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Polyharmonic equations on manifolds and asymptotic analysis of Hardy-Sobolev equations with vanishing
singularity
Saikat Mazumdar
To cite this version:
Saikat Mazumdar. Polyharmonic equations on manifolds and asymptotic analysis of Hardy-Sobolev equations with vanishing singularity. Analysis of PDEs [math.AP]. Université de Lorraine, 2016.
English. �NNT : 2016LORR0047�. �tel-01754652v2�
Equations polyharmoniques sur les vari´ et´ es et ´ etudes asymptotiques dans
une ´ equation de Hardy-Sobolev
TH` ESE
pr´esent´ee et soutenue publiquement le 27 juin 2016 pour l’obtention du
Doctorat de l’Universit´ e de Lorraine
(Sp´ecialit´e: Math´ematiques) par
Saikat Mazumdar
Composition du jury
Directeurs de th`ese: Fr´ed´eric Robert de Institut ´Elie Cartan de Lorraine, Universit´e de Lorraine
Dong Ye de Institut ´Elie Cartan de Lorraine, Universit´e de Lorraine
Rapporteurs : Emmanuel Hebey de Universit´e de Cergy-Pontoise Patrizia Pucci de Universit`a degli Studi di Perugia
Examinateurs : Yuxin Ge de Institut de Math´ematiques de Toulouse, Universit´e Paul Sabatier
David Dos Santos Ferreira de Institut ´Elie Cartan de Lorraine, Universit´e de Lorraine
Tobias Weth de Institut f¨ur Mathematik, Goethe-Universit¨at Frankfurt
Institut ´Elie Cartan de Lorraine
Key words and phrases. Higher order elliptic equations, Best constants on manifolds, Struwe Decomposition, Coron-Type Solution, Topological Method,
Hardy-Sobolev Equation, Pointwise control, Mass of the Green function, Pohozaev identity.
Acknowledgments
Foremost, I would like to express my sincere gratitude to Fr´ed´eric Robert and Dong Ye, my thesis supervisors, for their continuous support of my doctoral study and research, for their patient guidance, enthusiastic encouragement and insightful critiques of my work. They are a source of motivation and personal inspiration. I am infinitely grateful to Fr´ed´eric, my supervisor here at Nancy. Fr´ed´eric provided me with every bit of guidance, assistance, support and expertise that I needed, while also giving me the freedom to do whatever I wanted. His advice on both research as well as on my career have been invaluable.
I would like to express my sincere thanks to Emmanuel Hebey and Patrizia Pucci for taking a interest in my work and agreeing to be my thesis reporters. I would also like to thank the other members of my thesis committee: Yuxin Ge, David Dos Santos Ferreira and Tobias Weth. I am very thankful to Tobias Weth for inviting me to Frankfurt and for the discussions we had there.
I am very grateful to J´erˆome V´etois for taking an interest in my work and inviting me to McGill University. I deeply appreciate the enriching and rewarding experi- ence, both mathematical and non-mathematical, I had in Montreal.
I gratefully acknowledge the funding sources that made my Ph.D work possible. I was funded by F´ed´eration Charles Hermite (FR3198 du CNRS) and R´egion Lor- raine.
Very special thanks to Institut ´Elie Cartan de Lorraine for giving me the opportu- nity to carry out my doctoral research and for their financial support. I am grateful to the secretaries and librarians at IECL Nancy, for all their assistance.
I would like to thank Sandeep, my Master’s supervisor for his guidance, help and encouragement.. TIFR-CAM, Bangalore, India, my alma mater, has played a cru- cial role in my academic career. I am particularly thankful to Adimurthi, Mythily Ramaswamy and Sandeep for being the outstanding researchers, teachers and role models that they are and for all their encouragement and support during my time at TIFR-CAM and thereafter. I feel extremely privileged to have been a student there.
A heartfelt thanks to all my friends here. I particular I would I like to mention Akshay, Kanika, Krishnan, Marharyta (Rita) and Tejaswini for their amazing com- pany, constant support and for all the great times that we shared. I am deeply thankful to Pierre-William Martelli, with whom I shared my office all these years.
I would also like to say a big “Thank-you” to my friends from TIFR-CAM: my classmates Abhishek, Debayan and Swarnendu; my seniors Rajib da, Shyam da, Kaushik da, Debdip da; my juniors Ali, Debo (Debabrata), Indra (Indranil); and
iii
wonderful Deep and Saumya. We learned together, played together and grew up together. I cherish their friendship and support.
I would like to thank Barun, especially for the many wonderful trips to Germany.
I also wish to thank Saikat Das, my friend from high school who is now at Rutgers and has influenced me immensely.
Lastly, and most importantly, I wish to thank my family - my parents: Arundhuti and Bidyut Kumar Mazumdar, my grandparents and my brother Saugata. I thank them for all their love, support and encouragement. I dedicate this thesis to them.
Contents
Acknowledgments iii
Chapter 1. Introduction 1
Part 1 2
Part 2 9
13
Bibliography 15
Part 1. Polyharmonic operators on Riemannian manifolds 19 Chapter 2. GJMS-type Operators on a compact Riemannian manifold: Best
constants and Coron-type solutions 21
2.1. Introduction 21
2.2. The Best Constant 24
2.3. Best constant and direct Minimizaton 30
2.4. Concentration Compactness Lemma 33
2.5. Topological method of Coron 37
2.6. Positive solutions 44
2.7. An Important Remark 47
2.8. Appendix: Regularity 48
2.9. Appendix: Local Comparison of the Riemannian norm with the
Euclidean norm 51
Bibliography 55
Chapter 3. Struwe’s decomposition for a Polyharmonic Operator on a
compact Riemannian manifold with or without boundary 57
3.1. Introduction 57
3.2. Definition of Bubbles 60
3.3. Preliminary analysis 62
3.4. Extraction of a Bubble 64
3.5. Nonnegative Palais-Smale sequences 74
Bibliography 77
Part 2. Asymptotic Analysis of a Hardy-Sobolev elliptic equation
with vanishing singularity 79
Chapter 4. Blow-up Analysis For a Sequence of Solutions of the Critical
Hardy-Sobolev Equations 81
v
4.1. Introduction 81 4.2. Some results on Hardy, Sobolev and Hardy-Sobolev inequalities onRn 85 4.3. Hardy-Sobolev inequality on ⌦and the case of a nonzero weak limit 88
4.4. Preliminary Blow-up Analysis 91
4.5. Refined Blowup Analysis I 104
4.6. Refined Blowup Analysis II 129
4.7. Localizing the Singularity: The Interior Blow-up Case 135 4.8. Localizing the Singularity: The Boundary Blow-up Case 146
Bibliography 167
Introduction
Nonlinear problems such as those that naturally arise from geometry and physics like the study of geodesics, minimal surfaces, harmonic maps, conformal metrics with prescribed curvature, Hamiltonian systems, solutions of boundary value problems and Yang-Mills fields, can all be characterised as critical pointsu of some functionalF on an appropriate space X, i.e.,F0(u) = 0. So one is con- cerned with problems of existence, location, multiplicity and qualitative properties of critical points in such contexts and how they relate to the (weak) solutions they represent for the corresponding Euler-Lagrange equations.
The points of maxima or minima, if it exists, are the simplest example of critical points forF. In general the functional F maybe unbounded on X or it may not achieve maximum or the minimum value(s). Locating critical levels for a smooth functionalF on a spaceX essentially reduces to capturing the changes in the topol- ogy of the sublevel setsFa ={x2X:F(x)< a} asavaries in R. Under the right conditions onF, classical Morse theory states that a non-trivial topology between Fa and Fb should detect a critical level c between a and b. The next simplest example short of considering minimisation consists of taking two pointsu0 andu1
both lying below levelawhich are not connected in Fa, but become so if one can climb above that level. This means that the the sublevelsFa andFb have di↵erent topologies for some b > a and this yields a critical point c between levels a and b. This setting is often called the Mountain-Pass Principle since in practice one insures that the two villages are disconnected below level aby showing that they are separated by a mountain range with minimal altitude exceedinga.
The above proposition identifies a potential critical level. The problem of ex- istence of a critical point then reduces to proving that a sequence (xm) satisfying
m!1lim F(xm) =cand lim
m!1kF0(xm)k= 0 is relatively compact inX. This is usually where the hard analysis is needed. Any function possessing such a property is said to satisfy the Palais-Smale condition at levelc, in short (P S)c.
In this memoir we study some variational elliptic partial di↵erential equations with- out the compactness property as described above. In problems of these kind one encounters ablow-up phenomenon caused by scale and conformal invariance, which makes it non-compact. However, this lack of compactness is not always the final word and a finer analysis of the behavior of non-compact sequences may provide us with some new conditions that could prevent such an eventuality. As a model case, one can think of the well studied stationary Schr¨odinger equation
⇢ u+hu=|u|n42u in ⌦
u= 0 on @⌦
1
where := div(r) is the Laplacian operator with negative sign convention, ⌦ is a bounded smooth domain inRn or a closed Riemannian manifold of dimension n 3 andh2C1(⌦).
Broadly speaking, this memoir is divided into two parts.
Part 1: We analyse the question of existence for some Polyharmonic boundary value problems with critical Sobolev growth on a compact Riemannian manifold.
Part 2: Here we do a blow-up analysis of the nonlinear elliptic Hardy-Sobolev equation with critical growth and vanishing boundary singularity.
We give a quick overview of these topics, providing also an outline of the content of this memoir.
Part 1
Let M be a closed manifold of dimension n 3 and let k be a positive integer such that 2k < n. In recent years, there have been extensive study of the relation- ship between the conformally covariant operators, that is, operators which satisfy some invariance property under conformal change of metric onM, their associated conformal invariants, and the study of the related partial di↵erential equations.
In their celebrated work Graham-Jenne-Mason-Sparling [27] provided a systematic construction of a family of conformally covariant operators (GJMS operators for short) based on the ambient metric of Fe↵erman-Graham [18]. More precisely, let M be the set of Riemannian metrics on M, then for all g in M, there exists a local di↵erential operatorPg:C1(M)!C1(M) such thatPg= kg+l.o.twhere
g := divg(r), and, givenu >0,u2C1(M) and defining ˆg=un42kg, one has (1.1) Pˆg(') =u n+2kn 2kPg(u') for all'2C1(M).
Moreover,Pgis self-adjoint with respect to theL2 scalar product. A scalar invari- ant is associated to this operator, namely the Q-curvature, denoted asQg. When k= 1, Pg is the conformal Laplacian and the Q-curvature is the scalar curvature multiplied by a constant. When k = 2, Pg is the Paneitz operator introduced in [40]. TheQ-curvature was introduced by Branson and Ørsted [9] and later gener- alised by Branson[7,8]. In the specific casen >2k, we have thatQg:= n22kPg(1).
Then, taking '⌘1 in (1.1), we get that Pgu= n22kQgˆun+2kn 2k on M. Therefore, prescribing the Q curvature in a conformal class amounts to solving a nonlinear elliptic partial di↵erential equation of 2kthorder. Results for the prescription of the Q curvature problem for the Paneitz operator (namelyk= 2) are in Djadli-Hebey- Ledoux [16] and Esposito-Robert [17] for instance. Recently, Gursky-Malchiodi [28] proved the existence of a metric with constant Q curvature (still fork = 2) provided certain geometric hypotheses on the manifold (M, g) holds. These hy- potheses have been simplified by Hang-Yang [31].
This leads us to investigate the existence ofu2C1(M),u >0, givenf 2C1(M), such that
(1.2) P u=f u2]k 1 in M,
where 2]k:= n2n2k andP :C1(M)!C1(M) is a smooth self-adjoint 2k-th order partial di↵erential operator defined by
P u= kgu+
k 1
X
l=0
( 1)lrjl...j1 Ali1...il,j1...jlri1...ilu (1.3)
here the indices are raised via the musical isomorphism and for alll2{0, . . . , k 1}, Al is a smooth symmetricT2l0-tensor field on M (that is: Al(X, Y) =Al(Y, X) for allT0l-tensorsX, Y onM). WhenP :=Pg, then (1.2) is equivalent to saying that Qˆg=n22kf with ˆg=un42kg.
Equation (1.2) has a variational structure. Since P is self-adjoint in L2, we have that for allu, v2C1(M).
Z
M
uP(v)dvg= Z
M
vP(u)dvg= Z
M k/2
g u k/2g v dvg+
k 1
X
l=0
Z
M
Al(rlu,rlv)dvg
where
l/2 g u:=
⇢ m
gu ifl= 2mis even r mgu ifl= 2m+ 1 is odd
IfP is coercive and f >0, then, up to multiplying by a constant, any non-trivial solutionu2C1(M) to (1.2) is a critical point of the functional
u7!JP(u) :=
R
M
uP(u)dvg
✓ R
M
f|u|2]k dvg
◆2/2]k
(1.4)
The natural space to study JP is the Sobolev space Hk2(M); where for 1 l k, Hl2(M) is the completion of C1(M) with respect to the u 7! Pl
↵=0kr↵uk2. Equivalently (see Robert [43]), Hl2(M) can also be seen as the completion of the spaceC1(M) with respect to the norm
kuk2Hl2:=
Xl
↵=0
Z
M
| ↵/2g u|2 dvg
By the Sobolev embedding theorem we get a continuous but not compact embedding ofHk2(M) intoL2]k(M). The continuity of the embeddingHk2(M),!L2]k(M) yields a pair of real numbersA, B such that for allu2Hk2(M)
kuk2L2] k A
Z
M
| k/2g u|2dvg+Bkuk2Hk2 1
(1.5)
Following the terminology introduced by Hebey [32], we then define
A(M) := inf{A2R:9 B2Rwith the property that inequality (1.5) holds} As in the classical case k = 1, the value of A(M) depends only on k and the dimensionn. More precisely, we letDk,2(Rn) be the completion ofCc1(Rn) for the normu7! k k/2ukL2(Rn), and define K0(n, k)>0
(1.6) 1
K0(n, k) := inf
u2Dk,2(Rn)\{0}
R
Rn| k/2u|2dx
⇣R
Rn|u|2]kdx⌘2
2] k
as the best constant in the Sobolev’s continuous embeddingDk,2(Rn),!L2]k(Rn).
It follows from Lions [36], Ge-Wei-Zhou [20], that the extremal functions for the Sobolev inequality (1.6) exist and are exactly multiples of the functions
Ua, =↵n,k
✓
1 + 2|x a|2
◆n22k
a2Rn, >0 where↵n,k’s are explicit.
For polyharmonic operators on a compact Riemannian manifold we obtain the followingbest constant result:
Theorem 1.1. (Mazumdar[37], see Chapter 2)Let (M, g)be a smooth, com- pact Riemannian manifold of dimensionnand letk be a positive integer such that 2k < n. Then A(M) = K0(n, k) > 0. In particular, for any ✏ >0, there exists B✏2Rsuch that for all u2Hk2(M) one has
✓Z
M|u|2]kdvg
◆2
2]
k (K0(n, k) +✏) Z
M
| k/2g u|2 dvg+B✏kuk2Hk2 1
As a consequence of this result, we obtain a description of noncompact bounded families inHk2(M). This is the extension of the PL Lions concentration compactness lemma for Riemannian manifolds:
Theorem 1.2. (Mazumdar[38], see Chapter 2)Let (M, g)be a smooth, com- pact Riemannian manifold of dimension nand let k be a positve integer such that 2k < n. Suppose (um)be a bounded sequence inHk2(M)such that µm*µweakly inM
(a) um*u weakly inHk2(M)
(b) µm=| k/2g um|2g dvg*µ weakly in the sense of measures (c) ⌫m=|um|2]k dvg * ⌫ weakly in the sense of measures Then we have:
(i) There exists an at most countable index setI, a family of distinct points {xi 2 M : i 2 I}, families of nonnegative weights {↵i : i 2 I} and { i:i2I}such that
⌫ =|u|2]k+X
i2I
↵i xi
(1.7)
µ | k/2g u|2g+X
i2I i xi
(1.8)
where x denotes the Dirac measure concentrated at x 2 M with mass equal to 1 .
(ii) In addition we have for all i2I
↵2/2
] k
i K0(n, k) i
(1.9)
In particular P
i2I
↵2/2
] k
i <1.
(iii) Furthermore, if u⌘0 and⌫(M)2/2
]
k K0(n, k) µ(M), then⌫ is concen- trated at a single point.
Another consequence of A(M) = K0(n, k) is the existence of minimum energy solutions to (1.2) when the functional JP goes below a quantified threshold (see Theorem 1.3 below). In general the conformal covariance of the geometric operator Pgyields obstruction to the existence of solutions to (1.2). In particular, it follows from [12] that on the canonical sphere (Sn,can), there is no positive solution u2 C1(Sn) to the equationPcanu= (1 +✏')u2]k 1, for all✏6= 0 and all first spherical harmonic'.
We remark that any weak solution to equation (1.2) is infact a classical solution.
The proof (Mazumdar [37], see Chapter 2) is based on the ideas developed by Van der Vorst [49]. Concerning the existence of weak solutions to equation (1.2) we first look for minimizers of the functionalJP. The result we obtain in this direction (in the spirit of Aubin) is
Theorem 1.3. (Mazumdar[37], see Chapter 2)Let (M, g)be a compact Rie- mannian manifold of dimensionn >2k, withk 1. LetP be a di↵erential operator as in (1.3)and letf 2C0,✓(M)be a H¨older continuous positive function. Assume that P is coercive onHk,02 (M). Suppose that
u2Ninff
Z
M
uP(u)dvg< 1 (supMf)
2 2]
k K0(n, k) , where
Nf :={u2Hk2(M) : Z
M
f|u|2]kdvg= 1}
Then there exists a minimizeru2Nf. Moreover, up to multiplication by a constant u2C2k(M)is a solution to
P u=f u2]k 1 inM.
In addition, if the Green’s function of P on M with Dirichlet boundary condition is positive, then upto changing signu >0 is a classical solution to
P u=f u2]k 1 inM.
In a remarkable result first Coron [11], and then Bahri-Coron [5] showed that the topology of the domain plays a role in proving the existence of solutions to equations like (1.11) whenk= 1. Coron [11] showed that the equation
8<
:
u=un+2n 2 in ⌦ u >0 in ⌦
u= 0 on @⌦
always admits a solution if the bounded smooth domain ⌦ ⇢ Rn, n 3, has a sufficiently small hole. Here = P
i@ii is the Laplacian with minus sign convention. The idea of the proof is to argue by contradiction and to use a minimax method for the corresponding energy functionalJ, based on a setT of non-negative functions which are homeomorphic to the (n 1) dimensional sphere ⌃around a point in ⌦. The set T is contractible in the positive cone in H1,02 (⌦). So if the
above equation does not admit a solution, then under certain conditions such a contraction ofT in H1,02 (⌦) will induce a contraction of⌃in⌦, giving the desired contradiction.
Infact Bahri-Coron [5] showed that the e↵ect of topology is much stronger, and extended the Coron’s result for the case when ⌦ has a non-trivial topology (homology). We note that the solutions obtained by these topological methods are in general not a minimiser of the corresponding energy functionalJP. The result of Coron [11] has been generalised for the polyharmonic case by Ge and al. [20], and Weth and al. [6] for domains inRn. The next theorem proved in [37] is in this spirit:
Theorem 1.4. (Mazumdar[37], see Chapter 2)Let (M, g)be a smooth, com- pact Riemannian manifold of dimensionnand letk be a positive integer such that 2k < n. We letP be a coercive operator as in(1.3). Let◆g>0be the injectivity ra- dius of the manifoldM. Suppose that the manifoldM contains a pointx0 such that the embedded(n 1)dimensional sphereSx0(◆g/2) :={x2M/ dg(x, x0) =◆g/2}is not contractible inM\{x0}. Then there exists ✏02(0,◆2g)such that the equation
(
P u=|u|2]k 2u in ⌦M
D↵u= 0 on @⌦M for |↵|k 1 (1.10)
has a non-trivialC2k(⌦M)solution for⌦M :=M\Bx0(✏0). Moreover, if the Green’s Kernel of P on⌦M is positive, then we can choose u >0.
In the original result of Coron [11] and its subsequent generalisations by Ge and al. [20], and Weth and al. [6] (fork 1) the authors work with a smooth domain inRnand assume that it has a small “hole”. In the context of a compact manifold, this assumption is not enough: indeed, the entire compact manifold minus a small hole might retract to a point. In section 7 of Chapter 2, we show that, in the case of the canonical sphere the existence of a hole is not sufficient to get solutions to equation (1.11), showing that the hypothesis of Theorem 1.4 is necessary.
One can also let (M, g) to be a smooth, compact Riemannian manifold of dimension nwith boundary. By this we understand thatM is a compact, oriented submanifold of ( ˜M , g) which is itself a smooth, compact Riemannian manifold without boundary and with the same metricg and dimensionn. As one checks, this includes smooth bounded domains of Rn. When the boundary @M 6= ;, we let ⌫ be its outward oriented normal vector in ˜M. Then in addition to equation (1.2) one can also consider the following general boundary value problem onM
(
P u=f|u|2]k 2u in M
@⌫↵u= 0 on @M for |↵|k 1.
(1.11)
wheref 2C0,✓(M) is a H¨older continuous function.
The Hilbert space Hk,02 (M) is similarly defined as the completion of the space Cc1(M) with respect to the normk·k2H2kas defined earlier. We say thatu2Hk,02 (M) is a weak solution of equation (1.11) if
Z
M k/2
g u, k/2g 'dvg+
k 1
X
l=0
Z
M
Al(rlurl')dvg= Z
M
f|u|2]k 2u'dvg
for all ' 2 Hk,02 (M). The functional JP is well defined on Hk,02 (M)\ {0} and its critical points corresponds to weak solutions of (1.11). Any weak solution to equation (1.11) is again a classical solution(see Chapter 2: Regularity) .
One can also consider the free functional IP(u) := 1
2 Z
M
uP(u)dvg 1 2]k
Z
M
f|u|2]k dvg
onHk,02 (M). Critical pointsu2Hk,02 (M) ofIPare again weak solutions to equation (1.11).
Definition 1.0.1. Let(X,k·k)be a Banach space andF 2C1(X). A sequence (um)inX is said to be a Palais-Smale sequence for F if (F(um))mhas a limit in Rwhenm!+1, whileDF(um)!0strongly in X0 asm!+1.
In [38] we describe the lack of relative compactness of Palais-Smale sequences for IP, which is due to the noncompact embeddingHk,02 (M),! L2]k(M). We obtain a characterization of the Palais-Smale sequences forIP as a sum of bubbles plus a critical point ofIP (which can be trivial), a result in the spirit of Struwe’s celebrated 1984 result. We consider Riemannian manifolds with or without boundary. The main idea is that often a non-convergent Palais-Smale sequence (um) or a blown up version of it splits up into a piece that converges weakly to a solution of the original problemu0and another one that converges to solutions of a closely related limiting problem. This is a very powerful result which is often used to show the existence of solutions to a variational problem.
In our case, because of the higher order of the polyharmonic operator, unlike the classical case of the Laplace operator (k= 1), in general there might be bubbles approaching the boundary of the domain, and this generates special type of bubbles which are solutions to the rescaled equation in the half space.
For⌦ any open domain ofRn, we letD2k(⌦) be the completion of Cc1(⌦) for the normu7! k k/2uk2. The limiting equations of (1.11) are
(1.12) ku=|u|2]k 2uin Rn, u2D2k(Rn)
(1.13)
( ku=|u|2]k 2u inRn
@⌫↵u= 0 on@Rn )
, u2Dk2(Rn)
where := Eucl is the Laplacian onRn(with the minus sign convention) endowed with the Euclidean metric Eucl. Associated to the functional IP is the limiting functional
E(u) := 1 2 Z
Rn( k/2u)2dx 1 2]k
Z
Rn|u|2]kdxfor allu2Dk2(Rn).
The fullHk2 decomposition of the Palais-Smale sequences for the functionalIP is given by the following theorem
Theorem 1.5. (Mazumdar [38], see Chapter 3) Let (um) be a Palais-Smale sequence for the functional IP on the space Hk,02 (M). Then there exists d 2 N
bubbles [(x(j)m),(rm(j)), u(j)], j = 1, ..., d, there exists u1 2 Hk,02 (M) a solution to (1.11) such that, up to a subsequence,
um=u1+ Xd j=1
Bx(j)
m,rm(j)(u(j)) +o(1) where lim
m!+1o(1) = 0 inHk,02 (M) and
IP(um) =IP(u1) + Xd j=1
E(u(j)) +o(1) asm!+1.
for definition of bubbles see: section 2 of Chapter 3. As one checks, for any non- trivial weak solutionu2D2k(Rn) of (1.12) or (1.13)
(1.14) E(u) ]:= k
nK0(n, k) n/2k
When the Palais-Smale sequence is nonnegative, the bubbles are indeed positive and correspond to positive solutions of (1.12). We then have:
Theorem 1.6. (Mazumdar [38], see Chapter 3) Let (um) be a Palais-Smale sequence for the functional Ip on the space Hk,02 (M). We assume that um 0 for allm2N. Then there existsu12Hk,02 (M)a solution to (1.11), there existsd2N sequences : (x(1)m), . . . ,(x(d)m) 2M, (rm(1)), . . . ,(rm(d))2 (0,+1) such that r(j)m ! 0 andrm(j)=o(d(x(j)m,@M))asm!+1for all j= 1, ..., d, and up to a subsequence,
um=u1+ Xd j=1
⌘⇣
(˜r(j)m) 1exp 1
x(j)m
(·)⌘
↵n,k
r(j)m
(rm(j))2+dg(·, x(j)m)2
!n22k
+o(1) wherelimm!+1o(1) = 0 in Hk,02 (M), and⌘ is a smooth cut-o↵function and˜rm(j)’s are such that for allj= 1, ..., d
↵!lim+1
rj↵
˜ rj↵
= 0 andr˜j↵< dg(xj↵,@M) 2 Moreover,
IP(um) =I(u1) +d ]+o(1) asm!+1 where ] is as in (1.14).
Whenk = 1 andM is a smooth bounded domain of Rn, Theorem 1.5 is the pio- neering result of Struwe [46]. There have been several extensions. Without being exhaustive, we refer to Hebey-Robert [33] fork= 2 and manifolds without bound- ary, Saintier [45] for thep Laplace operator, El-Hamidi-V´etois [29] for anisotropic operators and Almaraz [2] for nonlinear boundary conditions. When the manifold is the entire flat spaceRn, the decomposition is in the monograph by Fieseler-Tintarev [48].
Palais-Smale sequence are produced via critical point techniques, like the Mountain- Pass Lemma of Ambrosetti-Rabinowitz [3] or other topological methods (see for instance the monograph Ghoussoub [21] and the references therein).
For general higher-order problems, we also refer to Bartsch-Weth-Willem [6], Pucci- Serrin [41], Ge-Wei-Zhou [20], the general monograph Gazzola-Grunau-Sweers [19]
and the references therein.
These works are the object of my following two papers (submitted)
[37] GJMS-type Operators on a compact Riemannian manifold: Best constants and Coron-type solutions. See Chapter 2 of this memoir.
[38] Struwe’s decomposition for a Polyharmonic Operator on a compact Rie- mannian manifold with or without boundary. See Chapter 3 of this mem- oir.
Part 2
Let⌦be a bounded smooth oriented domain ofRn,n 3, such that 02@⌦. We define the Sobolev spaceH1,02 (⌦) as the completion of the spaceCc1(⌦), the space of compactly supported smooth functions in⌦, with respect to the norm
kuk2H1,02 (⌦)= Z
⌦
|ru|2 dx
We let 2⇤ := n2n2 be the critical Sobolev exponent for the embeding H1,02 (⌦),! Lp(⌦). Namely, the embedding is defined and continuous for 1p2⇤, and it is compact i↵1p <2⇤. Leta2C1(⌦) be such that the operator +ais coercive in⌦, that is, there exists a constantA0>0 such that for all'2H1,02 (⌦)
Z
⌦
⇣|r'|2+a'2⌘
dx A0
Z
⌦
'2 dx (1.15)
Solutionsu2C2(⌦) to the problem (1.16)
8<
:
u+a(x)u=u2⇤ 1 in⌦
u >0 in⌦
u= 0 on@⌦
(often referred to as ”Brezis-Nirenberg problem”) are critical points of the func- tional
u7!
R
⌦ |ru|2+au2 dx
✓R
⌦|u|2⇤ dx
◆2/2⇤ ,
and a natural way to obtain such critical points is to find minimizers to this func- tional, that is to prove that
µa(⌦) = inf
u2H1,02 (⌦)\{0}
R
⌦ |ru|2+au2 dx
✓R
⌦|u|2⇤ dx
◆2/2⇤
(1.17)
is achieved. There is a huge and extensive litterature on this problem, starting with the pioneering article of Brezis-Nirenberg [10] in which the authors completely solved the question of existence of minimizers for µa(⌦) whenais a constant and n 4 for any domain, andn= 3 for a ball. Their analysis took inspiration from the contributions of Aubin [4] in the resolution of the Yamabe problem. The case whenais arbitrary andn= 3 was solved by Druet [13] using blowup analysis.
In [25] and [24], Ghoussoub-Yuan and Ghoussoub-Kang suggested to approach the minimisation problem by adding a singularity in the equation as follows. For any s2[0,2), we define
2⇤(s) := 2(n s) n 2
so that 2⇤= 2⇤(0). Weak solutions u2H1,02 (⌦)\{0}to the problem 8>
<
>:
u+a(x)u= u2⇤|x(s)|s 1 in⌦
u 0 in⌦
u= 0 on@⌦.
Note here that 0 2 @⌦ is a boundary point. Such solutions can be achieved as minimizers for the problem
µs,a(⌦) = inf
u2H1,02 (⌦)\{0}
R
⌦ |ru|2+au2 dx
✓R
⌦
|u|2⇤(s)
|x|s dx
◆2/2⇤(s) fors2(0,2) (1.18)
Consider a sequence of positive real numbers (s✏)✏>0 such that lim
✏!0s✏= 0. We let (u✏)✏>02C2 ⌦\{0} \C1 ⌦ such that
8>
<
>:
u✏+au✏= u2
⇤(s✏) 1
✏
|x|s✏ in⌦,
u✏>0 in⌦,
u✏= 0 on@⌦.
(1.19)
Moreover, we assume that the (u✏)’s are of minimal energy type in the sense that R
⌦ |ru✏|2+au2✏ dx
✓R
⌦
|u✏|2⇤(s✏)
|x|s dx
◆2/2⇤(s✏) =µs✏,a(⌦) 1
K(n,0)+o(1) (1.20)
as ✏!0, whereK(n,0)>0 is the best constant in the Sobolev embedding which can be characterised as
1
K(n,0) = inf
u2D1,2(Rn)\{0}
R
Rn|ru|2dx
✓ R
Rn|u|2⇤ dx
◆2/2⇤
(1.21)
Indeed, it follows from Ghoussoub-Robert [22,23] that such a family (u✏)✏exists if the mean curvature of@⌦at 0 is negative.
The lack of compactness of the critical Sobolev embeddings potentially generates a noncompactness of families of solutions to equations like (1.16). When a family is not relatively compact, we say that it is ablow up sequence. In the past years, there has been a considerable abundance of descriptions of blowing-up sequence, starting with the description in the sense of measures by Lions [36] and the description of Palais-Smale sequence by Struwe [46] that we have discussed in the first part of this memoir. Other classical references for the blow-up analysis of nonlinear critical elliptic pdes are Rey [42], Adimurthi- Pacella-Yadava [1], Druet-Robert-Wei [15], Han [30], Hebey-Vaugon [34] and Khuri-Marques-Schoen [35]. In particular, for
sequences of solutions, the optimal pointwise control of blow up is in Druet-Hebey- Robert [14]. The analysis of the 3D problem by Druet [13] and the monograph [14] by Druet-Hebey-Robert were important sources of inspiration.
Here, we are interested in studying the asymptotic behavior of the sequence (u✏)✏>0
as ✏ ! 0. As proved in Proposition 3.2 of [39], if the weak limit u0 of (u✏)✏ in H1,02 (⌦) is nontrivial, then the convergence is indeed strong and u0 is a minimizer of µa(⌦). In the spirit of the C0 theory of Druet-Hebey-Robert [14], our first result is the following:
Theorem 1.7. (Mazumdar [39], see Chapter 4) Let ⌦ be a bounded smooth oriented domain of Rn,n 3 , such that02@⌦, and let a2C1(⌦) be such that the operator +a is coercive in ⌦. Let (s✏)✏>0 2 (0,2) be a sequence such that
✏lim!0s✏ = 0. Suppose that the sequence (u✏)✏>02H1,02 (⌦), where for each✏>0,u✏
satisfies(1.19) and(1.20), is a blowup sequence, i.e
u✏*0 weakly inH1,02 (⌦) as ✏!0 We rescale and define
v✏(x) = u✏(x✏+k✏x)
u✏(x✏) forx2 ⌦ x✏
k✏
where
k✏=|x✏|s✏2µ
2 s✏
✏ 2
and
µ
n 2
✏ 2 =u✏(x✏) = max
x2⌦u✏(x).
Then there existsv2C1(Rn)such that v6= 0 and for any⌘2Cc1(Rn)
⌘v✏* ⌘v weakly inH12(Rn) as✏!0 and
v✏ !v in Cloc1 (Rn) as✏!0 Furtherv(0) = 1 and it satisfies the equation
⇢ v=v2⇤ 1 inRn
v 0 inRn
Next we obtain strong pointwise control
Theorem 1.8. (Mazumdar [39], see Chapter 4) Let ⌦ be a bounded smooth oriented domain of Rn,n 3 , such that02@⌦, and let a2C1(⌦) be such that the operator +a is coercive in ⌦. Let (s✏)✏>0 2 (0,2) be a sequence such that
✏!0lims✏ = 0. Suppose that the sequence (u✏)✏>02H1,02 (⌦), where for each✏>0,u✏
satisfies(1.19) and(1.20), is a blowup sequence, i.e
u✏*0 weakly inH1,02 (⌦) as ✏!0 Then, there exists C >0 such that
u✏(x)C
✓ µ✏
µ2✏+|x x✏|2
◆n22
for allx2⌦ where
µ
n 2
✏ 2 =u✏(x✏) = max
x2⌦u✏(x).
Theorem 1.8 asserts that the pointwise control is the same as the control of the classical problem withs✏= 0: however, to prove this result, we need to perform a very delicate analysis of the blowup with the perturbations✏>0.
With this optimal pointwise control, we are able to obtain more informations on the localization of the blowup point x0 := lim✏!0x✏ and the blowup parameter (µ✏)✏. We let Ga : ⌦⇥⌦\ {(x, x) : x2⌦} ! R is the Green’s function of the coercive operator +a in ⌦with Dirichlet boundary conditions. For anyx2⌦ we writeGaxas:
Gax(y) = 1
(n 2)!n 1|x y|n 2 +gax(y)
where!n 1 is the area of the (n 1)- sphere. In dimensionn= 3 or when a⌘0, one has that gxa 2 C2(⌦\ {x})\C0,✓(⌦) for some 0 < ✓ < 1, and ga is called the regular part of the Green’s function Ga. In particular, whenn = 3 ora ⌘0, mx(⌦, a) := gax(x) is defined for all x2 ⌦and is called the mass of the operator
+a.
Theorem 1.9. (Mazumdar [39], see Chapter 4) Let ⌦ be a bounded smooth oriented domain of Rn,n 3 , such that02@⌦, and let a2C1(⌦) be such that the operator +a is coercive in ⌦. Let (s✏)✏>0 2 (0,2) be a sequence such that
✏lim!0s✏ = 0. Suppose that the sequence (u✏)✏>02H1,02 (⌦), where for each✏>0,u✏
satisfies(1.19) and(1.20), is a blowup sequence, i.e
u✏*0 weakly inH1,02 (⌦) as ✏!0 We let(µ✏)✏2(0,+1)and(x✏)✏2⌦be such that
µ
n 2
✏ 2 =u✏(x✏) = max
x2⌦u✏(x).
We define x0:= lim✏!0x✏. Suppose
x02⌦is an interior point.
Then
✏lim!0
s✏
µ2✏ = 2⇤K(n,0)2⇤2⇤2dn a(x0) for n 5
✏!0lim s✏
µ2✏log (1/µ✏)= 256!3K(4,0)2 a(x0) for n= 4
✏lim!0
s✏
µn✏ 2
= nb2nK(n,0)n/2gxa0(x0) forn= 3 ora⌘0.
wheregxa0(x0)the mass at the pointx02⌦for the operator +a, dn=
Z
Rn
⇣ 1
1 + n(n|x|22)⌘n 2 dx forn 5 ; bn= Z
Rn
1
⇣1 + n(n|x|22)⌘n+22
dx
and!3 is the area of the 3- sphere.
Suppose
✏lim!0x✏=x02@⌦.
Whenn= 3 ora⌘0, then as ✏!0
✏lim!0
s✏d(x✏,@⌦)n 2 µn✏ 2
= nn 1(n 2)n 1K(n,0)n/2!n 1
2n 2 .
Moreover,d(x✏,@⌦) = (1 +o(1))|x✏| as✏!0. In particularx0= 0.
Indeed, we also tackle the general casen 4 ora6⌘0. The detailed results are in Theorems 4.3 and 4.10 of Chapter 4.
The main difficulty in our analysis is due to the natural singularity at 02@⌦. In- deed, there is a balance between two facts. First, sinces✏>0, this singularity exists and has an influence on the analysis, and in particular on the Pohozaev identity.
But, second, sinces✏!0, the singularity should cancel, at least asymptotically. In this perspective, our results are twofolds.
The influence and the role of s✏ > 0 is much more striking. Compared to the case s✏ = 0, there is an additional term in the Pohozaev identity involving s✏. Heuristically, this is due to the fact that the limiting equation u=|x| su2⇤(s) 1 is not invariant under the action of the translations whens >0.
This part is the subject of my work:
[39] Blow-up Analysis For a Sequence of Solutions of The Critical Hardy- Sobolev Equations. See Chapter 4 of this memoir.
Regarding notation, we tried to make it as much unified as possible. Nevertheless, the main specific notation will be introduced chapter by chapter.
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