Here,
and
v"(z,)
= 0
i s h o l o m o r p h i c w i t h a t most s i m p l e z e r o s i n
when
v(zo) = 0
H e showed t h a t
a).
(zo €
a
then t h e assumptions f o r t h e s o l v a b i l i t y of asymptotic equat i o n s of any order i s r e d u c e d t o a s i m p l e c o n d i t i o n , t h a t i s Det(6)
#
0,
where
6
€
solves
D
1 . 7 ) and
D e t ( 5 ) = lg'
(2.12)
I t i s known i n t h e complex f u n c t i o n t h e o r y t h a t
least a solution
6 6 D
=
g(6)t a NOW,
has a t
f o r a n y s i m p l y c o n n e c t e d domain
F u r t h e r i n t h e case o f a c o n v e x domain K
(1.7)
i s unique and
a,
6
a.
and hence
E D
D e t ( 6 ) > 0.
t h e genuine l a r g e s o l u t i o n s f o r
by a m o d i f i e d N e w t o n ' s i t e r a t i o n t a k i n g s o l u t i o n as a s t a r t i n g p o i n t .
If
n 2 3
( P ) are c o n s t r u c t e d n-th order asymptotic and
h
> 0
is suffi-
c i e n t l y s m a l l , t h e i t e r a t i o n scheme c o n v e r g e s t o p r o d u c e a genuine s o l u t i o n , e x c e p t f o r a " p a t h o l o g i c a l case".
W e can
show t h a t t h i s p a t h o l o g i c a l case d o e s n o t h o l d when
Det(6) > 0
([15] c . f .
[17]).
T h e s e c o n d i t i o n s c a n b e s t a t e d more s i m p l y
i f we n o t e t h a t a s o l u t i o n 6
= 0
by composing
$Yr)
D
6 =
5+6
h a n d s i d e : g N = g a y : D + 8.
of
to Then,
( 1 . 7 ) i s reduced t o
g = g(5)
from t h e r i g h t -
( 1 . 7 ) reads:
Nonlinear Eigenvalue Problem
(2.13) and
193
gi(0) = 0
Det(6)
$
0
is equivalent to
Thus, we have arrived at the point to illustrate our idea. From the complex analytic viewpoint, the equation (2.2) can be interpreted as follows.
Let
denote the Riemann
K
sphere with unit diameter, tangent to w-plane at the origin, and let
be the point on
w
corresponding to
K
Further, let the linear elements on
Q
and on
ponding points by
do
=
F
be denoted by
F(z). K
Idz( and
at
corresdT,
res-
do holds. In this 1+IFI situation, the equality (2.2) implies that the length of F ( a 0 )
pectively.
on
K
Then the relation
d
=
is equal to
Similarly, from (2.1) the area of
F(n)
on
K
is given by
(2.16) which plays a key role in our theory. When
Q
is a unit disc, we can show that all solutions
of
( P ) is parametrized by
T.
In fact, the value
s
u+(x) is equal to
s,
for u? = 2
s
where
log
varies from
:
C ,T
A 2
1+-c a + 1x1
2
0
to
Ken'ichi NAGASAKI and Takashi SUZUKI
194
(2.17)
s = q 2 1
*
c 1
-
-1.
Inspired by the above example, we conceive to analyze the so ution branch of ( 2 16).
( P ) by the parametrization with w
c R2
aw
and
To state our result, let
ed domain with a smooth boundary
s
in
be a simply connectlet
gl:D,w be a Riemann mapping such that ciently small
gY(0) = 0.
Then, for suffi-
Igl,
becomes a Riemann mapping such that
g N , E ' ' ( 0 ) = 0,
where
RE
- g N f E ( D ) . In fact, the univalentness follows from Dorboux's
theorem. tions,
Then, a criterion for the existence of large solu'I'
'gN,E
( 0 ) / g i r E ( O ) ( < 2 , holds when
IE I
is sufficiently
small. Further, Theorem 1.
The branch of Weston-Moseley's large solutions
connects to that of minimal solutions provided that
(€1
is
sufficiently small. 53. Lemmas and Proofs.
In this section, we present five lemmas and afterwards complete the proof of Theorem 1. to the simple connectedness of
These lemmas are irrelevant
a.
The first lemma concerning a priori estimates of solutions in question is due to C. Bandle [ 3 ] . another proof.
However we will give
Nonlinear Eigenvalue Problem
Lemma 1. 0 < s <
If a s o l u t i o n
where
A,
(3.1)
s
(u,
of
A)
195
(P)
i s t h e r i g h t - h a n d s i d e of
s a t i s f es t h a t (2.16
,
then
S
5 - 2 log(l-;)
IlUII, m
Proof. Ix
For
Qlu(x) = t l
6
r(t)
case
t 2 0,
r(t)
and
6
{x
of
Qlu(x) > t )
i s o b v i o u s l y t h e b o u n d a r y of
argument i n 5 2 , t h e l e n g t h D(t)
Q ( t )d e n o t e t h e s e t s
and
L(t)
F ( n ( t ) ) on t h e s p h e r e
of K
F(
respectively.
n(t).
r ( t ))
In t h i s
From t h e a n d t h e area
are d e f i n e d such as
and
where
F
is t h e function i n (2.1).
From t h e f a c t t h a t
(u, X)
solves
(P),
(3.2) n
b e i n g t h e o u t e r n o r m a l of
Q(t).
Owing t o t h e co-area f o r m u l a [ 3 ] ,
(3.3)
we have t h e r e l a t i o n
Ken'ichi NAGASAKI and Takashi SUZUKI
196
With Schwarz' inequality we get from (3.2) and (3.3) the inequality
The isoperimetric inequality on the sphere
K
with unit
diameter means that (3.5)
and
From (3.4
.
t) I
L(tI2 3.5), we find D'(t)
1
-n-D0)7 Integration from
( J u ( ( ~with respect to
to
0
t
yields
m
the desired estimate. For
p
Co(3i)
6
,
Note that
D(0)
m
The set
{p.(p) )j=l
(--m
3
denotes its eigenvalues. tial operator
- A -
s.
we introduce the eigenvalue problem
y = o
(3.7)
=
p
for < p1(p) < p , ( p )
x c an.
5 u3(p) 5
...+-)
Further, A
denotes the differenP with the zero Dirichlet condi-
on
tion. (P)
We note that the linearized eigenvalue problem of with respect to with
p = xeU
with
U
p = xe
.
p,(pl
u
at
(u,
x)
is reduced to the above problem
It is known about the least eigenvalue that > 0
(respectively, pl(p)
= 0)
P
l(p)
Nonlinear Eigenvalue Problem
for the minimal solution =
*
(u,
x ( a ) , (respectively, x
X)
*
of
197
(P) with
0 <
x
c X*
= x ),
for any non-minimal solution
(u,
For these facts, we
X).
refer to Crandall-Rabinowitz [ 4 ] . The following lemma is also due to C. Bandle [l], from which we find that the solution
(u,
X)
with small
is the
s
minimal one. Lemma 2 .
p = XeU
with
Proof.
If
(u, A )
solves
(P)
and
s <
,
then
. This is an immediate consequence of Proposition A .
In fact, set
q = eU ,
KO =
K M = 4 s 0
and
M =
n
eU dx,
then
< 2 n .
This means that the assumption of Proposition A is satisfied.
Therefore we know that the least eigenvalue
v1
of the
following problem:
- A + - v eU J I = o
for
x 6 n ,
0
for
x c an ,
J,=
is greater than assertion.
2K0
= A.
Obviously this fact proves the
Q
Ken'ichi NAGASAKI and Takashi SUZUKI
198
Lemma 3.
p = Ae
with
(u,
If
s < n,
then
.
U
Denote the eigenfunction of (3.6) with ( 3 . 7 )
Proof.
corresponding to
nl
domains
solves (1.1) and
A)
e2,
p2(p) by n2.
and
then it has two nodal
Either
or s 2 s A J02
holds for
s
=
eiqenvalue of
+
s
-
A
s2,
eU dx c
while
- p on
2
p2(p)
or
Ol
J
O2
coincides with the least under the Dirichlet
boundary condition. Moreover, the corresponding eigenfunction is
4 4 a,
or
9
I
I
vious one.
.
O2
Hence the lemma follows from the pre-
0
The next lemma is useful in parametrizing the solution (u,
A)
of
P
in terms of
implicit function theorem.
for
h = (
U A
) C
X ?
S
X
R
as
s
=
J
sz
eUdx
with the aid of
We introduce the nonlinear function
Nonlinear Eigenvalue Problem
(i
AU
o(h,
(3.8)
S )
=
Here
X = C
2+a -
(n)
n Co(a)
+ xeU
s = 75
and
Y = Ca(n) w i t h (u,
0 < a < 1.
1)
of
(1.1)
eudx.
i n Note t h a t i t s F r e c h e t d e r i v a t i v e
(h, s)
.
z ;
e d x - -
Then i t s z e r o c h a r a c t e r i z e s t h e s o l u t i o n such t h a t
Y
6 f
8f)
n
199
with respect to
dh@ :
f
?
---f
at
i s g i v e n by t h e m a t r i x
h
For t h i s o p e r a t o r , we claim t h e f o l l o w i n g . Lemma 4 .
of
0,
whenever
Proof. f =
(
V p )
(3.9)
dho :
f
f
--p
Supposing t h a t
w e have
AV
+ pv +
pe'
(h, s )
0.
p,(p)
?
is i n v e r t i b l e a t a n y z e r o
=
dh@(h, s)[f] = 0
o
(in
:}
dx = 0 ,
n),
v =
o
with
(on
an)
pl(p)
> 0,
and
i,
(3.10)
where
p = ieU ( h =
v = 2 A-'(p), A P
(r)
p{v
so t h a t
+
).
I n t h e case o f
we have
Ken’ichi NAGASAKI and Takashi SUZUKI
200
Further, A -1 ( p ) > 0 P v = 0.
v1
If
yl
where have
and hence we get
Q),
= 0
p
and
( p ) = 0, we get from (3.9) that
is the first eigenfunction.
> 0
= 0
p
(in
and hence
v = constant
x
p > 0,
Since
9,.
Now
we
v = 0
0
follows from (3.10). From the above proof, we see that lent to
Since
I
q*
=
= 1
0
A-l(p)
P
a(dh@)
h
0 $6 a(Ap),
in the case of
+
0
satisfies
hg*
+
is equiva-
where
pep*
=
o
in
a,
we
have
Here, we note Lemma 5.
In the case of as -
(3.12)
where
ax s
0
8i
o(Ap) , we have
1,
is the right-hand side of (2.16).
the relation Proof. By virtue of 0 L o(Ap)i - -1 A-l ( p ) is obtained by differentiating
X P
in
A.
AU
u =xe
Hence we have
(in a ) ,
u = 0
(on aa)
v
5
a ax
U
Noniinear Eigenvalue Problem
dx = & j Q p { l
+
A-'(p P
20 1
-0 1 I.
ldx =
W e p r o v e Theorem 1 u n d e r t h e s e p r e p a r t i o n s ;
O u t l i n e of P r o o f o f Theorem 1:
qN,€ - gN,E(s) = 5
+
D
:
cq1(5)
R e c a l l t h e Riemann mapping with
7 QE
qN,E1l(0) = 0.
From t h e Weston-Moseley t h e o r y , t h e r e e x i s t p o s i t i v e c o n s t a n t s and
E~
such t h a t t h e l a r g e s o l u t i o n s
6
-
(3.13)
arise for e
=
{ ( u "A,
Er
hu = k e
(in
and
aE)
I E ~<
and
0 < A < 6 A)IO
U
E
u
~
.
*
of
A I E
(on
u = 0
anE)
$:
W e set
< A < 61.
A s i s d e s c r i b e d above,
t h r o u g h t h e mapping
@E
:
( 3 . 1 3 ) i s r e - f o r m u l a t e d as
2
R -+ ?
d e f i n e d as
hU i L e u
L
S) =
for -
h =
U
,
( A )
where
C2+a -
and
(nE)r\C:(RE)
(3.14)
s*(x,
E)
=
2
I
eUdx
= XE x
IR,
YE = C a ( z E ) U
-
0s
PE
= YE
with
IR,
x
X
E
Putting
0 < a < 1.
*
(yAfE), *
e 'IEdx
and
h;,€
=
OE
we have
*
*
@ E ( h A , Es,
(A,
o f t h e diffeomorphism
€1)
-'
gN,€
= 0
( 0 < A < 6,
: RE
-+ no = D,
IEI
<
E
~
)
.
In t e r m s
t h e problem ( 3 . 1 3 )
Ken'ichi NAGASAKI and Takashi SUZUKI
202
on
Then,
go
x
R into
Po
x
R,
which is denoted by
*
*
X , E - ~ N , Eand
=
=
H X r ~
(Po),
As for
Step 1.
I
Let it be
(PE).
will be transformed into another mapping from
OE
* UX,E
no.
is transformed into that on
QE
( ~"'),
Then, setting
FE.
we have
every solution
is
= uf , X
U,
given explicitly as in (2.6) and is parametrized by
s =
n
eUdx
with
0 < s < n.
Let it be ( 0 < s < n).
We have
OO(hsfO,s )
(0 < s < n )
=
O < s < n
0
.
Further, d h ~ o ( h s , O ,S
is invertible by Lemmas 4 and 5, and the express
ion (2.17).
be
s = s+(X), -
where
s+(h)
-
X = X o ( s ) is two-valued.
The inverse mapping of
--+
n
s-(X)
and
Let it
0
as
< cO)
is
X * 0.
Step 2.
as
X
F (0,
Lm-sense, so that
bounded in (Po)
For each
E
+
0
CU:fEII~I
6),
U
*
X f E
converges to a solution of
from the elliptic estimate.
Lm-norm of the limit function goes to it ought to be
u + ~:
-
as
Further, the XtO,
and hence
Nonlinear Eigenvalue Problem
203
Consequently, *
as
* 0.
E
Step 3 .
On the other hand, a branch
of solutions of
OE(h, s )
originating from
-8,
.
-1
gN,E*
*E
( f:::)
-
-
I ( y S I E,
=
=
exists for
(h, s ) = (0, 0 )
is,)
10 < s <
‘0’
let
.
0
TI)
hslE
Then, we have
h =
.
bs
0 < s <
by Lemmas 1 and 3 .
We set
Through the diffeomorphism
be transformed into
FE
TI,
(11s ,E
H
-s
,E
( 0 < s < n).
s) = 0
In the same way as in Step 2, we see that
as
E
* 0
uniform in
(0,
such that Is
-
sol <
W S f Ef
8 )
Take an 6).
i
>O
.
Further, the convergence is
1s - s o l
such that
so
clo-e to
. s- ,\ ,
TI
IE I
5
p1/21
(0, n ) .
such that
-is - invertible - -- - - - - - f -o - -r
from Steps 1 and 3 .
as a solut on set of K
(0, n
Then, there exist constants
d - .F ~ -f’ -Hs , P
6
on each compact interval contained in
s
Step 4.
xo(so) t
s
for each
E~
IF I < I -
I
and F -
-1
pl>
0
and
Consequently, the branch
has a local uniqueness property
FE(H, s )
=
0.
Namely
5 ~,/2, Is1 - s o l
(p1/2,
there exists a FE(H1, s l ) = 0
Ken'ichi NAGASAKI and Takashi SUZUKI
204
and
(IH1-HsOfEI)<~ imply
(H1,S1) C t ( ~ s , E f ~ ) ~ ~ ~ - S O ~ ( ~ 1 / 2 1 .
On t h e o t h e r h a n d , S t e p 2 i m p l i e s t h a t
and
as
E
*
so t h a t
(as
E
IE(
(Hi
0
(s
0
) f
S
Hence
0).
+
*
F u r t h e r we have
0.
-*
*
(Ao(so)f
.d*
*
I
E ) )
f t($s,E~
connects with
s (x0(so),
-
S ) I IS
-8, ,
€ 1 ) = 0,
SoI
5 ~1/21
provided t h a t
i s s u f f i c i e n t l y small.
54.
C o n c l u d i n g Remarks.
1.
If
n
i s star-shaped,
o r i g i n , and i f
(u,
x)
f o r instance, with respect t o t h e
solves
(PI, t h e n R e l l i c h ' s i d e n t i t y
yields that
where
[l]).
e d x , A = In1 a n d B = n W e always have B 2n. I n case
do.
s =
B
5
-
XA,
4n
(Bandle
and
s
n,
we get
1 and hence
(
4
-~ El2 5
(4s
x 5 8n(B - 2n)/AB.
-
El2 5 B
I n o t h e r words,
s < n
holds
205
Nonlinear Eigenvalue Problem
when
and
5 4n
B
-
X > X :8n(B
From Lemmas 1 a n d
2n)/AB.
3 , w e c a n show t h a t
<
x
Theorem 2 .
*
=
i o n s on
x
*
(0).
(X,
*
Suppose t h a t
5
B
2n)/AB
They f o r m a b r a n c h w h i c h c o n n e c t s t o t h a t
o f minimal s o l u t i o n s and bends a t 2.
-
A = 8n(B -
(P) h a s e x a c t l y t w o s o l u t -
Then, t h e p r o b l e m ).
and
4n
.
=
I t i s c o n v e n i e n t t o regard t h e o p e r a t o r
+ xeU
A
(4.2)
'he
=[
J
L 2 (.) as t h a t i n
.
-
8s eU
e U dx
under t h e D i r i c h l e t c o n d i t i o n f o r t h e f i r s t
x
R
X
component, r a t h e r t h a n a s t h a t on
t h e domain
D(dh B) =
H
a positivity-preserving
H1 O(a).
1
2
( 0 ) n V
x
R
into
Y x
R
V x
R
Then, w i t h
i t becomes s e l f - a d j o i n t a n d h a s
X
R
a n d compact r e s o l v e n t ,
The a s s o c i a t e d s y m m e t r i c b i l i n e a r f o r m
d e f i n e d on
.
V
being
,
bL(
)
is
through t h e r e l a t i o n
Actually f o r
V
V
v x
and
g =
(r)~ V x
R
V
,
Ken’ichi NAGASAKI and Takashi SUZUKI
206
with
p = Xe
U
.
V
We identify T
x
IR
V* = tv f
with
being the unit tangent vector on
1 a H (n)l=v a$,
= 0
on
an),
through the isomor-
phism
Then, the identity ( 4 . 3 ) reads (4.5)
where
a(-,
0
)
is the symmetric bilinear form on
a(v, w)
=
SQ
(Vv-Vw- pvw)dx
(v, w C H1(Q)).
Now we can introduce the self-adjoint operator associated with the form
a1 *
*
v
x
=
0
H1(Q) :
A*
P
in
L2(n)
such as
v
Namely,
with D(A*) = {v f H 2 ( a ) I
a v
on
an
and
P
Because of the relation ( 4 . 3 ) , (4.5) and ( 4 . 6 1 , invertibility of
dh
4
in
L2 (n 1 x
R
a
v
= 0).
the
*
is equivalent to that of A. P
Nonlinear Eigenvalue Problem
in
201
L*
-< ....
denotes t h e set of e i g e n v a l u e s of
For t h e c o n s t a n t f u n c t i o n
.a(c, c ) < 0.
ln11/2 1
C =
*
.
P we have
On a c c o u n t o f t h e mini-max p r i n c i p l e , t h i s means
*
that
A
v1 ( p )
Since t h e codimension of
< 0.
V
in
V
*
is one,
we have
*
P,(Pl
1. P1(P)
*
and
! J 3 ( P ) 1. P , ( P )
a l s o f r o m t h e mini-max p r i n c i p l e . s <
*
by Lemma 3.
TI
P,(p)
*
p2(p)
p3(p) > 0
if
A l s o , b y v i r t u e o f Lemma 4 w e h a v e
F u r t h e r , we have
> 0
if
P l ( p ) 1. 0 .
$ o
is equivalent t o
O u t l i n e of Proof:
y*
*
Therefore,
= 1
(p)
I =
do
$
0.
We see t h a t
+
A-’(p) P
E
V*
satisfies
*
a ( y , yj) where
yj
= 0
denotes t h e
that I = a(cp
*
j-th eigenfunction of
, y*)= -
F u r t h e r , t h e e q u i v a l e n c e between follows d i r e c t l y .
... ) ,
( j = 1,2,
I n t h e case o f
A
P
and a l s o
p y*dx. 0 €
5
(dh 0 )
I > 0,
and
I = 0
t h e closed subspace
Ken'ichi NAGASAKI and Takashi SUZUKI
208
Inf
a(v, v ) > 0,
*
v tV1
*
so that
> 0.
p2
*
9, + C 2
v = C1
For this
v,
the case
p i
Appendix.
we have = 0
*
VIC V
any closed subspace element
I < 0,
In the case of
Lp*
on the other hand,
of codimension 1 contains an such that
a(v, v) < 0,
I
holds only if
=
5
(C1,
=
so that 0.
P2
C2) 6 R2\rOl.
5 0.
Hence we have
However, P *2 < 0.
Generalized Schwarz Symmetrization and the estimates
of eigenvalues. ~ , ( p ) in ( 3 . 6 ) and (3.7)
The estimate of the eigenvalue with
p = keU
was derived by
C.
Bandle as an application of
generalized Schwarz Symmetrization.
We review briefly the
theory for reader's conven ence. Definition A.l. Riemannian metric For real constants rized domain
is a disk
{
D
*
Kotb
Let
doL KO
=
D
be a domain in
p x)ds 2 ,
and
where
b(> O ) ,
ds2
such that
=
dxl 2
+ dx2. 2
the generalized symmet-
with Riemannian metric
x 1x1 < R 1
R2 with
0
Nonlinear Eigenvalue Problem
For a closed domain disk
*
(XI
we define
D
*
as a closed
Kotb
1x1 2 R ) .
The area elements of Riemann surfaces
(DKo,b'
do) are denoted by
d T = p(x)dx
and
curvatures of :
D,
209
'log -2p(xI
and
d;
d; = c(r; K O , b)dx.
(D, do) and
*
( DKo f b
and
respectively, that is, We note that Gaussian
d^o) are equal to k(x)
u : D -IR,
For a function
*
*
uK o f b : DKo,b
-+
JR
we define a
as follows:
*
The function
u ~ ~ , which ~ ,is radially symmetric, is
called a generalized symmetrized function of
*
Henceforth abbreviated to
and
respectively.
KO
Definition A.2. function
dr
(D, do)
DKO,bf c(r; K o f b ) and D*, c(r)
and
u*
u.
Uiof,
may be
for simplicity.
Among the many interesting properties of the generalized symmetrized function, we take up two of them, which will be needed later. Lemma A.l.
Let
g(x)
and
h(x)
be continuous in
-
D,
then
ho = min h(x), hl = max h(x) D D = { x 6 D,Ih(x) > t), then we have Proof.
Let
and
D(t)
Ken'ichi NAGASAKI and Takashi SUZUKI
210
and
the inequality (A.1) follows. Lemma
A.2.
Let
be a domain in E2 and
D
be Gaussian curvature of assumption that
(D, d u) = (D,
k(x) 5 K O and
in Definition A.1, and
M =
KOM
dT
holds for a real analytic function in
D
and
u(x) = 0
on
m)ds). ,
4
I, J =
k(x)
D
where
p(x)dx,
u(x)
Under the is that
KO
the inequality
such that
u(x) 1. 0
aD.
Before proving the above lemma, we recall an isoperimetric inequality on a Riemann surface, that is, Bol's inequality: For any subdomain =
J Bp(x)dx
and
L~ =
B
of
D,
let
MB =
JB
dT
=JaBm ds denote the area
211
Nonlinear Eigenvalue Problem
of
B
and the length of
aBl
then
For
t 2 0,
(A.3)
Proof of Lemma A . 2 . a(t)
and
of
D(t), T(t),
a f t ) as follows:
For a real analytic function t(a)
we define
a(t)
decreasing in
u,
the inverse function of
is well-defined because (0, urnax). Moreover,
a(t)
a(t)
is strictly
and
t(a)
differentiable. On account of
a
co-area formula
- da dt
[
-
ds
r(t)
and Schwarz‘s inequality, we have
From an application of Bolls inequality follows (A.
41
(vulds 2 { 4 n
- K,,a(t)l
a(t)
(-
dt
are
) .
Ken’ichi NAGASAKI and Takashi SUZUKI
212
Combining (A.4) with another co-area formula
and integrating in
u 3 max
[0
with respect to
t,
we get
r
In the right-hand side of (A.5), the substitution a
with the relation
a
d; = J 1x1< r
equality
2 nbr 1+-K r 4 0
and
we use the fact that M
=
i,*
d;
=
nbR
lvu
*
I
=
du = - dr
-
ID*
Vu * 1 2 dx ,
dt
da
2nbr b 2) 2 (l+;rKor
2
b 2 . 1+-K R 4 0
The assertion follows from (A.5) and (A.6). We derive some estimates of the minimal eigenvalues v,l
*
v1
of the following eigenvalue problems:
+
for
yields the
,
22nr dr =
where
r
= o
for
x 6 D
= o
for
x E aD
I
,
Nonlinear Eigenvalue Problem
The comparison of
w1
*
with
213
*
for
x E D
for
x E aD
*
,
is given in the follow-
v1
ing . Lemma A . 3 . in Lemma A . 2 ,
Proof.
If
K O M < 477,
where
KO
and
are the same
M
we have
We introduce Rayleigh quotients
and
According to the variational characterization of the minimal eigenvalue, we have w1
=
inf
1 YeHo(D)
R[q]
and
w;
=
inf R*[$] 1 * $€HO(D
.
On account o f the positivity of the eigenfunction corresponding to
v1
and the denseness in
analytic functions, the minimizing sequence can be chosen such that every assumptions in Lemma A . 2 .
4,
(n
=
Ci(E)
of real
[qnt of
1, 2 , . . . )
R[q]
satisfies the
Hence, the application of Lemma A.1
Ken’ichi NAGASAKI and Takashi SUZUKI
214
and A . 2 to
9,
yields the estimate
The assertion follows immediately from ( A . 7 ) . Next we will calculate the minimal eigenvalue of for particular Lemma A . 4 . vl(Bp)
p
and
D.
For a disk
B,
= {XI 1x1 < P I ,
we denote by
the minimal eigenvalue of the eigenvalue problem: 4c
A Y + v
,
(E.P)
2 2 q = 0
(l+r
9 where
(E.P)
r = 1x1
and
C
for
x 6 B,
,
for
x E aBp
,
)
= o
is a positive constant.
Then we have for
P
=
1
and
Proof.
In polar coordinates
(E. P),
can be expressed
as follows.
9
(A.9)
Setting q(r,
8)
=
u(r)O(e)
= O
in ( A . 8 ) ,
for we get
r =
P .
Nonlinear Eigenvalue Problem
@(el
+
= an sin n e
On the other hand, (A.10)
1 F(rur r
(A.11)
u(0
bn cos n 8
,
0, 1, 2,
=
... ) .
is a solution of
u(r)
rn
(n
215
.
u(p) = 0 z =
By the substitution of
r2-1 2
and
r +1
v(z) = u(r),
we
transform (A.lO,ll) into the associated Legendre equation: t(1-z 2 dv
(A.12) d
n2 v + - 1-2
v(-1) <
(A.13)
V(+)
m,
p
=
2v
=
P 2-1) z c (-1,-
for
0
p
2-1
P
In case
Y C
=
0
2+1
.
+1
1, the equation (A.12) is satisfied with
and v(z) = - z . This means that - is an C2 C2 eigenvalue of (E.P)l with the corresponding eigenfunction
n = O ,
$(X)
=
vl(B1)
v = -
2
1-r . Moreover
l+r
-
C2
must be the minimal eigenvalue n
because of the positivity of
Cp(x)
=
1-i-' 7 in
l+r
B1'
The latter part of the assertion is a consequence of the monotonity of the minimal eigenvalue of to the size of the domain
Bp,
that is,
(E.P) with respect P.
Lastly we reach the position to state the main proposition in this Appendix. Proposition A .
If
KO > 0
and
KOM < 2 n
,
where K O
and
Ken'ichi NAGASAKI and Takashi SUZUKI
216
M
are the same as in Lemma A.2., then "1 > 2KOt
being the minimal eigenvalue of
v1
Proof.
Setting
b
equal to
4Ki1
(E.P).
in
v(r: K O , b),
we
can reduce the problem ( E . P ) to the problem ( E . P ) R in g instead of C 2 . In this situation, Lemma A.4 with K -1
R < 1
follows from the assumption
KOM < 2n
.
The application of Lemma A.3 and A.4 provides that v 1 = vl(BR) > 2K0
v1
.
Note :
1) The complex number
tE(e)
=
gk,E(eie)/(gh,E(e ie )
I
indicates the unit normal vector of a n e at ie gN,€(e ) f a n E . Therefore, the univalentness of
(eie E S1 = a D ) the point
g N I E on the boundary
aD
follows if the winding number of the
mapping tE is
:
e i e .s
s1 + + ~ € ( e )
E
s1
+1 and
Both hold for small.
E
=
0
and hence when
/ E /
is sufficiently
Nonlinear Eigenvalue Problem
217
References [l]
Bandle, C., Existence theorems, qualitative results and a priori bounds for a class of a nonlinear Dirichlet problems, Arch. Rat. Mech. Anal., 58 (1975) 219-238.
[2]
Bandle, C., Isoperimetric inequalities for a nonlinear eigenvalue problem, Proc. Amer. Math. SOC., 56 (1976) 243246.
[3]
Bandle, C., Isoperimetric Inequalities and Applications, Pitman, Boston/London/Melbourne, 1980.
[a]
Crandall, M.G., Rabinowitz, P.H., Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems, Arch. Rat. Mech. Anal., 58 (1975) 207-218.
[5]
De Figueiredo, D.G., Lions, P.L., Nussbaum, R.D., A priori estimates and existence of positive solutions of nonlinear elliptic equations, J. Math. Pure Appl., 61 (1982) 41-63.
[6]
Fujita, H., On the nonlinear equations
U
Au+e = O
and av/at
V
=AV+e , Bull. Amer. Math. SOC., 75 (1969) 132-135. [7]
Gel'fand, I.M., Some problems in the theory of quasilinear equations, Amer. Math. SOC. Transl., l(2) 29 (1963) 295-381.
[a]
Gidas, B., Ni Wei-Ming, Nirenberg, L., Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979)'209-243.
[9]
Joseph, D.D., Lundgren, T.S., Quasilinear Dirichlet problems driven by positive sources, Arch. Rat. Mech. Anal., 49 (1973) 241-269.
Ken'ichi NAGASAKI and Takashi SUZUKI
218
[lo] Keener, J.P., Keller, H.B., Positive solutions of convex nonlinear eigenvalue problem, J. Diff. Equations, 16 (1974) 103-125. [ll] Keller
H.B., Cohen, D.S., Some positive problems
suggested by nonlinear heat generation, J. Math. Mech., 16 (1967) 1361-1376. [12] Laetsch, T., On the number of solutions of boundary value problems with convex nonlinearities, J. Math. Anal. Appl., 35 (1971) 389-404. [13] Liouville, J., Sur l'equation aux derivees partielles (a2 log x)/auav+2xa 2= O f J. de Math., 18 (1853) 71-72. [14] Moseley, J.L., Asymptotic solutions for a Dirichlet problem with an exponential nonlinearity, SIAM J. Math. Anal., 14 (1983) 719-735. [15] Nagasaki, K., Suzuki, T., in preparation. [16] Rabinowitz, P.H., Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math., 3 (1973) 161-202. [17] Wente, H., Counter example to a conjecture of H, Hopf, Pacific J. Math., 121 (1986) 193-244. [l8] Weston, V.H., On the asymptotic solution of a partial differential equation with an exponential nonlinearity, SIAM J. Math. Anal., 9 (1978) 1030-1053.