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250
and (3.1) follows
in
this case too. By Lemma 3.1 we deduce that condition (Co) holds.
(ii) Without loss of generality we can assume that
x
0
=
0, hence 5
0
(x)
=
\x\. Put
w =
\PM
and
wi =
w(l
- *) on
B
r
(0).
Assume first that
p ^ N.
Since
w
G W
o
1,p
(iJ
r
(0)) we get
from Hardy inequality (see [7]),
/ \Vw\" > |1
-
N/p\"
f
(\w\/\x\)",
(3.4)
JB
T
(0) JB
T
(0)
so that
w/\x\ e
U(B
r
{Q)). Since clearly w
x
/\x\
€
U(B
r
{Q))
it
follows that
u/\x\ £
LP(B
r
{Q)).
But we
have also
Vu €
V{B
r
{Q)). We deduce thus
as
above that (3.1)
holds,
hence condition (Co)
is
satisfied.
In
case
p
=
N
we argue as above, but with (3.4)
replaced
by
the following improved Hardy inequality due
to
Adimurthi, Chaudhuri and
Ramaswamy [1, Theorem 1.1],
/ \Vw\'>C
[
(\w\/\x\r(\ogR/\x\)-",
(3.5)
JB
r
(0) JB
r
(0)
V(0)
JB
r
(0)
with
R
>
re
2/p
.
It
follows then that,
u
e L"(B
r
(0)).
(3.6)
\x\\og(R/\x\)
In order
to
show that condition (C
0
) is satisfied we take
a = 1/2
and set,
rf
=
(l/2)"
+
'~\
j =
l,...,n,
for
n >
— j
5
^.
Since we always assume that
p > 1
we have
p = N > 2. By
Young
inequality,
(^)
2
|V
U
r<(2/
P
)^
+
(
l-2/p)|V<.
Therefore,
i E
/
(M/M)'+
i
E
/
iv«r
2
(i«i/ix|)
2
<
^/ (H/lxD'
+ c/
|Vur:=/{"
)
+
^"
)
.
W
V{2-
2
"<|:c|<2-»} ^{2-
2
"<|x|<2-»}
Since Vu
e
I^(B
r
(0))
it
is clear that liirin-Kx,/|
n)
=
0. Further,
/j
n)
<
— (lo
gj
R
+
2nlog2)
p
f (. ,,
*"[,.,
,.V -> 0 (by (3.6)).
It follows that condition (Co)
is
satisfied.
•
Remark 3.1.
A
comparison lemma related
to (ii) of
Proposition
3.1
was proved
by
Sreenadh [10].
251
Next
we
prove
a
uniqueness result
for
global solutions.
Theorem
3.1. Let a(x) €
L]
0C
(R
N
\
{0})
and
suppose that
u
G
C(R
N
\
{0})
n
WJ£(R
N
\
{0})
is a
positive solution
of
the equation
(1.3) inQ.
—
R
N
\
{0} which satisfies condition
(3.2),
and
for
some
a €
(0,1),
lim inf
r
-
>0
'{°r<\x\<r)
,
3?
.
f (\u\/\x\)"
< oo, if p€
{1,2),
f (M/My
+
lVur^M/liD^oo, tf
P
e(2,oo),
J
{<TT<\x\<r}
lim inf
r->0
or more generally, assume
u
satisfies conditions
(Cx,) and (Co)
with
T
0
= {0}. Let
v
e C(R
N
\
{0})
n
WJ£(R
N
\
{0}) be another positive solution
to
(1.3). Then,
v = cu for
some constant
c > 0.
(3.8)
Proof.
Consider
any x
r
G R
N
\ {0} and set a :=
vixj/uixi).
Put Qi = R
N
\
{0,X!}
and apply Theorem
1.3 for n = Qi and Ti = {x^} to
deduce that i)(a;)/M(a;)
> «i =
D(:TI)/U(:EI)
for
every
x ^ 0.
Since
x
t
and x are
arbitrary,
we
deduce that
u/v = c. •
Remark
3.2.
Theorem
3.1
generalizes Theorem
1.3 of
[8]
which treated
the
special case
a(x)
= |1
—
N/p\
p
\x\~
p
in
which
a
positive solution
to (1.3) is
given explicitly
by u(x) =
x
l-N/p
We close with
a
uniqueness result
in
bounded domains.
Theorem
3.2. Let O be a
bounded domain with boundary
T
0
. Let a(x) e
L}
0C
(Q)
and
suppose that
u e C(O) n W^(Q) is a
positive solution
of (1.3)
which satisfies (3.1),
or
more generally, condition
(Co). Let v € C(Q)
PI
W^(Q) be
another positive solution
to
(1.3).
Then,
v = cu for
some constant
c > 0.
Proof.
Simply repeat
the
argument
of
Theorem
3.1,
this time applying Theorem
1.3 to
the domain
£2
\
{ii}, whose boundary
is the
disjoint union
of
To
= dQ and r
x
= {^i},
for
any X\ €
Q.
•
Remark
3.3. Let Q be a
bounded domain
of
class
C
2
and let
r\
G
C(Q)
satisfy r\
> 0 in
CI
and
rj
= 0 on dQ. In
[6, Theroem 1.2]
it
was shown that
for a
certain critical value
A*,
there exists
a
positive solution
u
t
to the
equation,
-A
p
u
t
=
(y
V
+
(l-l/pr)
l
-^p±,
(3.9)
which satisfies
the
growth condition,
C-
1
6
1
-
1
'"<u
t
<C8
1
-
1/
'',inn,
for
some
C > 0,
(3.10)
where
we
denoted
5(x) =
dist(x,dil).
It is
easy
to
show that (3.10) implies
(3.1)
(with
8
0
= 8).
Indeed,
in
case
p £ (1,2]
this
is
immediate, while
for p > 2 we
need only
to remark that
by [9,
Lemma
A.3],
(3.10) implies that
|Vw,| <
C8~
l
^
v
in
Q,,
and the
conclusion follows
as
well. Therefore
we
infer from Theorem
3.2
that
u
t
is
actually
the
unique positive solution
to
(3.9),
up to a
multiplicative factor.
252
References
[1] Adimurthi, N. Chaudhuri and M. Ramaswamy, An improved Hardy-Sobolev inequality
and its application, Proc. Amer. Math. Soc. 130 (2002), 489-505.
[2] S. Agmon, Bounds on exponential decay of eigenfunctions, in "Schrodinger Opera-
tors",
ed. S. Gram, Lecture Notes in Math., Vol. 1159, Springer-Verlag, Berlin, 1985,
1-38.
[3] W. Allegretto and Y. X. Huang, A Picone's identity for the p—Laplacian and appli-
cations, Nonlinear Analysis TMA 32 (1998), 819-830.
[4] H. Brezis and M. Marcus, Hardy's inequality revisited, Ann. Sc. Norm. Pisa. 25
(1997),
217-237.
[5] M. Marcus, V. J. Mizel and Y. Pinchover, On the best constant for Hardy's inequality
in
R™,
Trans. A.M.S. 350 (1998), 3237-3255.
[6] M. Marcus and I. Shafrir, An eigenvalue problem related to Hardy's IP inequality,
Ann. Sc. Norm. Pisa. 29 (2000), 581-604.
[7] B. Opic, A. Kufner, "Hardy-type Inequalities", Pitman Research Notes in Math.,
Vol. 219, Longman 1990.
[8] A. Poliakovsky and I. Shafrir, Uniqueness of positive solution for singular equations
involving the p-Laplacian, preprint.
[9] I. Shafrir, Asymptotic behavior of minimizing sequences for Hardy's inequality, Com-
mun. Contemp. Math., 2 (2000), 151-189.
[10] K. Sreenadh, On the Fucik spectrum of
Hardy-Sobolev
operator, to appear in Nonlin-
ear Analysis TMA..
Supercritical variational problems with unique critical
points
Wolfgang Reichel
Mathematisches Institut der Universitat Basel, Rheinsprung 21
CH-4051 Basel, Switzerland
Email : reichel@math.unibas.ch
Abstract
We study how a simple class of one-parameter transformation-groups inter-
acts with functionals
jC[u]
=
j
n
L(x,u,
Vu) dx. We show how one can understand
Pohozaev's identity as a formula for the rate-of-change of £ under the action of the
transformation group. Uniqueness of the critical point of L will follow provided the
transformation group reduces the values of
C.
As an application we find a class of
conformally contractible domains, which includes the class of star-shaped domains,
and on which Pohozaev's uniqueness result remains valid.
1 Transformation groups
We begin our study with one-parameter transformation groups. Let Q C R
n
be a bounded
smooth domain with exterior unit normal v{x) for x 6 dil. We consider a one-parameter
family of maps G = {g
e
: dom<?
£
C fl x 1 -1 (1x1} enjoying the group-property
5ei °
&2
=
&1+E2!
ffo
= W. A particularly nice example of such a group arises when g
e
is
defined as the flow-map at time e of the dynamical system
x = £(x,u), u =
(f>(x,u),
(1)
with smooth functions £,
(/>.
To be precise, let (xe(x
0
,u
0
),ip
e
(xo,uo)) denote the solution
of (1) at time e with initial conditions (x
0
,uo). Then a one-parameter group G arises by
setting g
e
(x
0
,u
0
) = (x
£
(a:o,«o)^e(^o, Wj), for every (x
0
,u
0
) eSlxR, In a forthcoming
publication [5], we investigate transformation groups arising from the dynamical system
(1),
together with their interactions with variational problems.
In this paper, we study a simple type of transformation group arising from the following
dynamical system:
x = £(x), u = a(x)u. (2)
The system (2) is a special case of (1) and can be integrated semi-explicitly. Since
the two equations in (2) can be decoupled, we denote by Xc{
x
) the solution of the first
equation at time e, starting with the initial condition x at time e = 0. Then ip
e
(x,u) =
253
254
exp(/
0
£
a(x
T
(x))d,T)u solves the second equation, with the initial condition u at time e = 0.
Let us now take an initial function u(x). We can insert its graph (x,u(x)) into the
flow and obtain after time e the transformed graph (x
e
{x),i>
l
{x,u{x)). For sufficiently
small e, this will represent the graph of a new function (x,u(x)) with new independent
variable x =
Xe(
x
)-
Expressing x =
X-e{
x
)
we find the following explicit formula for the
transformed function
u(x)
expf / a(xr-
e
{x))dTJ u(x-
€
(x))
exp f / a(xr(x))dTJ u(x-
€
(x)),
(3)
where x belongs to Q
e
= Xe(0). We will write g
e
u for the new function u(x). In the
following, we will use summation convention, i.e., double indices will automatically be
summed from 1 to n.
Proposition 1.1 Let G = {<?
e
}e>o be a transformation group generated by (2). Then for
every i = 1,..., n and every x € O
e
, the following holds
am
. «,(/ *,,,»*)gfe^gw
f° da _ dx
j
( f° \ -
+
J dx"
(
'
XT
^~dx
Z
^
dTeXP
[j
a
(X
T
W>
dT
)
u
(X-e{i))-
Proof.
(4) follows directly from (3). D
In order to state the following Proposition, let us introduce the space CQ{D) of con-
tinuously differentiable functions on D with zero boundary values and the space C
0
' (D)
of uniformly Lipschitz-continuous functions on D with zero boundary values.
Proposition 1.2 Suppose £(x)
•
v(x) < 0 for all x € 80.. Then for
e
> 0 we have fi
e
C fi.
Extending u by zero outside fi, the transformed function g
e
u is zero in Q \ Q
e
. Moreover,
if the initial function u belongs to the space Cg(Q) or C
0
' (fi) then for small e > 0 we
have g
e
u €
CQ(Q
€
),C
0
'
(f2
e
), respectively. In both cases, we have g
e
u e C
0
' (fl) for small
e>0.
A transformation half-group {g
€
}
e
>o arising from a flow of type (2) and with the
property £(x)
•
v(x) < 0 on 80. will be called a contracting transformation half-group.
2 Rate of change of a functional under a transforma-
tion group
We consider a functional
C[u]
= J
Q
L(x, u, Vujdx, where L
:
Q x K"
+1
—»
R is a continu-
ously differentiable Lagrangian. Let us suppose w.l.o.g that L(x,0,0) = 0, for all x 6 fl-
it is simple to see that £ is Frechet-differentiable on the space
CQ'
1
(Q).
255
Our goal is to find a formula for the rate of change of C under the action of a contracting
transformation group. The theory of interaction of functionals with transformation groups
is well developed, see Olver [2]. In order to keep our point of view accessible for non-
experts, we decided not just to quote the following theorem from the literature, but to
include a proof for simple transformation groups generated by (2).
Theorem 2.1 Let G =
{g
e
}e>o
be a contracting transformation half-group generated by
(2).
Suppose the initial function u belongs to
CQ(0,).
Then the rate of change of the
functional £ under the transformation group can be computed as follows:
—C[gM\^o
f dL
_.
dh dL (da du d& du\
r
,. ,, (
5
)
= J
a
^
+
Tu^
+
^ fe
M +
fl
&T ft^&J
+ M
^
dX
'
where the Lagrangian L and its derivatives are evaluated at (i,u(i),Vti(i)), and the
functions a, £, u and their derivatives are evaluated at x.
Remark 2.2 It turns out that
a
formula similar to (5)
holds
for more general transforma-
tion groups generated by (1). The proof
requires
more in-depth analysis of transformation
groups and can
be
found in Olver [2].
Proof.
Since L(x, 0,0) = 0 and since supp(g
e
M) c O
e
, we may write
C[g
e
u]
= / L(x,u,Vxu)dx.
If we consider x = Xe(
x
) as
a
change of variables, we obtain
C[g
€
u]
= f L(x
e
(x),u(x,(x)),(V
4
u)(x,(i)))det(£>
Xe
(s))dx.
Jet
Differentiation w.r.t. e at e = 0 yields
j
e
£[geu}\
€=0
f 8L
ri
dL dL d (du , . ,,\ i
T
,.
AJ
(6)
where we have used that ^ det Dx
e
{x) = div£(x
e
(x)) det Dxe(x) and xo = Id. It remains
to express du/dxi in terms of Xe{
x
) by the help of (4), and to differentiate the result w.r.t.
e. Doing so we obtain first by (4)
J|(X.(*))
= exp
(/W))*) -g(*)^(Xe(*))
f° da dx
J
f f
e
\
+
/
faTMT+e{x))-Q^;{Xt{x))dTexp
( /
Xr(x)dr
\ u{x),
°i It is important to have u continuously differentiable. For Lipschitz functions, the formula does not
hold in general.
256
and after differentiation w.r.t. e at e = 0 and using \o = Id, dxb/dxi = 6
t
j we get
A
t)
(JJL
{z)
+
pf
{x
^fjM\ W*
w
„
w
.
(7)
ctey \ axi dxidxi de
IE=O/
cte*
The last relation simplifies since d
2
xb/dxidxi = 0. The result follows by inserting (7)
back into (6). •
Corollary 2.3 Let G = {g
e
}
c
>o be a contracting transformation half-group generated by
(2).
Suppose that the initial function u
belongs
to C
2
(fi)nCQ(f2). Then the rate of change
of the functional under the transformation group satisfies
±C
M
U = J
a
(au - *
•
Vu)£
[u]
dx
+
J J
.U)(L-
£ J£) da, (8)
where
£[u]
= ""^(ffO + f^ i
s
the Euler-operator associated to the functional C. As before,
the Lagrangian L and its derivatives are evaluated at (x, u(x), Vu(i)), the functions a, u,
£ and their derivatives at x.
Proof.
The proof hinges on a formula of E. Noether, which she used in the derivation
of her famous theorem on symmetry and conservation laws, cf. Olver [2], Section 4.4.
Noether's formula expresses the integrand in (5) as a multiple of the Euler-equation plus
a divergence-term, i.e.
dL <9L dL_(da_ ^_^_^_\
T
A-C
dxi du dui \dxi dx
t
dxidxjj
= {au - £
•
Vu)£[u] + 4~ (?L +{au-£- Vu)f^
aXi \ oUi
(9)
Formula (9) is verified by explicit calculation. If we recall that u = 0on
dQ.
and hence
Vu = vdu/dv, an integration by parts results in (8) and completes the
proof.
Remark 2.4 For solutions of the Euler-equation £[u] = 0, the identity between the vol-
ume integral (5) in Theorem 2.1 and the surface integral (8) in Corollary 2.3 is known
as Pohozaev's identity, cf. Pohozaev [3] and Pucci and Serrin [4]. It has long been
noted,
cf. Pucci, Serrin [4] and Van der Vorst
[7],
that some kind of connection between
one-parameter transformation groups, Noether's formula and Pohozaev's identity exists.
For yet another variational
approach
for deriving Pohozaev's identity, which is especially
useful for free-boundary problems, see Wagner [8].
3 Variational sub-symmetries and uniqueness of crit-
ical points
We can now begin to formulate our main result. Loosely speaking it states under mild as-
sumptions on the Lagrangian that a contracting transformation half-group, which reduces
the values of L, implies the uniqueness of the critical point of C.
257
Definition 3.1 Let the functional £[u] = J
n
L(x,u,Vu)dx be defined on C%
A
(Q). A
contracting transformation half-group G
~_{g
€
}
€Z0
generated by (2) is called a variational
sub-symmetry for £ if for all u €
C^
l
(Q)
there exists e
0
(u) > 0 such that £[g
e
u] < £[u]
for all e E (0, e
0
(u)).
Proposition 3.2 Let G = {g
e
}
€
>o be a contracting transformation half-group generated
by (2). Then the condition
~C[g
£
u}\e=o < 0, for all u E C^U), (10)
is necessary and sufficient for G to be a variational sub-symmetry. If (10) holds with
strict inequality for all u E Cg(fi) unless u = 0 then G is called a strict variational
sub-symmetry w.r.t 0.
The importance of the proposition lies in the fact that one can use the explicit formula
for ^£[g
e
u]\
e=0
given in (5) to verify that G is a variational sub-symmetry. We are now
ready to state our main uniqueness result, which follows directly from Corollary 2.3.
Theorem 3.3 Suppose G is a variational sub-symmetry generated by (2) for the func-
tional £[u] =
f
fri
L(x,u,
Vu)dx on the
space
CQ'
1
(ft). Suppose also that L(x, 0, p) is convex
in p € M
n
for all x e dQ. Then either of the following two conditions implies that u
—
0
is the only critical point of £:
(i) G is a strict variational sub-symmetry
w.r.t.
0,
(ii) if u €
C
2
(fi)
n
CQ(Q)
is a solution of £[u] = 0 and
(£
•
u)(L(x, 0, Vu) - !^!^(x,0, V«)) = 0, on dft,
then necessarily u = 0.
Remark 3.4 1.
2.
For a strict variational sub-symmetry the theorem can be understood heuristically:
since a strict variational sub-symmetry strictly reduces the energy £\u] everywhere
except at u = 0 it is only compatible with a (unique) global minimizer at u = 0.
For example, a strict variational sub-symmetry applied to a mountain-pass reduces
the energy everywhere along a minimizing path, which contradicts the definition of
a mountain-pass.
3.
A verification of condition (ii) is usually achieved via a unique continuation princi-
ple, cf. Section J^.l.
258
4 Applications
4.1 A generalization of Pohozaev's result
In 1965 S.I. Pohozaev proved a remarkable identity [3], which allowed him to conclude
that the problem
Au + \u\
p
'
l
u = 0, in ft,
M
= 0on<9ft, (11)
has only the trivial solution if ft C R", n > 3 is a bounded, star-shaped domain and p >
jj^§.
Theorem 3.3 recovers this result if we consider the transformation group generated
by
n-2
x = —x, u = —-—u,
i.e., g
e
u(x) = e~f~
c
u(e
e
x) with x €
e
_6
ft
and if we assume w.l.o.g that ft is star-
shaped w.r.t. 0. It turns out that G = {p<J
£
>o is a variational sub-symmetry for
£[u] = /o(ol^
u
l +rl
u
l
p+1
)^
a;
- The functional verifies the conditions of Theorem 3.3.
In particular property (ii) is fulfilled, since there exists a subset of 9ft of positive measure
with x
•
v{x) > 0 and therefore the vanishing of x
•
v(x)\Vu\
2
on 0ft as assumed in (ii)
implies that |Vu|
2
= 0 on a subset of 3ft of positive measure. By the unique continuations
theorem u = 0 follows. In case p >
T
~ we could argue without the unique continuation
theorem, since in that case G is a strict variational sub-symmetry.
Now we want to investigate more systematically what kind of vector-fields £(x) and
what kind of functions a(x) can generate a variational sub-symmetry for the functional
£[u] via
x = t;(x), u = a(x)u.
To verify the infinitesimal criterion of Proposition 3.2, we use (5) and calculate
d
Je
£[ffew]|e=o = / -a\u\
p+1
+ uVa
•
Vu + a\Wu\
2
- VuD^Vudx
J
a
+
/
i)<
.iv
{
(i|v„p-
;r
i
T
i„r)*
+ J \Vu\
2
fa + ^H - VuDiVudx,
where the integral over div(Vau
2
/2) vanishes. In Pohozaev's original result with £ = —x
the term VuD^Vu turned out to be just — |Vw|
2
which equals ^|Vu|
2
. In order to
obtain similar best possible results for more general £ we are interested in those vector
fields £ such that
bD£b = ^
|b|
2
,
for all b 6 K". (13)
259
Lemma 4.1 In R", n > 3 all smooth vector-fields £ which satisfy (13) are (up to a
constant shift) given as linear combinations of the vector-fields
Y
t
j =
XjCi —
x
t
ej, for 1 < i < j < n,
z
i = (
x
i-Y,rfi
x2
j)ei + 2Y,rti
x
i
x
j
e
j, fori =
l...n,
where e; denotes the unit-vector in the x
t
coordinate-direction. The linear combinations
of these vector-fields are called conformal vector-fields.
The proof is obtained by showing first that all component-functions of £ are second-
order polynomials. It is then easy to find the precise form of £. In dimension n = 2 one
can show that the conformal vector-fields are exactly those vector-fields £ = (f
*,
£
2
) such
that ^(x,y) + i£
2
(x,y) is holomorphic in x + iy. Applications of this observation are
discussed in Reichel [5].
We use such a conformal vector-field £ and return to (12) to find
|«u = /
n
(-
fl
-^i)i«r-^
In order to achieve ^C[g
e
u]\
e=0
< 0 it is helpful to choose a = ^f div : £. If we note
that all our conformal vector-fields satisfy Adiv : £ = 0 we finally obtain
S
£
MI- -/
n
(^-
p
-Tl)w
P+ld
^-
Hence the sub-symmetry criterion of Proposition 3.2 will be fulfilled provided div£ < 0
and p > ^|
2
. Uniqueness of the solution « = 0of(ll) now follows from Theorem 3.3.
We sum up our results: .
Definition 4.2 A bounded domain Q C M", n > 3 is
called
conformally contractible if
there exists a conformal vector-field £ such that £(x)
•
v(x) < 0 on dQ with strict inequality
on a subset of 90 of positive measure. The vector-field £ is called an associated vector-
field to fi.
Theorem 4.3 (i) Letp > ^~. IfQ, C R
n
, n > 3 is a bounded conformally contractible
domain with associated vector-field £ and if additionally div£ < 0 in fi then (11)
has only the zero-solution.
(ii) In the special case p = ^| the additional assumption div£ < 0 is not
needed.
Remark 4.4 Part (i) of the theorem holds if
we
replace the non-linearity \u\
p
~
l
u in (11)
by f(u) and assume that F(tu)/t"-i is non-decreasing in t e R for all u e R, where
F(s) =
^f(T)dT.
°* The alternative div : £ > 0 and p < ^| must be discarded for bounded domains since we also
require £
•
v < 0 on d£l. It does however make sense if
Q,
is the complement of a bounded domain.