Non-stability of Paneitz-Branson equations in arbitrary dimensions

0.29 MB·

by  Laurent Bakri

Non-stability of Paneitz-Branson type
equations in arbitrary dimensions.
4 Laurent Bakri Jean-Baptiste Casteras §
∗† ‡
1
0
2

n
Abstract
a
J Let (M,g) be a compact riemannian manifold of dimension n 5.

8 ≥
We consider a Paneitz-Branson type equation with general coefficients
1
∆2u div (A du)+hu = u2∗−2−εu on M, (E)
] g g g
− | |
P
A 2n
whereA isasmoothsymmetric(2,0)-tensor,h C∞(M),2∗ =
g
. ∈ n 4
h and ε is a small positive parameter. Assuming that there exists a p−os-
t
a itive nondegenerate solution of (E) when ε = 0 and under suitable
m
conditions, we construct solutions u of type (u BBl ) to (E) which
ε 0 ε
[ blow up at one point of the manifold when ε tend−s to 0 for all dimen-

1 sions n 5.
v ≥
9
4 Keywords: Paneitz-Branson type equations, blow up solutions, Liapunov-
5
Schmidt reduction procedure.
4
.
1
0 Mathematics Subject Classification (2010): 35J30,35J60,35B33,35B35
4
1
:
v 1 Introduction and statements of the results
i
X
r Let (M,g) be a compact riemannian manifold of dimension n 5. We will
a ≥
be interested in solutions u C4,θ(M), θ (0,1), of the following equation
∈ ∈
P u := ∆2u div (A du)+hu = u 2∗−2u, (1.1)
g g − g g | |
∗Departamento de Matemática, Universidad Técnica Federico Santa María, 1680 Av.
españa Valparaíso,Chile. E-mail :[email protected].
†The first author was supported by PROYECTO BASAL PFB03 CMM, Universidad
de Chile, Santiago (Chile).
‡UFRGS,InstitutodeMatemática,Av. BentoGoncalves9500,91540-000PortoAlegre-
RS, Brasil Phone : [phone]. E-mail [email protected]
§The second author was supported by the CNPq (Brazil) project 501559/2012-4.
1
2n
where A is a smooth symmetric (2,0)-tensor, h C∞(M) and 2∗ = .
g
∈ n 4
Followingtheterminologyintroducedin[3], theoperatorP hasbeenrefe−rred
g
to as a Paneitz-Branson type operator with general coefficients. When A is
g
given by
(n 2)2 +4 4
A = A := − R g Ric , (1.2)
g paneitz g g
2(n 1)(n 2) − n 2
− − −
where R (resp. Ric ) stands for the scalar curvature (resp. Ricci curvature)
g g
n 4
with respect to the metric g, and h = − Q where Q is the Q-curvature
g g
2
with respect to the metric g which is defined by
1 n3 4n2 +16n 16 2
Q = ∆ R + − − R2 Ric 2,
g 2(n 1) g g 8(n 1)2(n 2)2 g − (n 2)2| g|g
− − − −
then P is the so-called Paneitz-Branson operator and equation (1.1) is re-
g
ferred to as the Paneitz-Branson equation. It is well known that the Paneitz
4
operator is conformally invariant, i.e. if g˜ = ϕn−4g then, for all u C∞(M),
∈
we have
Pn(uϕ) = ϕnn−+44Pn(u).
g g˜
We also point out that if (M,g) is Einstein (Ric = λg, λ R), then the
g
∈
Paneitz-Branson operator takes the form
P u = ∆2u+b∆ u+cu, (1.3)
g g g
n2 2n 4 n(n 4)(n2 4)
where b = − − λ and c = − − λ. More generally, when
2(n 1) 16(n 1)2
− −
b and c are two real numbers, the operator P defined in (1.3) is referred
g
to as a Paneitz-Branson type operator with constant coefficients. Existence,
compactness and stability of solutions to (1.1) when P is a Paneitz-Branson
g
type operator with constant coefficients, have been widely investigated this
last decade (see for example [5, 7, 8, 16, 19] and the references therein).
However, less isknown for solutions of (1.1) inthecase where P is aPaneitz-
g
Branson type operator with general coefficients. Esposito and Robert [4]
proved the existence of a non trivial solution to (1.1) under the hypothesis
that n 8 and minTr (A A ) < 0. In [18], Sandeep studied the
g g paneitz
≥ M −
stability of equation (1.1) in the following sense : he considered sequences of
positive solutions (u ) of
α α
∆2u div (A du )+a u = u2∗−1, u C4,θ,
g α − g α α α α α α ∈
2
where A are smooth (2,0) symmetric tensors and a are smooth functions.
α α
Sandeep proved that if A converges in C1(M) to a smooth symetric tensor
α
A , a converges in C0(M) to a smooth positive function a and u converges
g α α
weakly in H2(M) to a function u , then u is nontrivial provided that A
0 0 g
−
A is either positive or negative definite (generalizing a result of [9]).
paneitz
Recently, Pistoia and Vaira [15] studied the stability of (1.1) when it is the
Paneitz-Branson equation, namely they considered the following equation
∆2u div ((A +εB)du)+Q u = u 2∗−2u, (1.4)
g − g paneitz g | |
where ε is a small positive parameter and B is a smooth symmetric (2,0)
tensor. They proved that if (M,g) is not conformally flat, n 9 and there
≥
exists ξ M a C1 stable critical point (see below for the definition) of the
0
∈
Tr B(ξ)
g
function ξ , such that Tr B(ξ ) > 0, then equation (1.4) is
g 0
→ Weyl (ξ)
g g
| |
not stable, i.e. there exists ε > 0 such that, for any ε (0,ε ), equation
0 0
∈
(1.4) admits a solution u such that u (ξ ) + .
ε ε 0
ε−→→0+ ∞
The aim of this paper is to investigate the stability in the sense of Deng-
Pistoia of (1.1). We say that (1.1) is stable if, for any sequences of real
positive numbers (ε ) such that ε 0 and for any sequences of solutions
α α α
α−→→∞
(u ) C4,θ(M), θ (0,1), of
α α
∈ ∈
∆2u div (A du )+hu = u 2∗−2−εαu , (1.5)
g α − g g α α | α| α
bounded in H2(M), then up to a subsequence, u converges in C4(M) to
α
some smooth function u solution of (1.1). Deng and Pistoia [2] proved that
(1.1) is not stable if
a. n 7,Tr (A A )isnotconstantandminTr (A A ) > 0,
g g paneitz g g paneitz
≥ − M −
b. or n 8 and ξ M a C1 stable critical point of Tr (A A )
0 g g paneitz
≥ ∈ −
such that Tr (A A )(ξ ) > 0.
g g paneitz 0
−
Our main result shows that under suitable assumptions, equation (1.1) is not
stable for any n 5. In fact, inspired by the recent result of Robert and
≥
Vétois [17] on scalar curvature type equations, we investigate the existence
of families (u ) C4,θ(M) of blow-up solutions to (1.5) of type (u BBl ).
ε ε 0 ε
∈ −
Following the terminology of Robert and Vétois, we say that a blow-up se-
quence (u ) is of type (u BBl ) if there exists u C4,θ(M) and a bubble
ε ε 0 ε 0
− n−∈4
BBlε(x) = [n(n 4)(n2 4)]n−84 µε 2 , where x,xε M and
− − µ +d (x,x )2 ∈
(cid:18) ε g ε (cid:19)
µ R+ is such that µ 0, such that
ε ε
∈ −ε→→0
u = u BBl +o(1),
ε 0 ε
−
3
where o(1) 0. Before stating more precisely the results, we would like
−ε→→0
to recall that a solution of (1.5) is called nondegenerate if the kernel of the
linearization of the equation is trivial (see (2.3)). Let φ C1(M), we also
∈
recall that a critical point ξ of φ is said C1 stable if there exists an open
0
neighborhood Ω of ξ such that, for any point ξ Ω¯, there holds φ(ξ) = 0
0 g
∈ ∇
if and only if ξ = ξ and such that the Brower degree deg( φ,Ω,0) = 0.
0 g
∇ 6
We obtain :
Theorem 1.1. Let (M,g) be a compact riemannian manifold of dimension
n, A and h be such that P is coercive. Let u C4,θ, θ (0,1), be a
g g 0
∈ ∈
positive nondegenerate solution of (1.1). Assume in addition that one of the
following condition holds:
a. 5 n < 7,
≤
b. 8 n 13 and there exists ξ M a C1 stable critical point of
0
≤ ≤ ∈
(n 1)
ϕ(ξ) = − (Tr (A A ))(ξ)
g g paneitz
(n 6)(n2 4) −
− − (1.6)
2nu (ξ)ω
0 n−1
+ 1 , ξ M,
(n+2)(n(n 4)(n2 4))n−84ωn n=8 ∈
− −
such that ϕ(ξ ) > 0,
0
c. n > 13 and minTr (A A ) > 0,
g g paneitz
M −
then, for any ε > 0, there exists a solution u of type u BBl to (1.5). In
ε 0 ε
−
particular, (1.5) is not stable.
Let us notice that in the geometric case i.e. when A = A , the pre-
g paneitz
vious theorem only applies if 5 n 8. However, with a small modification
≤ ≤
of the proof, we can construct a solution of type u BBl to (1.5) when
0 ε
−
5 n 11 and A = A . More precisely, we prove the following result :
g paneitz
≤ ≤
Theorem 1.2. Let (M,g) be a compact riemannian manifold of dimension
n, A and h be such that P is coercive. Let u C4,θ, θ (0,1), be a positive
g g 0
∈ ∈
nondegenerate solution of (1.1). Assume that A = A . Then, for any
g paneitz
5 n 11 and any ε > 0, there exists a solution u of type u BBl to
ε 0 ε
≤ ≤ −
(1.5). In particular, (1.5) is not stable.
The proof of the theorems relies on a well known Lyapunov-Schmidt re-
duction procedure which permits to reduce the problem to a finite dimen-
sional one for which we defined a reduced energy. The solutions to (1.5) will
4
then be obtained as critical points of this reduced energy. We refer to [1]
and the references therein for more information on the Lyapunov-Schmidt
reduction procedure. We would like to emphasize that the proof of Theorem
1.1 is inspired by the previous work of Robert and Vétois [17]. Thus we will
keep their notations. We also want to point out that we use without proof
computations done in Deng and Pistoia [2] (for more details on these compu-
tations, see their paper). The plan of the paper is the following : in section
2 we introduce notations and perform the finite dimensional reduction. In
section 3 we study the reduced problem and prove Theorem 1.1. The error
estimate and the C1 uniform asymptotic expansion of the reduced energy are
done in the appendix.
Acknowledgements : The authors would like to thank F. Robert for his
comments and suggestions on a preliminary version of this paper.
2 Finite dimensional reduction.
Let (ξ ) be a sequence of points of M. In all the following, we will suppose
α α
up to extracting a subsequence that, for α large enough, all the points ξ
α
belong to a small open set Ω of M in which there exists a smooth orthogonal
frame. Thus, we will identify the tangent spaces T M with Rn for all ξ Ω.
ξ
∈
We recall that we suppose that P is coercive.
g
In all the following, we will denote by .,. , the scalar product, for
h iPg
u,v H2(M),
∈
u,v = ∆ u∆ vdV + A ( u, v)dV + huvdV,
h iPg g g g ∇g ∇g
ZM ZM ZM
where here and in the following dV stands for the volume element with
respect to the metric g, and . , for the associated norm which is then
k kPg
equivalent to the standard norm of H2(M). We denote by i∗ : Ln2+n4(M)
→
H2(M) the adjoint operator of the embedding i : H2(M) Ln2−n4(M), i.e.
→
for all ϕ Ln2+n4(M), the function u = i∗(ϕ) H2(M) is the unique solution
∈ ∈
of ∆2u div (A du) + hu = ϕ. Using this notation, we see that equation
g − g g
(1.5) can be rewritten as, for u H2(M),
∈
u = i∗(f (u)),
ε
where f (u) = u 2∗−2−εu. Before proceeding we recall some basic facts. It is
ε
| |
well known (see [11]) that all solutions u H2(Rn) of the equation
∈
∆2 u = u2∗−1 = unn−+44 in Rn
eucl
5
are given by
Uδ,y(x) = δ4−2nU(x−y), δ > 0, y Rn
δ ∈
where
n−4 n−4
U(x) = [n(n 4)(n2 4)]n−84 1 2 = αn 1 2 . (2.1)
− − 1+ x 2 1+ x 2
(cid:18) | | (cid:19) (cid:18) | | (cid:19)
It is also well known (see [12]) that all solutions v H2(Rn) of
∈
∆2 v = (2∗ 1)U2∗−2v
eucl −
are linear combinations of
n 4 x 2 1
V (x) = α − | | −
0 n 2 (1+ x 2)n−22
| |
and
x
i
V (x) = α (n 4) , i = 1,...,n.
i n − (1+ x 2)n−22
| |
Let χ : R R be a smooth cutoff function such that 0 χ 1, χ(x) = 1
r→r ≤ ≤
if x [ 0, 0] and χ(x) = 0 if x R ( r ,r ). We define, for any real δ
0 0
∈ − 2 2 ∈ \ −
strictly positive, ξ M and x M,
∈ ∈
Wδ,ξ(x) = χ(dg(x,ξ))δ4−2nU(δ−1exp−ξ1(x)),
where d (x,ξ) stands for the distance from x to ξ with respect to the metric
g
g and exp is the exponential map with respect to the metric g. We also
ξ
define, for any real δ strictly positive, ξ M and x M,
∈ ∈
n−4 d(x,ξ)2 δ2
Zδ,ξ(x) = χ(dg(x,ξ))δ 2 − n−2,
(δ2 +d(x,ξ)2) 2
and, for ω T M,
ξ
∈
exp−1x,ω
n−2 ξ g
Zδ,ξ,ω(x) = χ(dg(x,ξ))δ 2 n−2.
(δ2(cid:10)+d(x,ξ)2)(cid:11) 2
We denote by Π respectively Π⊥ the projection of H2(M) onto
δ,ξ δ,ξ
K = span Z ,(Z )
δ,ξ { δ,ξ δ,ξ,ei i=1..n}
6
respectively
K⊥ = φ H2(M)/ φ,Z = 0 and φ,Z = 0, ω T M .
δ,ξ ∈ h δ,ξiPg h δ,ξ,ωiPg ∀ ∈ ξ
n o(2.2)
We recall that a solution u of (1.5) is nondegenerate if the linearization of
0
the equation has trivial kernel, that is
K = ϕ C4,θ(M)/P ϕ = (2∗ 1) u 2∗−2ϕ = 0 . (2.3)
g 0
∈ − | | { }
(cid:8) (cid:9)
We are looking for solution u to (1.5) of the form
u = u W +φ ,
0 − δε(tε),ξε δε(tε),ξε
where u is a nondegenerate positive solution of (1.5), φ K⊥
0 δε(tε),ξε ∈ δε(tε),ξε
and
√t ε if n 8
ε
δ (t ) = ≥ , t > 0. (2.4)
ε ε 2 ε
(cid:26) (tεε)n−4 if 5 ≤ n ≤ 8
It is easy to see that equation (1.5) is equivalent to the following system
Π (u W +φ i∗(f (u W +φ ))) = 0, (2.5)
δε(t),ξ 0 − δε(t),ξ δε(t),ξ − ε 0 − δε(t),ξ δε(t),ξ
and
Π⊥ (u W +φ i∗(f (u W +φ ))) = 0. (2.6)
δε(t),ξ 0 − δε(t),ξ δε(t),ξ − ε 0 − δε(t),ξ δε(t),ξ
We begin by solving (2.6).
Proposition 2.1. Let u C4,θ(M) be a nondegenerate positive solution of
0
∈
(1.5). Given two real numbers a < b, there exists a positive constant C
a,b
such that for ε small, for any t [a,b] and any ξ M, there exists a unique
∈ ∈
function φ K⊥ which solves equation (2.6) and satisfies
δε(t),ξ ∈ δε(t),ξ
φ C ε lnε . (2.7)
δε(t),ξ Pg ≤ a,b | |
(cid:13) (cid:13)
Moreover, φ is cont(cid:13)inuously(cid:13)differentiable with respect to t and ξ.
δε(t),ξ
In order to prove the previous proposition, we set, for ε small, for any
positive real number δ and ξ M, the map L : K⊥ K⊥ defined by,
∈ ε,δ,ξ δ,ε → δ,ε
for φ K⊥,
∈ δ,ε
L (φ) = Π⊥ (φ i∗(f′(u W )φ)).
ε,δ,ξ δ,ξ − ε 0 − δ,ξ
We will first prove that this map is inversible for δ and ε small.
7
Lemma 2.1. There exists a positive constant C such that for ε small, for
a,b
any t [a,b], any ξ M and any φ K⊥, we have
∈ ∈ ∈ δ,ε
L (φ) C φ .
εα,δεα(tα),ξα Pg ≥ a,bk kPg
(cid:13) (cid:13)
Proof. Assumebycontradictionthatthereexisttwosequencesofpositivereal
(cid:13) (cid:13)
numbers (ε ) and (t ) such that ε 0 and a t b, a sequence of
α α α α α α
α−→→+∞ ≤ ≤
points (ξ )) of M and a sequence of functions (φ ) such that
α α α α
φ K⊥ , φ = 1 and L (φ ) 0. (2.8)
α ∈ δεα(tα),ξα k αkPg εα,δεα(tα),ξα α Pg α−→→∞
(cid:13) (cid:13)
To simplify notations, we set Lα = Lε(cid:13)α,δεα(tα),ξα, Wα =(cid:13)Wδεα(tα),ξα, Z0,α =
Z and Z = Z for i = 1,...,n where e is the i-th vector
δεα(tα),ξα i,α δεα(tα),ξα,ei i
in the canonical basis of Rn. By definition of L , there exist real numbers
α
λ , i = 0,...,n such that
i,α
n
φ i∗(f′ (u W )φ ) L (φ ) = λ Z . (2.9)
α − εα 0 − α α − α α i,α i,α
i=0
X
Standard computations give
Z ,Z ∆ V 2 δ , (2.10)
h i,α j,αiPg α−→→∞ k eucl ikL2(Rn) ij
where δ stands for the Kronecker symbol. Therefore, taking the scalar
ij
product of (2.9) with Z , using the previous limit and recalling that φ and
i,α α
L (φ ) belong to K⊥ , we deduce that
α α δεα(tα),ξα
n
f′ (u W )φ Z dV = λ ∆ V 2 + λ o(1),
εα 0 − α α i,α − i,αk eucl ikL2(Rn) | i,α|
ZM i=0 !
X
(2.11)
where, here and in the following, o(1) 0. It is easy to see using the
α−→→+∞
definition of W and Z and a change of variables that, for α large enough,
α i,α
f′ (u W )φ Z dV
εα 0 − α α i,α
ZM
= f′ (W )φ Z dV +o(1) (2.12)
εα α α i,α
ZM
= (2∗ −1−εα)δεα(tα)εαn+24 χα2∗−2−εαU2∗−2−εαViφ˜αdVg˜α +o(1),
Rn
Z
˜ n−4
where χα = χ(δεα(tα)|x|), φα(x) = δεα(tα) 2 χαφα(expξα(δεα(tα)x)) and
g˜ (x) = exp∗ g(δ (t )x). Since (φ ) is bounded in H2(M), passing to
α ξα εα α α α
8
˜
a subsequence if necessary, we can assume that (φ ) converges weakly to a
α α
function φ˜ H2(Rn). Letting α + in (2.12), we deduce that
∈ → ∞
f′ (u W )φ Z dV (2∗ 1) U2∗−2V φ˜dV = 0, (2.13)
ZM εα 0 − α α i,α α−→→∞ − ZRn i geucl
n+4
where we used that V is solution of ∆2 V = U2∗−2V in Rn and
i eucl i n 4 i
φ K⊥ to obtain the last equality. Therefore,−from(2.11) and (2.13),
α ∈ δεα(tα),ξα
we have
n
λ = o(1)+o( λ ).
i,α i,α
| |
i=0
X
From (2.9), this implies
φ i∗(f′ (u W )φ ) L (φ ) 0.
α − εα 0 − α α − α α α−→→∞
Since by assumption L (φ ) 0, we finally obtain that
εα,δεα(tα),ξα α Pg α−→→∞
(cid:13) (cid:13)
φ(cid:13) i∗(f′ (u W(cid:13) )φ ) 0. (2.14)
α − εα 0 − α α Pg α−→→∞
(cid:13) (cid:13)
Since (φ ) isbound(cid:13)edinH2(M), uptotaking(cid:13)asubsequence, wecanassume
α α
that φ converges weakly in H2(M) to a function φ H2(M). Then, using
α
∈
(2.14), we get, for any ϕ H2(M),
∈
ϕ,φ f′ (u W )ϕφ dV = ϕ,φ i∗(f′ (u W )φ )
h αiPg − εα 0 − α α α− εα 0 − α α Pg
(cid:12) ZM (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)(cid:10)ϕ φ i∗(f′ (u W )φ(cid:11) )(cid:12)
(cid:12)(cid:12) (cid:12)(cid:12) ≤ (cid:12)k kPg α − εα 0 − α α (cid:12)Pg
= o( ϕ (cid:13)). (2.(cid:13)15)
k kP(cid:13)g (cid:13)
We deduce from this that φ is a weak solution of P φ = (2∗ 1)u2∗−2φ. Since
g − 0
u is a nondegenerate solution of (1.5), we obtain that φ = 0. Therefore,
0
φ ⇀ 0 weakly in H2(M). Now we will show that φ˜ ⇀ 0 weakly in
α α
α→∞ α→∞
H2(Rn). Let ϕ˜ be a smooth function with compact support in Rn, we will
use (2.15) with, for x M,
∈
ϕ(x) = χ(dgξα(x,ξα))δεα(tα)4−2nϕ˜(δεα(tα)−1exp−ξα1(x)).
Thus, applying (2.15) to the previous ϕ and using a change of variable, we
9
have,
∆ φ˜ ∆ ϕ˜dV +δ (t )2 A ( φ˜ , ϕ˜)dV
g˜α α g˜α g˜α εα α g˜α ∇g˜α α ∇g˜α g˜α
Rn Rn
Z Z
+ δ (t )4 h(exp (δ (t )x))φ˜ ϕ˜dV (2.16)
εα α ξα εα α α g˜α
Rn
Z
= δ (t )4 f′ (u W (exp (δ (t )x)))φ˜ ϕdV +o(1),
εα α εα 0,α − α ξα εα α α g˜α
Rn
Z
where u (.) = u (exp (δ (t ).)). Nowit is easy tosee that, letting α
0,α 0 ξα εα α → ∞
in (2.16),
∆ φ˜∆ ϕ˜dV = (2∗ 1) U2∗−2φ˜ϕ˜dV .
eucl eucl g g
eucl − eucl
Rn Rn
Z Z
n+4
Thus φ˜ is a weak solution of ∆2 φ˜ = U2∗−2φ˜. So, from [12], we
eucl n 4
know that there exists λ R, i = 0,...,n, −such that φ˜ = n λ V . Since
i ∈ i=0 i i
φ K⊥ , using the same argument as in (2.13), we deduce that φ˜ 0.
α ∈ δεα(tα),ξα P ≡
Using one more time (2.15) with ϕ = φ , a change of variables and since
α
φ ⇀ 0 weakly in H2(M) and φ˜ ⇀ 0 weakly in H2(Rn), we get
α α
α→∞ α→∞
φ 2 = (2∗ 1 ε ) u W 2∗−2−εαφ2dV +o(1)
k αkPg − − α | 0− α| α
ZM
C φ2dV +C W 2∗−2−εαφ2dV +o(1)
≤ α | α| α
ZM ZM
C φ2dV +C U 2∗−2−εαφ˜2dV +o(1) 0.
≤ α | | α g˜α α−→→∞
ZM ZM
This yields to a contradiction with (2.8).
Proof of Proposition 2.1. It is easy to see that equation (2.6) is equivalent to
L (φ) = N (φ)+R ,
ε,δε(t),ξ ε,δε(t),ξ ε,δε(t),ξ
where
N (φ) = Π⊥ (i∗(f (u W +φ)) f (u W )
ε,δε(t),ξ δε(t),ξ ε 0 − δε(t),ξ − ε 0 − δε(t),ξ
f′(u W )φ),
− ε 0 − δε(t),ξ
and
R = Π⊥ (i∗(f (u W )) u +W ).
ε,δε(t),ξ δε(t),ξ ε ε 0 − δε(t),ξ − 0 δε(t),ξ
10