UNIVERSIDADE ESTADUAL DE CAMPINAS Instituto de Matemática, Estatística e Computação Científica
ALEXANDRE STEHLICK MARCONDES NORONHA TOMEI
EQUAÇÕES ELÍPTICAS COM POTENCIAL SINGULAR CRÍTICO
ELLIPTIC EQUATIONS WITH CRITICAL SINGULAR POTENTIAL
ALEXANDRE STEHLICK MARCONDES NORONHA TOMEI
ELLIPTIC EQUATIONS WITH CRITICAL SINGULAR POTENTIAL
EQUAÇÕES ELÍPTICAS COM POTENCIAL SINGULAR CRÍTICO
Thesis presented to the Institute of Mathematics, Statistics and Scientific Computation of the University of Campinas in partial fulfilment of the requirements for attainment of degree of Doctor in the area of Mathematics.
Tese apresentada ao Instituto de Matemática, Estatística e Computação Científica da Universidade Estadual de Campinas em atendimento parcial aos requerimentos para obtenção do grau de
Doutor na área de Matemática.
Supervisor: Djairo Guedes de Figueiredo
ESTE EXEMPLAR CORRESPONDE À VERSÃO FINAL DA TESE DEFENDIDA PELO ALUNO ALEXANDRE STEHLICK M. N. TOMEI, E ORIENTADA PELO PROFESSOR DJAIRO GUEDES DE FIGUEIREDO.
CAMPINAS 2017
Ficha catalográfica
Universidade Estadual de Campinas
Biblioteca do Instituto de Matemática, Estatística e Computação Científica Ana Regina Machado - CRB 8/5467
Tomei, Alexandre Stehlick Marcondes Noronha,
T594e TomElliptic equations with critical singular potential / Alexandre Stehlick Marcondes Noronha Tomei. – Campinas, SP : [s.n.], 2017.
TomOrientador: Djairo Guedes de Figueiredo.
TomTese (doutorado) – Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica.
Tom1. Equações elípticas singulares. 2. Teoria assintótica - Teoria da estimativa. 3. Equações diferenciais elípticas. 4. Regularidade ótima. I. Figueiredo, Djairo Guedes de, 1934-. II. Universidade Estadual de Campinas. Instituto de Matemática, Estatística e Computação Científica. III. Título.
Informações para Biblioteca Digital
Título em outro idioma: Equações elípticas com potencial singular crítico Palavras-chave em inglês:
Singular elliptic equations
Asymptotic theory - Estimation theory Elliptic differential equations
Optimal regularity
Área de concentração: Matemática Titulação: Doutor em Matemática Banca examinadora:
Djairo Guedes de Figueiredo [Orientador] Gabriela del Valle Planas
Lucas Catão de Freitas Ferreira João Marcos Bezerra do Ó Ederson Moreira dos Santos Data de defesa: 03-10-2017
Programa de Pós-Graduação: Matemática
pela banca examinadora composta pelos Profs. Drs.
Prof(a). Dr(a). DJAIRO GUEDES DE FIGUEIREDO
Prof(a). Dr(a). GABRIELA DEL VALLE PLANAS
Prof(a). Dr(a). LUCAS CATAO DE FREITAS FERREIRA
Prof(a). Dr(a). JOÃO MARCOS BEZERRA DO Ó
Prof(a). Dr(a). EDERSON MOREIRA DOS SANTOS
Agradeço ao professor Djairo Guedes de Figueiredo, meu orientador durante os anos do dou-toramento, pelos preciosos diálogos em nossos encontros semanais. Menciono o grande valor que suas diretrizes na aquisição da cultura matemática fundamental tiveram nos primeiros anos de minha formação doutoral, ora pela recomendação da mais fina literatura, ora pela sua própria exposição acerca dos temas.
Em 2015 estive sob orientação de Massimo Grossi, professor na Universidade de Roma ’La Sapienza’, a quem agradeço imensamente pela proposição do tema que viria a ser o assunto de minha tese doutoral. Além disso, devo a ele minha iniciação a diversas técnicas matemáticas de grande valor.
Agradeço a José Vitor Pena e Daniel Fadel pela grande amizade, pelo suporte e pelo incen-tivo que sempre manifestaram.
Agradeço a FAPESP pelo suporte financeiro que possibilitou este trabalho.
Agradeço a Universidade Estadual de Campinas e ao Instituto de Matemática e Computação Científica pelo ambiente acadêmico proporcionado.
Finalmente, obrigado a minha mãe pelo apoio e confiança que permitiram o desenvolvi-mento da minha personalidade, da qual este trabalho é atestado de uma curva biográfica.
O presente trabalho diz respeito ao estudo de equações elípticas lineares e não-lineares envol-vendo o operador
=jxj2;
onde é o operador Laplaceano em algum aberto de RN contendo a origem, com N 3, e é um parâmetro real. Para D 0, existe uma teoria linear bem estabelecida e há muitas ferramen-tas para abordar questões como existência, unicidade e regularidade de soluções para problemas não-lineares. Para o operador =jxj com 2 < < 2 e ¤ 0, muito da teoria linear
e não-linear do caso D 0 pode ser aplicada, incluindo a teoria padrão de regularidade elíptica. Neste caso, o operador também possui uma teoria espectral em L2 bem estabelecida, o que é de particular interesse em Física. O caso > 2 é interessante, claramente, somente quando o domínio do operador é ilimitado, visto que neste caso não haveria singularidade no infinito. Além disso, a Transformada de Kelvin reduz o caso > 0 ao caso < 0 desde que admitamos o domínio ilimitado.
O caso D 2 é, portanto, um caso limite quando apenas algumas propriedades do caso 2 < < 2 permanecem e outras são perdidas. De fato, a maioria dos resultados para D 2 são sensíveis ao sinal de , como será visto.
No primeiro capítulo, deduzimos o modelo físico por trás do operador em questão - uma síntese de axiomas da Mecânica Quântica e a dedução da equação de Schrodinger. Na opinião do autor, os matemáticos deveriam ter ao menos uma noção básica do modelo físico por trás do problema matemático, uma vez que este conhecimento abre uma janela para intuições sobre os problemas que de outra forma poderiam solicitar muito tempo para serem obtidas. Após o modelo físico, apresentamos algumas observações sobre os aspetos matemáticos do caso em que o potencial varia com o inverso do quadrado: V .x/D =jxj2, para que fiquem claras suas peculiaridades.
O capítulo 2 apresenta uma teoria básica para equações da forma u u=jxj2 D f . Ao
contrário do caso D 0, a existência, a unicidade e a regularidade não são totalmente compre-endidas, especialmente a última. Apresentamos alguns resultados alcançados por outros autores e também algumas demonstrações alternativas produzidas por nós. Existe um resultado padrão, creditado a Baras e Goldstein ([?]), referente à inexistência de soluções, resultado que é freqüen-temente citado por autores que abordaram o potencial 1=jxj2. No entanto, naquele artigo, os autores trabalharam com uma equação parabólica e a demonstração é bastante técnica e longa. Restringindo-nos ao caso elíptico, conseguimos obter uma demonstração diferente (Proposition 2.3.7). Usando uma técnica similar, também fomos capazes de provar uma generalização sutil de uma estimativa a priori dada por Dupaigne em [?] (Proposition 2.3.4).
por vários autores. No capítulo 4, transcrevemos um artigo completo escrito em conjunto com Massimo Grossi, onde desenvolvemos questões de regularidade e comportamento assintótico em relação ao parâmetro para soluções positivas. A regularidade (Theorem 4.1.1) é apresen-tada sob a forma de uma estimativa que fornece informações sobre o comportamento da solução perto da origem para todos os valores de , incluindo valores negativos. Esta estimativa genera-liza um resultado anterior de P. Han ([?]) (a mesma estimativa para > 0) e um resultado de [?] (soluções radiais). No mesmo artigo, provamos uma estimativa assintótica muito precisa para as soluções radiais W01;2quando ! 1 (Theorem 4.1.2) - esse resultado também generaliza os anteriores obtidos em [?].
This work is devoted to the study of linear and non-linear elliptic equations involving the oper-ator
=jxj2;
where is the Laplace operator in some open subset of RN containing the origin, with N 3, and is a real parameter. For D 0 we have a well established linear theory, and there are many tools to approach questions as existence, uniqueness and regularity of solutions for non-linear problems. For operators =jxj , with 2 < < 2 and ¤ 0, much of the linear
and non-linear theory of the case D 0 can be applied, including regularity theory. In this case, the operator also have a well-established spectral theory in L2, which is of special interest for Physics. The case 2 is interesting, clearly, only when the domain of the operator is unbounded, since then we have no singularity at infinity. Furthermore, Kelvin Transform reduces the case > 0 to the case < 0 since we allow the domain to be unbounded.
The case D 2 is, therefore, a limit case when some proprieties of case 2 < < 2 remain true and others are lost. In fact, most of the results for D 2 are sensitive to the value , as we will see.
In the first chapter we deduce the physical model behind the operator in question - it is basically a synthesis of axioms of Quantum Mechanics and the derivation of the Schrodinger equation. In author’s opinion, mathematicians should have at least a basic notion of the model behind the mathematical problem, since this knowledge opens a window for insights on the problems that otherwise could request an unknown large time to be realized. After the physical model, we present some mathematical concerns about the case of the inverse square potential V .x/D =jxj2 in order to explain its peculiarities.
Chapter 2 presents some basic theory for equations of the form u u=jxj2 D f . Unlike the case D 0, existence, uniqueness and regularity are not fully understood, specially the last one. We present some results achieved by other authors and also some alternative proofs given by us. There is a standard result, credited to Baras and Goldstein ([?]), concerning non-existence of solutions, that is usually cited by authors that worked with this operator - however, in that article the authors work with a parabolic equation and the proof is rather technical and long. Restricting ourselves to the elliptic case, we were able to get a different proof (Proposition 2.3.7). Using a similar technique we were also able to prove a subtle generalization of an a priori estimate given by Dupaigne in [?] (Proposition 2.3.4).
Chapters 3 and 4 concerns the Dirichlet problem for the non-linear equation u u=jxj2 D jujp 1u, with p > 0. In chapter 3 we present previous results achieved by several authors. In
gives information about the behaviour of the solution near 0 for all values of , including neg-ative values. This estimate generalizes a previous result from P. Han ([?]) (the same estimate for > 0) and a result from [?] (radial solutions). In the same article we prove a very precise asymptotic estimate for the W01;2radial solutions when ! 1 (Theorem 4.1.2) - this result also generalizes previous ones obtained in [?].
1 Introduction 12
1.1 Physical background: Schrodinger operators . . . 12
1.2 The inverse square potential . . . 18
2 Linear theory 21 2.1 A transformation . . . 21
2.2 Maximum principle . . . 23
2.3 The Dirichlet problem . . . 28
2.4 Strong solutions . . . 34
3 Some results on the non-linear problem 35 3.1 Non existence results . . . 35
3.2 Solutions in the whole space and balls . . . 38
4 Regularity and asymptotic approach to semilinear elliptic equations with singular pot. 41 4.1 Introduction . . . 41
4.2 Presentation of the method . . . 45
4.3 Proof of Theorem 1 . . . 50
Introduction
The role played by the Laplace operator is of known uttermost importance in Mathematics and in Sciences. Its major role is played in Physics, and it is due to to the fact that is the part of Hamiltonian operator associated with the kinetic energy. The full operator associated with the Hamiltonian takes the form
V;
where V D V .x/ is the potential energy. The results proved in this work concern a specific case of the potential energy, actually a limit case of it in some sense that will be explained afterwards, namely, when V is proportional tojxj 2. The Hamiltonian operator appears in the Schrodinger
equation, and its derivation is based in the formula for position and momentum operators. In the first section we make a brief exposition of the physical model that leads us to operators V . In the second part we present mathematical concerns on the inverse square potential case.
1.1 Physical background: Schrodinger operators
Only in this section we will use bold letters as x to refer to points of the euclidean space. Also, we will concern about regularity or degree of differentiability of functions only if matters for the specific situation.
In Hamiltonian viewpoint of Mechanics, position x and momentum p are seen as indepen-dent variables for the description of the motion of a particle (or system of particles), and so we turn attention to classical observables, i.e, scalar quantities that can be expressed in terms of x and p. Among the most important classical observables is the Hamiltonian, defined by
H.x; p/D 1 2mjpj
2
C V .x/;
wherejpj2=2m is the kinetic energy and V .x/ is the potential energy, which we are assuming do
not depend on time. One should see that it is nothing but the mechanical energy of the particle.
The time evolution of the position and momentum is described by the equations: dpj dt D @H @xj ; dxj dt D @H @pj
where j D 1; 2; :::; N and pj; xj denotes the components of x and p, respectively. These
equa-tions, called Hamilton’s equaequa-tions, are equivalent to Newton’s Second Law. Using Chain Rule and Hamilton’s equations we obtain the time evolution equation for any observable f .x; p/:
df dt D ff; H g; where ff; gg WD N X j D1 @f @xj @g @pj @f @pj @g @xj
is called the Poisson bracket of f and g. Ifff; H g D 0 then f is conserved along .x.t/; p.t//. From mathematical viewpoint, the Hamiltonian formulation has the obvious advantage of do not relies at the empiric concept of force, and so is more suitable fore an theoretical approach. Often we cannot obtain an explicit formula for the the solution(s) .x.t /; p.t // of Hamilton’s equations, and it is then due to Poincare (among others) the development of techniques that allows us to get qualitative information about the shape of the curves .x.t /; p.t //2 RN RN. A great deal of these information are obtained by looking to the linear Taylor approximation of the function .x; p/ ! . rxH;rpH /. The reader interested in learn something about these
techniques could give a look at [?].
In Quantum Mechanics, position and momentum of a particle are no longer seen as vectors in the euclidean space RN, but as random variables X and P . The reason behind the pro-posal of this new approach are empiric propositions as de Broigle’s hypothesis, which seems to imply some insurmountable uncertainties on measurements of position and momentum of a particle. In this fashion, we should expect the physical laws to appear, for example, as axiomatic propositions concerning the density functions of X and P , by analogy with Hamilton’s equa-tions. Canonically, however, it is assumed the existence of an intermediate meaningful function W .x; t/ 2 RN Œ0; 1 ! .x; t/ 2 C which encodes all the information necessary in order to work out the density functions of X and P . The density function of the position X is, then, defined as
fX.x; t / D .x; t/ .x; t /D j .x; t/j2:
And so it is naturally requested that .; t/ 2 L2
.RN/ for all t 0, actually,R
RN j .x; t/j
2d xD
1. Physically, .x; t / is thought as the wave function associated with the particle. The choice of C as co-domain of is pragmatic: any function D .x/ 2 L2 can be spanned by
fei kx W k 2 RNg in the sense that, for in a dense subspace as Schwartz space, we have
.x/D .2/ 1=2R
RN .k/eO
i kxd k; if we take the co-domain of to be R, then we would have
to work with sines and cosines, leading to big and boring computations. The wave functions x! ei kx=p2, besides not square integrable in RN (for this we can imagine the particle living in the unit ball), are taken as prototypes for the physical laws in the fashion we will now see.
The density of the momentum P is defined by taking de Broigle’s hypothesis as back-ground physical law, i.e, by assuming that a particle with the prototype wave function .x; t /D .x/ D ei kx=p2, 0 x 2, with k integer, has momentum „k, where „ is the Plank’s constant. It is also taken as a “natural superposition assumption" that if perform the measure the momentum of a particle with wave function .x/D pa1
2e
k1xCpa2
2e
k2x, where a2
1C a22D 1,
we obtain either P D „k1with probability a21 or P D „k2 with probability a22. This
assump-tion is further assumed for countable combinaassump-tions of ei kx and for “uncountable combinations" in the sense of .x/ D p1
2
R
R .k/eO
i kxd k. We also remark that R
Rj j 2 D R
Rj Oj 2, by
Plancherel’s theorem. Therefore the natural candidate to be the density of the momentum (di-vided by„) is
fP =„.k; t / D j O .k; t /j2:
One can see that if we take .x/D ei k0x=p2, with k
0 2 R, then O .k/ is the distribution ı.k
k0/, which implies the particle has momentum„k0with probability 1. Formally speaking, until
now the only physical ingredients we put in our development are the de Broigle’s hypothesis and some sort of superposition principle that gives sense to the momentum of a linear combination of “standard waves”; the remains are just statistical technology. The next physical ingredient to be added is the behaviour in time of the states .x; t /. It is nothing but assume that a particle in a standard time-dependent state .x; t /D 0.x/e i !t has constant energy E D „!,
the “quantum" of energy suggested by Plank’s model of blackbody radiation. So t.x; t / D
i !e i !t 0.x/ D i„ 1E .x; t /, which means that solves a linear differential equation.
Since most states of interest can be obtained by linear combinations of waves 0.x/e i !t by
using the formula .x; t / D .2/ 1RRRRN K.k; !/ei.kx !t/d k d!, where K is obtained by taking the Fourier transform of with respect to x and then the inverse Fourier transform with respect to t , it is reasonable to assume that every possible state of a particle with constant energy E must solve the Schrodinger equation:
d
dt D
E i„ :
Nevertheless, since the Hamiltonian H D P2=2m C V .X/ is also a random variable, the
case of constant energy E is merely the case where H has density fH.h/ D ı.h E/. If
the state evolve as .x; t / D .a1e i !1t C a2e i !2t/ 0.x/, with !1 ¤ !2, then we invoke a
superposition principle to postulate that measurements for the energy of the particle could only give the results E1 D „!1 or E2 D „!2, and in additional, t.x; t / D . i!1a1e i !1t
i !2a2e i !2t/ 0.x/WD i„ 1.E1a1 1.x; t /C E2a2 2.x; t //, where j.x; t /D e i !jt 0.x/.
Still, more generally, for .x; t /D R
R O2.x; !/e i !td! we have t.x; t / D R R O2.x; !/ . i !/e i !td! D i„R RE.!/ O2.x; !/e
i !td! . These computations highlight the role of
some linear operator OH , whose eigenvalues are the allowed values of energy of the particle and such that
d
dt D
1 i„H :O
This can be seen as a hint to introduce an operator language in quantum mechanics. The possible values of some observable f would appear as eigenvalues of certain operator Of which could be derived from f in some way. Furthermore, it should be possible to derive the language of probability distributions from the operator description, i.e., the operator Of should encode the probability density for measurements of the observable f . One of the great advantages of the operator language, despite of the concise writing, is that it allows a more general description of the states, as we will see. Before further commentaries on operator language, let us summarize the ingredients of the model that we assumed so far:
(i) Observables have a statistical description in terms of probability distributions, in opposi-tion to the deterministic point of view in classical mechanics.
(ii) de Broigle’s hypothesis and superposition principle motivate a relation between position and momentum.
(iii) Plank’s relation and superposition principle motivate the time evolution equation. As said before, the idea of the operator language is to think the possible states of observables as eigenvalues of some operator. A priori, there may be more than one operator that encodes the same probability distribution for a given observable, so it is wise to choose, being possible, a self-adjoint operator in a Hilbert Space, since then spectral theorems to derive existence of spectra and basis of eigenvectors would be at disposal. It lead us to the first “axiom":
A1 - The (kinematic) state of a particle (or system of particles) is represented by a non-zero vector in a fixed complex Hilbert Space H. Any two linear dependent vectors always represent the same state.
We denote by h; i the inner product on H. Since linear dependent state represents the same vector, being preciser, the states actually lives in the projective space P .H/. We usually take a representative with norm 1, i.e, the normalization condition k kH D 1 inspired by
R j j2 D 1 of the probabilistic description. Now that we have prepared the Hilbert space, the
next step is to assign to any given observable f D f .x; p/ an operator Of , and the probability distribution of values of f should be possible to be derived from Of ; still this choice should be consistent with our observations for the Hamiltonian OH , namely, (i) if is an eigenvector of
O
f with eigenvalue (means Of D ), then the only possible value of f at the state is . From this one could see how good would be H has a basis of eigenvectors of Of , and this is why we will request Of to be a self-adjoint operator on H. Suppose H has an denumerable orthonormal basis f 1; 2; :::g and let 2 H so that D Pj 1aj j. Then h ; Of i D
P
i;j 1aiajh i; Of ji D Pj 1jaj2, which isPi 1jprob.f D j/, the expected value of
f at state (denoted byhf i ). Analogously, for higher momentsh ; Ofm i D hfmi . These
remarks motivates the following axiom
A2 - To each observable f D f .x; p/ there is assigned a self-adjoint operator Of in the Hilbert spaceH. The probability distribution of measurements of an observablef for a particle (or system) in a state represented by an unit vector 2 H satisfies
hfmi D h ; Ofm i; mD 1; 2; 3; :::
By using the spectral theorem, it may be shown the existence of only one probability measure fitting those moment conditions. For a particle moving in the real line R, whose probability den-sity for position was defined asj j2, let us compute the operator X induced by the observable f .x; p/D x. In this case the Hilbert space is taken as L2.R/ and for a state 2 L2,R 2D 1, we havehxmi D R
Rx m
j .x/j2dx, and by axiom A2 it has to be equal toR
R
Xm . So the
obvious choice is
X .x/ D x .x/:
It is immediate from this definition that every x for which .x/ ¤ 0 is an eigenvalue of X. One can also check that X is in fact a symmetric operator. Observe that X do not maps L2into itself, in fact, x .x/ may fail to be square integrable. In these cases the operator X has to be defined just in a dense subspace of H (as for example Schwartz functions or smooth functions with compact support), i.e, X is faced as an unbounded self-adjoint operator on H. For these operators the spectral theorem gives a continuum spectrum, as expected for position, which in most classical examples can assume a continuum of values (the eigenvalues). In order to derive the momentum operator P for a particle moving in the real line R, we first recall the density of momentum fP =„.k; t / D j O .k; t /j2. Call k D p=„ and let be, for example, a Schwartz function. Then hkmi D R
Rk
mj O .k/j2d k D R
R .k/O
km .k/d k, which by Par-O
seval’s identity is equal toR
R .x/ ../m .// L .x/ dx DO R R .x/ . i /m@m x .x/ dx, and so hpmi DR R .x/ . i„/m@m x .x/ dx D R R .x/ . i„@ x/m .x/ dx. It follows that P .x/D i„ @x .x/:
So P is a symmetric operator densely defined in L2. If the position and momentum operators X and P are known, then we can invoke the functional calculus that is corollary of the spectral theorem in order to conclude that one variable observables like f .x/ and g.p/ are mapped into f .X / and g.P /, respectively (provided f , g are mensurable). In this way, there is no ambiguity when we define the Hamiltonian operator to be OH D P2=2m C V .X/, for example. The
problems start when we first look to the observable f .x; p/ D xp. First we notice that X and P do not commute: XP .x/ D X. i„@x /.x/ D i„x@x .x/ D i„. .x/ @x.x // D
i„ .x/ C P .x /.x/ D i„ .x/ C PX .x/. So
XP PX D i„I:
This is nothing but the operator version of the famous Principle of Uncertainty. The standard version of it is written as
xp „
2;
where x2 and p2 are the variances of the probability distributions of measures of position and momentum, respectively. The standard version can be deduced from the operator ver-sion as follows: x22
p D h.x hxi /2i h.p hpi /2i D h ; .X hxi I /2 ih ; .P
hpi I /2 i. The operators X0WD X hxi I; P0WD P hpi I are symmetric because X and P do; furthermore by Cauchy-Schwartz inequality we have x2p2 D hX0 ; X0 ihP0 ; P0 i jhX0 ; P0 ij2 jImhX0 ; P0 ij2 D jhX0 ; P0 i hX0 ; P0 ij2=4, and so x2p2 jh ; .X0P0 P0X0/ ij2=4 D jh ; .XP PX / ij2=4D jh ; i„ ij2=4D „2=4, which gives the result. The non-commutation of X and P immediately produces the doubt of which op-erator should be associated with the observable f .x; p/ D xp (also, neither XP or PX are self-adjoint). This discussion lead us to the topic of quantization schemes, i.e, trials to associate operators to general observables - for more information we recommend Chapter 13 of [?].
The time evolution of the state .x; t / is described by the axiom concerning the Schrodinger equation, motivated before:
A3 - Let .x; t / be state of a given particle at time t . Then for x fixed, .x; t / satisfies the Schrodinger equation,
d
dt D
1 i„H :O
whereH is the Hamiltonian. For a particle moving in the euclidean space RN, let Xj and Pj
be the operators associated with the observables xj and pj, the j -th component of position and
momentum, respectively. Since Pj D i„@xj, we get P
2
j D „ 2@2
xjxj and so the operator associated with p2 DP j pj2is 1 „2P 2 D : The Laplace operator D P
j@2xjxj is, therefore, the quantization of the kinetic energy, and so a differential operator of central role in quantum mechanics. The standard quantization for the potential energy V .x/ gives OV .x/D V .x/ .x/, and so the Hamiltonian operator takes the form
O
H D „
2
2mC V .x/
Of great importance is the knowledge of the eigenvectors of the operator OH , since it is a self-adjoint in a Hilbert Space. The problem of finding these eigenvalues consists exactly of find
pairs .E; / solving
O
H D E
This eigenvalue problem is often called time independent Schrodinger equation, and it is nothing but the linear elliptic equation
V .x/ D E :
Observe that, besides its name, this equation is not a case of the Schrodinger equation, but just the problem of eigenvalues for the Hamiltonian.
When the particle (or system) is absolutely isolated from any interference, the time evolution is fully described by the Schrodinger equation, a continuous evolution which depends on the initial state of the system. However, if the system is not completely isolated we must introduce a new axiom to tell us what happens to the state of the system.
A4 (Collapse) - Assume that a particle (or system) is evolving according to the Schrodinger equation until a timet0 when a measurement of an observable f is performed. If the result
of the measurement is a number 2 R, then the wave function discontinuously changes to a random eigenfunction (relative to the eigenvalue) of Of .
This assumption led to several debates in the 20th century, manly because highlights heavy non-determinism. Besides observables are seen as random variables, the Schrodinger equation gives a deterministic time evolution for these random variables; the Axiom 4, however, says that even the state of the system (which encodes the probabilities) can suffer random alter-ations. Second to Copenhagen interpretation, the act of measuring an observable makes the wave function immediately collapse to an eigenvector of the operator, so that we measure the related eigenvalue. Then in some extent, it is like the mere definition of an observer changes the system, which is quite bizarre. There were many criticizers of this interpretation, as Einstein when he says about God does not plays dices, etc. A very reasonable critic is the one about the nature of the observer, since it seems to be considered as classical object by Copenhagen -if we consider the observer and anything else around the system as as a new quantum system, the collapse hypothesis could be avoided perhaps. An alternative way to understand the col-lapse became popular in the last decades of XX century: when a quantum system is isolated, it keeps coherent, and so preserves its quantum characteristics, but when it starts to interact with environment, the coherence is broken giving rise to classical behaviours.
1.2 The inverse square potential
We have seen that the quantization of the Hamiltonian H D p2=2m C V .x/ leads to the so called Schrodinger operator OH D C V .x/, where is the Laplace operator and V .x/ is the multiplication operator by the potential function. In this section we will present the subject of
our concern in the present work, namely, the potential V.x/D
jxj2;
were is a real parameter.
The first issue regarding this case, more known among theoretical physicists, concerns the “time independent" Schrodinger equation, i.e., the eigenvalue problem u V .x/u D Eu in RN. One remarkable difference between Quantum Mechanics and other previous physical models is the level of accuracy of the mathematical language. The choice to address Q.M. by naive mathematics as finite dimensional linear algebra or didactic simplifications can lead to paradoxes. In this way, a careful description of the proprieties of the operator V is necessary; in special, how things change when we deform the potential V . Of particular interest in Physics are the singular potentials of the formjxj d, with d > 0. From the point of view of Spectral Theory, these potentials are well behaved provided d < 2 (see [?]). For d D 2 the Hamiltonian is no longer self-ajoint in L2.RN/ and a much more careful approach by by theory of unbounded operators in necessary.
Regarding existence of solutions of elliptic equations, the inverse square potential is also a case of special issues. Consider the problem
8 ˆ < ˆ : u jxjdu D f in ˝nf0g uj@˝ D 0 (1.2.1)
where d 2 R, ˝ RN is open and bounded, N 3 and 0 2 ˝. For f 2 L2.˝/, we can consider the problem in the generalized sense, i.e, critical points, in W01;2.˝/, of the functional
J ŒuD 1 2 Z ˝ jruj2 jxjdu 2 Z ˝ f u:
The first basic concern is about well-definition of this functional, i.e., if it is in fact finite for every u 2 W01;2.˝/. By Holder inequality, R u
2=
jxjd .R jxj dN=2/2=N
kuk2=22 , where 2D 2N=.N 2/ is the critical Sobolev exponent. ThenR u2=
jxjd <1 provided dN=2 < N , i.e, provided d < 2. So for d < 2 existence of generalized solutions is assured. If d D 2, however, there is an inequality that help us to address the existence problem. Hardy’s inequality (which will be proved afterwards, in Proposition 2.3.1) will imply that
N 2 2 2Z ˝ u2 jxj2 Z ˝ jruj2
for all u2 W01;2.˝/. Therefore the functional J is well defined provided < N, where
N D N 2 2
2
(1.2.2)
will be an important parameter in our work. We will prove in next chapter that if > N and f is integrable non-negative function, then the Dirichlet problem for u u=jxj2D f has no
positive solution even in the distributional sense (L1 solutions). Existence of solutions for the case D N is still not well comprehended.
Another problem concerning the inverse square potential is the regularity of solutions at 0. Far from the origin the standard elliptic theory clearly applies and so over there nothing is new. Consider problem the problem (1.2.1) with f 2 Lq for q > 1. By Holder inequality, u=jxj2in q-integrable at 0 provided q < N=d . If q then lies on the interval .N=d; N=2/, elliptic theory could be applied and global boundedness would be deduced. If d D 2, however, the standard Moser iteration argument doesn’t apply and a new approach is necessary. Actually, in the case > 0 the solution is unbounded near 0 (see [?]).
In chapter 4, we present an article where the author, in cooperation with Massimo Grossi, studied positive solution of the non- linear Dirichlet problem for u u=jxj2 D up, p > 1. We proved a regularity estimate near the origin for solutions for every value of . For negative values, it was obtained that every generalized solution of the problem vanishes at 0. Moreover, when goes to 1 the solution lacks analyticity at 0. Our proof uses an iteration procedure based on estimates in weighted Sobolev Spaces and an iteration procedure using the Caffarelli-Kohn-Nirenberg inequality.
Linear theory
Let ˝ RN be a smooth bounded domain containing 0, with N 3. We will study some proprieties of the operator
L D
jxj2
where 2 R. A natural first question to ask is what is the function space on which Lwill be
operating over. The first classical space that may comes to mind is C2.˝/\ C.˝/, whereupon the Laplace operator is well defined. However, the presence of the singular potentialjj 2seems
to make Lu more singular than u at the origin, and therefore it makes sense to consider also
the space C2.˝n0/ \ .˝n0/. Our technique will be described in the next section, and for this to work, will be necessary to impose some degree of integrability at 0. The basic assumption here will be u=jxj2 2 L1.˝/, such that L can de well defined in the distributional sense:
Lu D
Z
˝
uL ; 2 C01.˝/:
Inequalities will be consider in the following sense:
Lu 0 ” Lu 0 8 2 C01.˝/; 0;
and analogously for Lu 0. It follows that linear PDEs are understood in the distributional
sense: Lu D f ” Lu D Z ˝ f 8 2 C01.˝/: 2.1 A transformation
We will exhibit a change of variables by means of which we can deduce several proprieties of the operator L. Most of these results can be found spread in several papers in which the authors
use different techniques. This transformation is a modification of the transformation that we use at chapter 4 to prove results concerning a non-linear problem. Our transformation acts just on
spherical shells and so we prefer to look for the Laplace operator in spherical coordinates: uD @ 2u @2 C N 1 @u @ C 1 2SN 1u; u D u.x/; x 2 R N;
where D jxj and SN 1means the Laplace-Beltrami operator on the unit sphere SN 1 RN of the function !2 SN 1 ! u.!/. We consider the invertible transformation
u.x/D jxj a=bv.jxj1=b 1x/
where a2 R and b > 0. Note that the domain of the function v is ˝b D fy D jxj1=b 1x W x 2
˝g. Let us introduce the notations y D jxj1=b 1x, r D jyj. By using the previous formula for u, straightforward computations yields
uC u jxj2 D b 2 jyj 2b aC2 @2v @r2 C .bN C 1 2a 2b/ 1 r @v @r C .a 2 C 2ab abN C b2/v r2 C b2 r2SN 1v : In order to avoid weighted spaces it will be convenient to require bN C 1 2a 2b D N 1, which gives aD .b 1/.N 2/=2. It follows
uC u jxj2 D b 2 r.1 b/.N C2/=2 @2v @r2 C .N 1/ 1 r @v @r C b2 r2SN 1vC .N C b 2 . N// v r2 :
If < N we shall also request NC b2. N/ D 0, i.e,
b D bD p N p N . < N/ Define LD @2 @r2 N 1 r @ @r b2 r2SN 1:
For our purposes it will be better to rewriteLin the following way:
L D .1 b2/ @2 @r2 C N 1 r @ @r b2
which follows from the formula for v in spherical coordinates. As we will see, a remarkable parameter in the linear theory of the operator Lis given by
The the final shape of our change of variables is
u.x/D jxj cv.jxj1=b 1x/; (2.1.2)
v.y/ D jyjbcu.jyjb 1y/: (2.1.3)
Also, if u is C2or at least two times weak differentiable, we have seen that
Lu.x/D b 2jyj.1 b/.N C2/=2Lv.y/: (2.1.4)
One can easily check that this previous formula also holds in the sense of distributions, just by using the C2 version on test functions. It can also be seen, by working out some calculation, thatL is an uniformly elliptic operator outside any small ball containing 0, for all < N (i.e.,
all b > 0).
2.2 Maximum principle
The maximum principle for elliptic operators is a classical and well known tool used in several instances, as for example provide data on existence of solutions for linear and non-linear equa-tions. It can also be improved to give results on symmetry proprieties of solutions (see [?]). In this section we will prove a sort of Strong Maximum Principle for the operator L, by means of
the transformation introduced in the last section. As remarked,Lis uniformly elliptic outside
any ball containing 0, and so the standard maximum principle could already provide relevant information. However, by avoiding the origin we don’t get knowledge about the singularity that may exist (or not) at 0 due to the singular potential of the operator L. In order to cover this
issue, we provide a version of the maximum principle which supports more general operators: Lemma 2.2.1. Let˝ RN, N 1, a smooth bounded open set containing 0. We consider a linear differential operatorL D L ;`given by
L D .1 / @2 @r2 C ` 1 r @ @r D @ 2 @r2 ` 1 r @ @r r2SN 1v;
where` and are real parameters subject to the conditions .N `/C ` 0; > 0: Letv 2 C2.˝/\ C.˝/ and assume L v 0 in ˝n0. Then if inf
˝v is attained at some point
Proof. We use a standard recipe for maximum principle. In a first step we consider the special caseL v C for some positive constant C . In the second step we consider the general case L v 0.
Step 1.First we show that ifL v C > 0, then the results follows. Assume, by contradic-tion, that x02 ˝n0 is a local minimum for v. Then
@v
@r.x0/D 0;
@2v
@r2.x0/ 0; SN 1v.x0/ 0 8i;
and soL v.x0/ 0, which is false. Now if 0 is a local minimizer for v, then we argue in the
following way: assume, without loss, that vx1x1.0/ is the minimum among the non negative numbers vxixi.0/, i D 1; :::; N . Letting t > 0 and setting e1 D .1; 0; :::; 0/ we have
L v.te1/D .1 / @2v @x21 .t e1/C .` 1/ 1 t @v @x1 .t e1/ v.t e1/:
Passing to the limit as t ! 0, we get lim
t !0L v.te1/D . ` `/
@2v
@x12.0/ v.0/
where we used L’Hôpital’s Rule. By our choice of direction x1 we have v.0/ N vx1x1.0/. Therefore
lim
t !0L v.te1/ . ` N `/
@2v
@x12.0/ 0; which is false sinceL v C > 0 in ˝n0.
Step 2. Let m D inf˝v and ˝C D fx 2 ˝ W v.x/ > mg. Assume, by contradiction, that
@˝C\˝ ¤ ;. Select y 2 ˝Cthat is closer to @˝Cthan to @˝. Setting R D dist.y; @˝C/, we
obtain the ball BR.y/ ˝Cover which v > m and whose boundary contains a point x0 2 ˝
such thatrv.x0/D 0.
Define .x/D R˛C jx yj˛, where ˛ > 0 will be chosen. So @ @xi D ˛jx yj˛ 2.xi yi/; 8i @2 @xi@xj D jx yj˛ 4f˛.˛ 2/.xj yj/.xi yi/C ˛jx yj2ıijg; 8i; j:
Straightforward computations give 1 r @ @r D 1 jxj2 X i xi @ @xi D ˛jxj 2jx yj˛ 2fjxj2 .x; y/g; @2 @r2 D 1 jxj2 X i;j xixj @2 @xi@xj D jxj 2jx yj˛ 4f˛.˛ 2/Œjxj2 .x; y/2C ˛jxj2jx yj2g; Djx yj˛ 2f˛.˛ 2/C N˛g:
Notice that each of the previous expression are bounded in ˝, since we are assuming ˝ bounded and provided ˛ > 4. ThenL . /.x/ D ˇ2.x/˛2 ˇ1.x/˛ where ˇ1and ˇ2are bounded. The
expression for ˇ2is
ˇ2.x/WD jxj 2jx yj˛ 4fŒjxj2 .x; y/2C Œjxj2jyj2 .x; y/2g:
Fix any radius r < R. Then there is a constant C D C.r/ > 0 such that L . / C > 0 over the annular domain A.y/ D fx W r < jx yj < Rg when ˛ is sufficiently large. In fact, the equations .x; y/ D jxj2 and j.x; y/j D jxjjyjj hold together only when x D y, and then Œjxj2 .x; y/2 C Œjxj2jyj2 .x; y/2 C0 > 0 in A.y/, by continuity. It further implies jˇ2.x/j C0.R r/˛ 4jxj 2 C0 > 0. So over A we have L . / D ˛.ˇ2˛ C ˇ1/
˛.C0˛ C00/ C0 C C00 for ˛ C00=C0 C 1 > 0. Since v > m in BR.y/, we have
v m C > 0 in @Br.0/, which further implies v v.x0/ 0 over @Br.0/ for some
small > 0. Summarizing,
L .v v.x0/ / C > 0 over A.y/
v v.x0/ 0 over @Br.y/
v v.x0/ 0 over @BR.y/:
By Step 1 we conclude that v v.x0/ 0 in A.y/. If denotes the outer-pointing normal
to @BR.y/, then @v @.x0/ @ @.x0/D ˛R ˛ 1< 0;
which is false since x0is an interior critical point.
With the previous result on hands, we are now able to derive a maximum principle for the operator L. For our own convenience, introduce the following notation:
u.x/D u.x/:jxjc: Proposition 2.2.2. Let < N. Assume
(i)u2 C2.˝/\ C.˝/ and L
u 0 over ˝n0;
(ii)inf˝uis attained in˝.
Thenuis constant, i.e.,u.x/D C jxj c for some real constantC .
Proof. Let v given by (2.1.3). Note that u.x/D v.jxj1=b 1x/, and so v2 C2.˝
b/\ C.˝b/.
Since Lu 0 over ˝n0, by (2.1.4) we get Lb;Nv D Lv 0 over ˝bn0. Still, condition (ii)
implies that inf˝bv is achieved in ˝b, and therefore last lemma implies that v is constant, and uas well.
Remark. (I) Unlike the Strong maximum Principle for uniformly elliptic operators, the last result does not implies that u is constant, but shows that it assumes some precise shape. This is clearly related with the fact that annihilates constant functions while Lannihilatesjxjc.
Actually, more generally, for any c2 R direct calculation gives Lfjxj g D p. /jxj 2;
where
p. /WD 2C .N 2/ C :
For < N, p has roots c and Qc D p N C pN . Then jxjQc also annihilates L. If
D N, the roots are equal and jxj .N 2/=2log x also annihilates L. If > N, then cand Qc
are complex and, hence, we can consider the combinationsjxjc C jxjQc andjxjc jxjQc to conclude that
jxj .N 2/=2cos.p N log jxj/ and jxj .N 2/=2sin.p N log jxj/
annihilate L. The role of the polynomial p when studying Lhas already been observed by
some authors as in [?], [?] and [?].
(II) The result implies that for 0 < < N and u2 C2.˝/\ C.˝/ the Dirichlet inequality Lu 0 in ˝; uj@˝ D 0
has only the trivial solution u 0. If < 0, however, there are at least some remarkable cases, to be seen later, where the singularity at 0 is removable.
We have following immediate consequence from Proposition 2.2.2: Corollary 2.2.3. Let < N. Assume 02 C2.˝n0/ \ C.˝n0/, uj
@˝ D 0 and Lu 0 in ˝n0.
Then for all˝0 ˝n0, there is a constant C > 0 such that u C in ˝0. If, in additional,
D2uis continuous at 0, then we can assume02 ˝0.
Proof. Since u 2 C2.˝n0/ \ C.˝n0/, we have u 2 C2.˝n0/ \ C.˝n0/, and then then
applied to L, outside a small ball about 0). It gives the first part of the corollary. Now, if
D2u is continuous at 0, then u 2 C2.˝n0/ \ C.˝n0/, and Proposition 2.2.2 implies the
second part of the corollary.
A weak versions of Proposition 2.2.2 for (sub)super-solutions of the equation Lu D 0,
uj@˝ D 0 in the sense distributional sense were given by Dupaigne in [?]:
Proposition 2.2.4 (Dupagine, [?]). Let 0 < < N. Assume u=jxj2Cc 2 L1and Z
˝
uL 0; 8 2 C2.˝/; 0:
Thenu 0 a.e. in ˝.
Remark. (I) The conditionR˝uL 0 8 2 C2.˝/; 0 encodes u D 0 at @˝ in
some weak way. In fact, if u2 C2.˝/ and uj@˝ D 0, the Green identity gives
Z ˝fu ug D Z @˝ u@ @ @u @ D 0;
and soR˝ Lu 0 8 2 C2.˝/; 0, which implies Lu D 0. Observe that u D 0 at
@˝ pointwisely does not make sense for u2 L1since @˝ has null measure. Another possible
way to sate the Dirichlet boundary condition is to require lim !0 N Z ˝\B.y/ juj D 0: And if one wants to state u 0 at @˝, can require
lim !0 N Z ˝\B.y/ u D 0:
where u is the negative part of u.
(II) For < N and if u2 L1, as required in Proposition 2.2.2, then Dupaigne’s hypothesis
u=jxj2Cc 2 L1is satisfied. In fact, u=jxj2Cc D u=jxj2C2c and 2C 2c
D N p N <
N .
(III) The condition u 2 L1cannot be weekend with respect to the power, i.e., the power c
ofjxj can’t be decreased. In fact, consider ˝ D B1.0/ and E.x/D Eı.x/D jxj c ıjxj Nc,
where 0 < ı < 1. Then Eı is a sign-changing function satisfying LE D 0 pointwisely in
B1.0/n0 and Eı. Let us show that Lu D 0 over B1.0/ in the sense of distributions. By Green’s
second identity (the same used in part (I)), we have Z B1nB fE Eg D Z @B E@ @ @E @ :
for every 2 C2.B
1/ with j@B1 D 0. Also, j R
@BfE@ @Eg j C
N 2 Nc D Cc, where C is a constant. Making ! 0, we get that R
B1nB fE Eg ! 0 as ! 0. On the other handj fE Eg .x/j C jxj Nc 2, which is integrable about 0. By dominated convergence theorem, we getRB
1fE Eg D 0, which shows that LE D 0 in the sense of distributions.
2.3 The Dirichlet problem
Here we will deal with the classical Dirichlet sort of problems associated with the operator L,
i.e, problems of the sort
( LuD f in ˝
uj@˝ D 0
: (2.3.5)
We will speak of two different senses for solutions the Dirichlet problem: The generalized sense: f 2 L2, u2 W1;2
0 .˝/ and Z ˝ ru r u jxj2 D Z ˝ f 8 2 W01;2.˝/:
The sense of distributions: f 2 L1, u is mensurable such that u=jxj2 2 L1 and
Z ˝ uL D Z ˝ f 8 2 C2.˝/; j@˝ D 0:
The generalized sense is well established thanks to the following classical inequality by Hardy:
Proposition 2.3.1 (Hardy’s inequality). If p < N then for every u2 W1;p
.RN/ we have Z RN jujp jxjp CN;p Z RN ˇ ˇ ˇ ˇru x jxj ˇ ˇ ˇ ˇ p ; whereCN;p D Œp=.N p/p is the sharp constant.
Proof. Assume u smooth with compact support in RN. Thenju.x/jp D R1
1 d dtju.tx/j p dt and so Z RN jujp jxjp dx Z RN Z 1 1 pju.tx/j p 1 jxjp 1 ˇ ˇ ˇ ˇru.tx/ x jxj ˇ ˇ ˇ ˇ dt dx: By Fubini’s theorem and changig variables y D tx we get
Z RN jujp jxjp dx p Z 1 1 tp 1 N dt Z RN ju.y/jp 1 jxjp 1 ˇ ˇ ˇ ˇru.y/ y jyj ˇ ˇ ˇ ˇ dy:
By Holder inequality, Z RN jujp jxjp dx p Z 1 1 tp 1 N dt Z RN ju.y/jp jyjp dy pp1Z RN ˇ ˇ ˇ ˇru.y/ y jyj ˇ ˇ ˇ ˇ p dy p1 CN;p Z RN ju.y/jp jyjp dy pp1 Z RN ˇ ˇ ˇ ˇru.y/ y jyj ˇ ˇ ˇ ˇ p dy p1
which gives the inequality.
In order to prove that CN;p is the sharp constant, let us consider functions of the sort
Uk;a.x/ D minfk a;jxj ag, a > 0, k 0. So Uk;a 2 W1;p.RN/ provided jxj a 1 is
p-integrable at infinity, i.e., provided a > .N p/=p. Then Z RN ˇ ˇ ˇ ˇrU a;k x jxj ˇ ˇ ˇ ˇ p D Z jxj>k D ap Z jxj>k jxj ap p D ap!N k .paCp N / paC p N;
by integration over spherical shells, where !N is surface area of the unit ball in RN.
Analo-gously, Z RN jUk;ajp jxjp D !N k .paCp N / paC p N C !N kN a p N p : Then R RN jrUa;k x=jxjj p R RN jUk;aj p=jxjp D .N p/ap N pC .pa C p N /kap a:
The result follows making a& .N p/=p and k ! C1.
Remark. Our interest is in the case p D 2 and N 3, for which we have
N 2 2 2Z RN u2 jxj2 Z RN jruj2 8u 2 C01.R N /
By density the inequality remains true for u 2 W01;2.˝/. Not by coincidence, as we will see,
the sharp constant for Hardy’s inequality coincides exactly with the value of N. Still, for < N we see that kuk WD Z ˝ jruj2 Z ˝ u2 jxj2
is equivalent to the standard norm of the Sobolev Space W01;2.˝/. Hardy’s inequality has several generalizations, but one that it worth to comment here is due to Brezis and Vazquez ([?]): Z ˝ jruj2 N 2 2 2Z RN u2 jxj2 C ˛p Z ˝ jujp 2=p
for all u2 W01;2.˝/, 1 < p < 2N=.N 2/, where ˛p > 0 depends on the measure of ˝. This
inequality into an equality.
Proposition 2.3.2. Iff 2 L2.˝/ and u 2 W01;2.˝/ is a solution of 2.3.5 in the distributional
sense, then it is also a solution in the generalized sense.
Proof. Since u 2 W01;2, we get u=jxj 2 L2 by Hardy’s inequality. Holder’s inequality then
givesR juj=jxj2 R u2=jxj21=2
R 1=jxj21=2
<1, so u=jxj 2 L1. Integrating by parts the
equation for distributional sense, we get the equation for the generalized sense.
The basic existence and uniqueness result for generalized solutions of 2.3.5 is given below. Lemma 2.3.3. Assume < N. If f 2 L2, then there is exactly one u 2 W1;2
0 .˝/ that is
solution of (2.3.5) in the generalized sense. Proof. Let us consider the bilinear form
B. ; /D Z ˝ r r jxj2
defined for ; 2 W01;2.˝/. Hardy’s inequality implies that B is coercive. Holder’s inequality
followed by, again, Hardy’s inequality, gives that B is continuous. On the other hand, 2 W01;2.˝/ ! R
˝f is continuous since f 2 L 2
. Then by Lax-Milgram theorem there is an unique u2 W01;2.˝/ such that B.u; / D
R
˝f for all 2 W 1;2
0 .˝/, which is precisely the
desired result.
Our next result provides a necessary condition for existence of solutions in the sense of distributions. A very similar result can be found in [?] (its Lemma 1.5), where the author uses approximated problems. Here we present an estimate that is a little bit more general. Our proof uses some subtle test functions. The choice of these test functions is inspired by part (I) of Remark for Proposition 2.2.2. We remind that L annihilatesjxj c andjxj c.
Proposition 2.3.4. Assume0 < < N, u=jxj2 2 L1, f 2 L1 andf 0. Then a necessary
condition in order to haveLuD f is the estimate
p. / Z ˝ jxj 2u Z ˝ jxj f: for every < c, wherep. /D 2C .N 2/ C . Moreover
Z
˝
jxj cf <1: Proof. Define Eı W RNn0 ! R by
Setting r D r.ı/ D ı1=. Nc c/we get that E
ı.x/ > 0 forjxj > r, Eı.x/ D 0 for jxj D r
and Eı.x/ < 0 forjxj < r. The idea will to submit Eı to some subtle smoothing process in
order to use it as test function in Lu D f and pass to limit as ı ! 0. Let r > 0 be such
that Br.0/ ˝ and 2 C01.˝/ satisfying 1 at Br.0/. Assume the range of ı such
that r.ı/ < r, i.e, ı < rNc c. Let QE
ı D EıC where EıC is the positive part of Eı – then
Q
Eı 2 C0.˝/. Observe that QEı is smooth everywhere except at the spherejxj D r.ı/ (where
it is 0). In order to fix this issue and properly get an appropriate test function, we shall mollify Q
Eı by a process described next.
Let gı WD QEı. Hence gı 2 C.˝/ (in fact, the only worrisome place is the sphere
jxj D r.ı/, but it is inside B
r.0/ where QEı.x/ D EıC.x/ D EıC.x/=jxj2, which is
continuous). Let ' W Œ0; 1/ ! R be a smooth increasing function such that
'.r/D( 0 if r 1
r if r 2 ;
and for > 0 small define ' D '.=/. Set g;ı WD '.gı/. Then g;ı 2 C01.˝/ and
sup˝jg;ı gıj ! 0 as ! 0. Finally, let ;ı be the solution of the problem
(
;ı D g;ı in ˝
;ı D 0 on @˝
:
Then ;ı 2 C1.˝/. Using it as test function, we get
Z ˝ u g;ı jxj2 ;ı D Z ˝ f ;ı
Since the inverse of the Dirichlet Laplacian . / 1 W C.˝/ ! C.˝/ is continuous, we have ;ı ! QEı uniformly in ˝ as ! 0. Passing to the limit as ! 0, we get
Z ˝ u QEı jxj2EQı D Z ˝nBr.0/ u QEı jxj2EQı D Z ˝ f QEı
In the limit as ı! 0, QEı QEı=jxj2converges uniformly to .jxj c/ jxj c 2in
˝nBr.0/. On the other hand, QEı converges monotonically tojxj c in ˝. By the monotone
convergence theorem, we get that Z
˝
jxj cf <1:
Now we show thatR jxj 2u < 1 for < c. Take f ng C2.˝/, with nj@˝ D 0, an
Fatou’s Lemma implies Z ˝ uLfjxj g Z ˝ jxj f:
But Lfjxj g D p. /jxj 2(part (I) of Remark for Proposition 2.2.2). Therefore p. /R jxj 2u
R jxj f <1.
Remark. The estimate cannot be improved to support the limit case D c. For take ˝ D
B1.0/ and E.x/D jxj Nc jxj c– hence LE D 0 in the sense of distributions (same
compu-tation of part (III) of last remark). However,RB
1E=jxj 2Cc DR B1jxj N R B1jxj 2 2c D 1 (the first integral isC1 and the second is finite since 2 C 2c < N .
A remarkable consequence of the last proposition is that, for 0 < < N, the only non-negative function that is annihilated by Lis the identically zero function:
Corollary 2.3.5. Assume0 < < N. If u=jxj2 2 L1,u 0, satisfies L
uD 0, then u 0.
Our next main result in a non-existence theorem often credited to Baras and Goldstein ([?]). However, Baras and Goldstein actually proved a corresponding result for a parabolic version of equation Lu D f . References to that article usually make use or words like “the following
can be deduced from [?] in an elliptic case", which is a little sketchy in our modest opinion. Furthermore, the proof that is presented in [?] is long and rather technical. Seeking for a differ-ent method, we were able deduce a short proof that works directly for the elliptic case. In our proof we use the following distributional version of the strong maximum principle:
Lemma 2.3.6. Let˝ be an smooth open domain and v 2 L1.˝/, v 0. Assume v 0, vj@˝ D 0 in the following sense of distributions:
Z
˝
v 0
for all 2 C2.˝/ with j
@˝ D 0. If v is identically zero in some open subset of ˝ then it is
null everywhere.
Proof. We use the standard argument of mollification: Define the classical function W RN !
R,
.x/D( c expŒ1=.jxj
2 1/
if jxj 1
0 if jxj 1 ;
where c is such thatR
RN D 1, which is non-analytic on the sphere jxj D 1. Then for > 0,
D N.x=/ also satisfies
R
RN D 1 and has support jxj . Set
v.x/D
Z
RN
.x y/v0.y/ dy;
(i) v 2 C1.RN/;
(ii) v ! v0a.e. in RN (and therefore v ! v a.e. in ˝).
Furthermore, since v 0 in some open subset of ˝, we can find an open subset U compactly contained in ˝ and such that v 0 in U for all > 0 sufficiently small. Fix V ˝ any
open subset containing U and let us show that vjV 0. For > 0 small enough we have
V C B.0/ ˝ and so .x / has compact support in ˝ for all x 2 V . Therefore for
x2 V we have
v.x/D
Z
˝
.x y/v.y/ dy 0;
by the hypothesis of the lemma. Summarizing, we have v 2 C2.V /, vj 0, vjU 0 and,
provided is small enough, v 0 in V . The classical strong maximum principle then
implies v 0 in V for small . Taking the limit as ! 0, we get v 0 a.e. in V .
Proposition 2.3.7. Assume > N. If f 2 L1,f 0 is non identically zero, then there is no u, u 0, u=jxj2 2 L1, solution ofLu D f .
Proof. We will show that any function u satisfying the hypothesis of the proposition must van-ishes at some open subset of ˝. With this in hands, since is positive, the previous lemma will imply u 0 everywhere, contradicting LuD f ¥ 0.
Let 0W .0; 1/ ! R given by
0.r/D r .N 2/=2cos
p
a log r;
where 0 < a < is fixed. The inspiration to consider this function comes from the observations on the role of the polynomial p in the remark of Proposition 2.2.2 - for short,
because L 0.x/ D a=jxj2 formally. The proof consists basically in show that 0 can be
used as test function in the distributional definition of LuD f provided we perform a cut-off
and smoothness.
Observe that 0 is a limit point of the zeros of 0. Let 0 < r1 < r2 two consecutive zeros
of 0 such that Br2.0/ ˝ and 0.r/ > 0 for r1 < r < r2. Define Q 0.x/ as the extension of 0.jxj/ to ˝ by setting Q 0 0 outside Br2.0/nBr1.0/. Then g DW Q 0 2 C.˝/ since 0 D . a/ 0=jxj2 and 0 D 0 at @Br1.0/[ @Br2.0/. Let ' W Œ0; 1/ ! R be a smooth increasing function such that
'.r/D( 0 if r 1
r if r 2 ;
and for > 0 small define ' D '.=/. Finally, setting g D '.g/, we have that g 2
C1.˝/, supp g ˝nBr1.0/ and sup˝jg gj ! 0 as ! 0. Now let be the unique
solution of the problem
(
D g in ˝
D 0 on @˝
Then 2 C1.˝/. Using as test function in the definition of the Dirichlet problem for LuD f we get Z ˝ u g jxj2 D Z ˝ f
Since the inverse of the Dirichlet Laplacian . / 1 W C.˝/ ! C.˝/ is continuous, we have ! Q 0uniformly in ˝ as ! 0. So we can pass the limit in the previous equation to get
Z ˝ u Q 0 jxj2 Q0 D Z ˝ a jxj2u Q 0D Z ˝ f Q 0: It impliesR˝f Q 0 D R ˝u Q 0=jxj
2 D 0, which in particular gives u D 0 a.e. in B
r2.0/nBr1.0/.
2.4 Strong solutions
Not much is known about the regularity of solutions of the Dirichlet problem for Lu D f
when f 2 LP.˝/, p > 1. In [?] we find the following result concerning radial solutions: Proposition 2.4.1. Assume0 < < N and assume that q is a real number satisfying
N
N c q
N 2C c
: LetB1be the unit ball about 0 in RN and consider the space
E D W2;q.B1/\ W01;q.B1/\ fu W u=jxj2 2 Lq.B1/g:
Then if f 2 Lq.B
1/ is a radial function, the Dirichlet problem for Lu D f has an unique
radial solution in the spaceE.
The proof given by Dupaigne relies in differentiation of the Green representation of the solution. The generalization to non-radial solutions in general domains is still open, as far as the author knows.
Some results on the non-linear problem
In the last chapter we addressed some remarkable results concerning the linear theory for the operator L. Non-linear problems of power behaviour as uD jujp 1u for p > 0 was widely
studied by many authors in last decades – topics as sense of solutions, regularity, existence and uniqueness, asymptotic behaviours, blow-ups, ... were studied in several papers. A natural question to ask is how the non-linear theory changes when we consider a different operator from . Here we speak about some remarkable results concerning the Schrodinger operator V , where V .x/ is the inverse square potential jxj 2. More precisely, results concerning
the Dirichlet problem 8 ˆ < ˆ : uD jxj2uC juj p 1 u in ˝nf0g uj@˝ D 0 (3.0.1)
where p > 0, 2 R and ˝ is an open and smooth smooth subset of the euclidean space RN, N 3.
3.1 Non existence results
A non-existence result for equation (3.0.1), relatively to the range of p, can be achieved by considering the classical result due to Pohozaev.
Lemma 3.1.1 (Pohozaev, [?]). Let ˝ a smooth domain, g W ˝ R ! R continuous and suppose thatw 2 C2.˝/ satisfies
(
w.x/D g.x; w.x// in ˝
wj@˝ D 0
: (3.1.2)
Ify 2 RN is a fixed vector andn.x/ denotes the outward normal to @˝ at x, then w satisfies: Z @˝ .x y/ n.x/jrw.x/j2 dS.x/D 2N Z ˝ G.x; w/ dxC C 2 N X i D1 Z ˝ .xi yi/Gxi.x; w/ dx .N 2/ Z ˝ g.x; w/w dx: whereG.x; s/DRs 0 g.x; s/ ds.
Proof. Let us multiply the differential equation for w by .x y/ rw, where y is any fixed point in RN, and integrate by parts:
Z ˝ X i;j .xi yi/ @w @xi @ @xj @w @xj dxD Z ˝ jrwj2dx Z @˝ .x y/ n.x/jrw.x/j2dSC CX i;j Z ˝ .xi yi/ @2w @xi@xj @w @xi dxDX i Z ˝ .xi yi/ @w @xi g.x; w/ dxD N Z ˝ G.x; w/ dx X i Z ˝ .xi yi/Gxi.x; w/ dx: But X i;j Z ˝ .xi yi/ @2w @xi@xj @w @xi dxD 1 2 Z @˝ .x y/ n.x/jrw.x/j2dS N 2 Z ˝ jrwj2:
The result follows observing thatR˝jrwj2 dxDR
˝wg.x; w/ dx.
Classically, this lemma is used to show non-existence of solutions to the problem (
w D jwjq 1w in ˝
wj@˝ D 0
: (3.1.3)
where ˝ is star shaped and q .N C 2/=.N 2/. Setting g.x; s/ D jsjp 1s and y D 0 in
Lemma 3.1.1 we get Z @˝ x n.x/jrw.x/j2dS D 2N qC 1 .N 2/ Z ˝jwj qC1:
Since ˝ is star shaped, the left hand side of the last identity is non-negative. Consequently, since q > .NC2/=.N 2/ then we must have w 0. If q D .N C2/=.N 2/ and ˝ is bounded we getrw D 0 on @˝, and so w 0 over all ˝ by Hodge’s Lemma. So if q > .N C 2/=.N 2/ and ˝ is star shaped, the only solution for (3.1.3) is w 0. If q D .N C 2/=.N 2/ and ˝
is, in additional bounded, then the same conclusion holds. Let us show that the same Pohozaev identity can be derived for solutions of 3.0.1.
Proposition 3.1.2. Assumep .N C2/=.N 2/ and ˝ a smooth bounded star shaped domain containing 0. Ifu 2 W01;2.˝/ solves (3.0.1) in the generalized sense, i.e.,
Z ˝ ru r jxj2u D Z ˝ up 8 2 W01;2.˝/; thenu 0.
Proof. Let Br.0/ a small ball about 0 such that Br.0/ ˝ and set ˝r D ˝nBr.0/. So 1=jxj2
is bounded in ˝r and standard elliptic theory, together with bootsrapping argument, applies
to the weak form stated. It gives u 2 C2.˝r/ and u D u=jxj2 C up in ˝r. Setting
g.x; s/D s=jxj2C sp we have, Z ˝r G.x; u/ dxD Z ˝r u2 jxj2 dxC 1 pC 1 Z ˝r upC1dx; X i Z ˝r xiGxi.x; u/ dxD Z ˝r u2 jxj2; Z ˝r g.x; u/uD Z ˝r u2 jxj2 C Z ˝r upC1 Setting y D 0 and using ˝r as domain in Lemma 3.1.1 we get the identity
Z @˝r x n.x/jru.x/j2dS D 2N pC 1 .N 2/ Z ˝r upC1: (3.1.4)
It is remarkable that the integrals of u2=jxj2cancel each other and we obtain the same identity as for the classical non-linear problem (3.1.3). So (3.1.4) holds for all r > 0, and sinceR˝jruj2< 1, we can pass to the limit as r ! 0 to obtain
Z @˝ x n.x/jru.x/j2dS D 2N pC 1 .N 2/ Z ˝ upC1:
So if p > .NC2/=.N 2/, we must have u 0 since ˝ is star shaped. If p D .N C2/=.N 2/, then we just haveru D 0 over @˝, but since u D u=jxj2C up 0 in ˝, Hopf’s Lemma applies and therefore u 0 in ˝.
The previous computation shows that the singularity 1=jxj2is special in the sense that equa-tion (3.0.1) has the same Pohozaev identity as the classical equaequa-tion (3.1.3), and so the same non-existence results relatively to the range of p. However, if we allow solutions in a more gen-eral sense (namely L1solutions in the sense of distributions), then the range of p for existence may depend on . It seems there is a special role played by the parameter
where cis given in (2.1.1). The role of this parameter as a critical exponent can be seen in the
following result.
Proposition 3.1.3 (Brezis-Dupaigne-Tesei, [?]). Assume 0 N and p > 1. Let B be any ball centred at 0 andu W B ! R, u 0, be such that both up andu=jxj2 belongs toL1loc.B/. Ifp < p then equationLu D up, uj@B D 0 has a solution in the sense of distributions, i.e,
in the sense that
Z B u jxj2 D Z B up for all 2 C2.B/ with j
@B D 0.
Remark. We stress that p D 1 when D 0, which means that the classical problem
(3.1.3) with ˝ D B1.0/ has a positive solution in the sense of distributions for all 1 < p <1.
In fact, consider a function of the form w.r /D Ar ˇ. Choosing ˇ D 2=.p 1/ and
Ap 1 D ˇ2C .N 2/ˇ D 2.N 2/ .p 1/2 p N N 2 ;
then w.r / is solution. Furthermore, we can assume A > 0 (i.e., p > N=.N 2/), since for p < .N C 2/=.N 2/ we always have the Mountain-Pass solution.
In the same work ([?]) it was proved that if p pthen we have no solution in the sense of
distributions, even if we do not assume integrability hypothesis at the origin:
Proposition 3.1.4 (Brezis-Dupaigne-Tesei, [?]). Assume 0 < N and p p. LetB any
ball centred at 0 andu 2 Lploc.Bn0/, u 0, solves the inequality Lu u
p inBn0, uj
@B D 0,
in the sense of distributions, i.e., Z B u jxj2 Z B up
for all 2 C2.Bn0/ with j
@B D 0. Then u 0.
The non-existence of W01;2solutions in the range p .N C2/=.N 2/ given in Proposition (3.1.2) seems to be strongly related to the singularity 1=jxj2, which is absorbed in W1;2 norm
by Hardy’s inequality. If, however, we are no more in W01;2, then we can allow more singular solutions. But even very singular solutions seems to be not possible if p p.
3.2 Solutions in the whole space and balls
The article [?] by Terracini often appears as reference for elliptic problems dealing with the singular potential 1=jxj2 because she was one of the first to address the question from a
results we consider the space
D1;2.RN/D fu 2 LN2N2.RN/W ru exists in the weak sense and jruj 2 L2.RN/g: We will say that u2 D1;2.RN/ solves (3.0.1) in the weak sense if
Z RN ru r jxj2u D Z RN up for all 2 D1;2.RN/.
Proposition 3.2.1 (Terracini, [?]). The following statements holds true for D1;2.RN/ weak so-lutions of problem(3.0.1).
(i) There exist no solution belonging toLpC1.RN/ for p ¤ .N C 2/=.N 2/;
(ii) If0 < < and p D .N C 2/=.N 2/ then there is an unique solution in D1;2.RN/ (up to re-scaling);
(iii) There is a < 0 such that for every < the problem admits at least two positive solutions, one is radial and the other is not.
Remark. The class of radial solutions that uniquely solve (3.0.1) in RN for p D .N C 2/=.N 2/ and 0 < < is given by
Uı.x/D
ıN22
jxjN22.1 /.1C ı2jxj2/N22
The techniques used by Terracini includes moving plane method and analysis of the phase space. The results of Terracini were later reobtained by Dancer, Gladiali and Grossi ([?]), by using a subtle change of variables.
When one speaks about solutions of elliptic problems in a ball, one of the first questions arising is the symmetry of solutions. In the classical article [?] the authors show that any positive solution of the canonical problem (4.4.50) must be radially symmetric about 0, radial for short. The same question, relatively to positive solutions of of problem (3.1.2) when > 0, was first addressed by Terracini in the RN case, and later by Chaves and Garcia-Azorero in the case of a ball:
Proposition 3.2.2 (Chaves, Garcia-Azorero, [?]). Assume ˝ D B1.0/ and 0 < . Then
any positive weak solutionu2 W01;2.˝/ of (3.1.2) is radially symmetric about 0.
The proof uses moving plane method and phase plane analysis. As for the RN case, we may expect to have no uniqueness when < 0 and ˝ D B1.0/. In fact, Dancer, Gladiali and Grossi
have shown the following:
(i) The problem(3.1.2) has only one positive radial weak solution in W01;2.˝/;
(ii) Denote by U the radial solution of (3.1.2). There is a sequence 0 > 1 > 2 > :::
and maximum intervalsIj . 1; 0/, with j 2 Ij, such that a non-radial bifurcation occurs
at.; u/ for 2 Ij. Moreover if j is even, there are at least ŒN=2 continua of non-radial
solutions bifurcating from .; u/ for 2 Ij. The first branch is invariant by O.N 1/
transformations, the second byO.N 2/ O.2/, the third by O.N 3/ O.3/ and so on. The lack of monotonicity for the case < 0 will be turned explicit in next chapter, when we present Theorem (4.1.1). From it we deduce that u vanishes at 0 for every < 0. Moreover, the theorem implies that u lacks analyticity at 0 as ! 1.
Regularity and asymptotic approach to
semilinear elliptic equations with singular
potential
4.1 Introduction
In this paper we present results concerning non-trivial solutions of the elliptic problem 8 ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ : uD jxj2uC u p in ˝nf0g u 0 in ˝nf0g uj@˝ D 0 (4.1.1)
where 02 ˝ RN is a smooth domain, N 3, p > 1 and 2 R. This is a case of a non-linear Schrodinger equation with singular potential. The term jxj 2 is called Calogero potential in Physics literature, where positive and negative sign for means attractive and repulsive field, respectively.
A weak solution of the problem (4.1.1) is supposed to be critical point of the functional J Œu D 1 2 Z ˝ jruj2 jxj2u 2 1 pC 1 Z ˝ upC1; (4.1.2)
defined on the Sobolev space W01;2.˝/. Hardy’s inequality allows to use variational methods to prove existence of non-trivial solutions when N, where N D .N 2/2=4 is the best constant. On the other hand, jxj 2 is the smallest singularity for which the functional J can be well-defined in W01;2.˝/. Furthermore, this singularity is critical in the sense that
boot-strapping argument does not improves integrability of u at 0, and so regularity near zero must be investigated by other tools. Concerning the range of the parameters in (4.1.1), we observe