On Schrödinger equations with multisingular inverse-squareanisotropic potentials
Veronica Felli
Dipartimento di Matematica ed Applicazioni
University of Milano–Bicocca
joint works with Elsa M. Marchini and Susanna Terracini
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.1/23
The Schrodinger equation with a dipole-potential
In nonrelativistic molecular physics, the Schrödinger equation forthe wave function of an electron interacting with a polar molecule(supposed to be point-like) can be written as
(
−~
2
2m∆ + e
x · D
|x|3− E
)
Ψ = 0,
where
e = charge of the electron
m = mass of the electron
D = dipole moment of the molecule.
See [J. M. L evy-Leblond, Electron capture by polar molecules, Phys. Rev. (1967)].
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.2/23
Dipole Schr odinger operators:
Lλ,d := −∆ −λ (x · d)
|x|3, x ∈ R
N , N ≥ 3,
λ = 2me~
|D| ∝ magnitude of the dipole moment
d = D/|D| orientation of the dipole
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.3/23
Dipole Schr odinger operators:
Lλ,d := −∆ −λ (x · d)
|x|3, x ∈ R
N , N ≥ 3,
λ = 2me~
|D| ∝ magnitude of the dipole moment
d = D/|D| orientation of the dipole
Plan of the talk: asymptotics near the singularity of solutionsto equations associated to Lλ,d
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.3/23
Dipole Schr odinger operators:
Lλ,d := −∆ −λ (x · d)
|x|3, x ∈ R
N , N ≥ 3,
λ = 2me~
|D| ∝ magnitude of the dipole moment
d = D/|D| orientation of the dipole
Plan of the talk: asymptotics near the singularity of solutionsto equations associated to Lλ,d
ւ
positivity properties(localization of binding)
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.3/23
Dipole Schr odinger operators:
Lλ,d := −∆ −λ (x · d)
|x|3, x ∈ R
N , N ≥ 3,
λ = 2me~
|D| ∝ magnitude of the dipole moment
d = D/|D| orientation of the dipole
Plan of the talk: asymptotics near the singularity of solutionsto equations associated to Lλ,d
ւ ↓
positivity properties(localization of binding)
essentialself-adjointness
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.3/23
Dipole Schr odinger operators:
Lλ,d := −∆ −λ (x · d)
|x|3, x ∈ R
N , N ≥ 3,
λ = 2me~
|D| ∝ magnitude of the dipole moment
d = D/|D| orientation of the dipole
Plan of the talk: asymptotics near the singularity of solutionsto equations associated to Lλ,d
ւ ↓ ց
positivity properties(localization of binding)
essentialself-adjointness
study of nonlinearSchrodinger equations
with multi-singularpotentials
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.3/23
Schrodinger operators with dipole-type potentials
Dipole potentials have the sameorder of homogeneity as inverse
square potentials 1/|x|2
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.4/23
Schrodinger operators with dipole-type potentials
Dipole potentials have the sameorder of homogeneity as inverse
square potentials 1/|x|2;
no inclusion in the Kato class, validityof a Hardy-type inequality, invariance
by scaling and Kelvin transform
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.4/23
Schrodinger operators with dipole-type potentials
Dipole potentials have the sameorder of homogeneity as inverse
square potentials 1/|x|2;
no inclusion in the Kato class, validityof a Hardy-type inequality, invariance
by scaling and Kelvin transform
We consider a more general class of Schrödinger operators withpurely angular multiples of radial inverse-square potentials:
Lh := −∆ −h(x/|x|)
|x|2, in R
N , N ≥ 3,
where h ∈ L∞(SN−1).
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.4/23
Schrodinger operators with dipole-type potentials
Dipole potentials have the sameorder of homogeneity as inverse
square potentials 1/|x|2;
no inclusion in the Kato class, validityof a Hardy-type inequality, invariance
by scaling and Kelvin transform
We consider a more general class of Schrödinger operators withpurely angular multiples of radial inverse-square potentials:
Lh := −∆ −h(x/|x|)
|x|2, in R
N , N ≥ 3,
where h ∈ L∞(SN−1).
Natural setting to study the properties of operators Lh:
D1,2(RN ) := C∞c (RN )
‖·‖, ‖u‖ = ‖∇u‖L2 .
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.4/23
References
• problems with radially inverse-square singular potentials1/|x|2 (h ≡ const, i.e. isotropic case):
Jannelli, Ferrero–Gazzola, Ruiz–Willem, Baras–Goldstein,Vazquez–Zuazua, Garcia Azorero–Peral, Berestycki–Esteban, Smets,F.–Schneider, Abdellaoui–F.–Peral, F.–Pistoia, F.–Terracini,Brezis-Dupaigne-Tesei, Kang-Peng, Han, Chen, Dupaigne, ...
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.5/23
References
• problems with radially inverse-square singular potentials1/|x|2 (h ≡ const, i.e. isotropic case):
Jannelli, Ferrero–Gazzola, Ruiz–Willem, Baras–Goldstein,Vazquez–Zuazua, Garcia Azorero–Peral, Berestycki–Esteban, Smets,F.–Schneider, Abdellaoui–F.–Peral, F.–Pistoia, F.–Terracini,Brezis-Dupaigne-Tesei, Kang-Peng, Han, Chen, Dupaigne, ...
• S. Terracini [Advances in Differential Equations (1996)]: howthe presence of the singular potential affects
−∆u = h(x/|x|)u
|x|2+ u
N+2
N−2 , h ∈ C1(SN−1)
concerning existence, uniqueness, and qualitativeproperties (symmetry) of positive solutions.
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.5/23
References
• F.-Marchini-Terracini[Discrete Contin. Dynam. Systems (2008)]estimate of the asymptotic behavior of solutions toSchrödinger equations with anisotropic inverse-squaresingular potentials near the singularity.
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.6/23
References
• F.-Marchini-Terracini[Discrete Contin. Dynam. Systems (2008)]estimate of the asymptotic behavior of solutions toSchrödinger equations with anisotropic inverse-squaresingular potentials near the singularity.
• F.-Marchini-Terracini [Indiana Univ. Math. J., to appear]positivity, essential self-adjointness, and spectral propertiesof Schrödinger operators with multiple locally anisotropicinverse-square singularities.
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.6/23
References
• F.-Marchini-Terracini[Discrete Contin. Dynam. Systems (2008)]estimate of the asymptotic behavior of solutions toSchrödinger equations with anisotropic inverse-squaresingular potentials near the singularity.
• F.-Marchini-Terracini [Indiana Univ. Math. J., to appear]positivity, essential self-adjointness, and spectral propertiesof Schrödinger operators with multiple locally anisotropicinverse-square singularities.
• F. [Preprint 2008]Existence of ground state solutions to a class of nonlinearSchrödinger equations with critical power-nonlinearities andpotentials exhibiting multiple anisotropic inverse squaresingularities.
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.6/23
Hardy-type inequality
∫
RN
h(x/|x|)
|x|2u2(x) dx ≤ΛN (h)
∫
RN
|∇u(x)|2 dx ∀u ∈ D1,2(RN )
Best constant ΛN (h):= supu∈D1,2(RN )\0
∫
RN |x|−2h(x/|x|)u2(x) dx∫
RN |∇u(x)|2 dx
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.7/23
Hardy-type inequality
∫
RN
h(x/|x|)
|x|2u2(x) dx ≤ΛN (h)
∫
RN
|∇u(x)|2 dx ∀u ∈ D1,2(RN )
Best constant ΛN (h):= supu∈D1,2(RN )\0
∫
RN |x|−2h(x/|x|)u2(x) dx∫
RN |∇u(x)|2 dx
Classical Hardy inequality: ΛN (1) = 4(N−2)2
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.7/23
Positivity properties of Lh = −∆ − h(x/|x|)|x|2
Consider the quadratic form associated to Lh
Qh(u) :=
∫
RN
|∇u(x)|2dx−
∫
RN
h(x/|x|)u2(x)
|x|2dx.
The following conditions are equivalent:
• Qh is positive definite, i.e. infu∈D1,2(RN )\0
Qh(u)R
RN |∇u(x)|2 dx> 0
• ΛN (h) < 1
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.8/23
Positivity properties of Lh = −∆ − h(x/|x|)|x|2
Consider the quadratic form associated to Lh
Qh(u) :=
∫
RN
|∇u(x)|2dx−
∫
RN
h(x/|x|)u2(x)
|x|2dx.
The following conditions are equivalent:
• Qh is positive definite, i.e. infu∈D1,2(RN )\0
Qh(u)R
RN |∇u(x)|2 dx> 0
• ΛN (h) < 1
µ1(h) is the 1st eigenvalue of the operator −∆SN−1 − h(θ) on SN−1 , i.e.
µ1(h) = minψ∈H1(S
N−1)ψ 6≡0
R
SN−1
h
|∇SN−1ψ(θ)|2 − h(θ)ψ2(θ)
i
dV (θ)R
SN−1 ψ2(θ) dV (θ).
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.8/23
Positivity properties of Lh = −∆ − h(x/|x|)|x|2
Consider the quadratic form associated to Lh
Qh(u) :=
∫
RN
|∇u(x)|2dx−
∫
RN
h(x/|x|)u2(x)
|x|2dx.
The following conditions are equivalent:
• Qh is positive definite, i.e. infu∈D1,2(RN )\0
Qh(u)R
RN |∇u(x)|2 dx> 0
• ΛN (h) < 1
• µ1(h) > −“N − 2
2
”2
µ1(h) is the 1st eigenvalue of the operator −∆SN−1 − h(θ) on SN−1 , i.e.
µ1(h) = minψ∈H1(S
N−1)ψ 6≡0
R
SN−1
h
|∇SN−1ψ(θ)|2 − h(θ)ψ2(θ)
i
dV (θ)R
SN−1 ψ2(θ) dV (θ).
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.8/23
Remark: Let Ω ⊂ RN be a bounded open set such that 0 ∈ Ω.
If ψh1 is the positive L2-normalized eigenfunction associated toµ1(h)
−∆SN−1ψh1 (θ) − h(θ)ψh1 (θ) = µ1(h)ψh1 (θ), in S
N−1,∫
SN−1 |ψh1 (θ)|2 dV (θ) = 1,
and
σh = σ(h,N) := −N − 2
2+
√
(
N − 2
2
)2
+ µ1(h),
it is easy to verify that ϕ(x) := |x|σhψh1 (x/|x|) ∈ H1(Ω) satisfies(in a weak H1(Ω)-sense and in a classical sense in Ω \ 0)
Lhϕ(x) = −∆ϕ(x) −h(x/|x|)
|x|2ϕ(x) = 0.
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.9/23
Asymptotics of solutions to perturbed dipole-type equations
How do solutions to equations associated to linear and nonlinear perturbationsof operator Lh behave near the singularity?
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.10/23
Asymptotics of solutions to perturbed dipole-type equations
How do solutions to equations associated to linear and nonlinear perturbationsof operator Lh behave near the singularity?
Linear perturbation: if Lh is perturbed with a linear term which isnegligible with respect to the inverse square singularity, thensolutions behave as ϕ(x) := |x|σhψh
1(x/|x|) near 0
(in the spirit of the Riemann removable singularity theorem)
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.10/23
Asymptotics of solutions to perturbed dipole-type equations
How do solutions to equations associated to linear and nonlinear perturbationsof operator Lh behave near the singularity?
Linear perturbation: if Lh is perturbed with a linear term which isnegligible with respect to the inverse square singularity, thensolutions behave as ϕ(x) := |x|σhψh
1(x/|x|) near 0
(in the spirit of the Riemann removable singularity theorem)
Theorem 1 [F.-Marchini-Terracini, Discrete Cont. Dyn. Systems (2008) ]
For h ∈ L∞(SN−1) such that ΛN (h) < 1, q ∈ L∞loc(Ω \ 0) such that
q(x) = O(|x|−(2−ε)) as |x| → 0 for some ε > 0, let u ∈ H1(Ω), u ≥ 0a.e. in Ω, u 6≡ 0, be a weak solution to Lhu = q u. Then the function
x 7→u(x)
|x|σhψh1 (x/|x|)
is continuous in Ω.
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.10/23
A Cauchy’s integral type formula ∀R > 0 such that B(0, R) ⊂ Ω
lim|x|→0
u(x)
|x|σhψh1(
x|x|
) =
∫
SN−1
(
R−σhu(Rθ)+
∫ R
0
s1−σh
2σh+N−2 q(s θ)u(s θ) ds
−R−2σh−N+2
∫ R
0
sN−1+σh
2σh+N−2 q(s θ)u(s θ) ds
)
ψh1 (θ) dV (θ)
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.11/23
A Cauchy’s integral type formula ∀R > 0 such that B(0, R) ⊂ Ω
lim|x|→0
u(x)
|x|σhψh1(
x|x|
) =
∫
SN−1
(
R−σhu(Rθ)+
∫ R
0
s1−σh
2σh+N−2 q(s θ)u(s θ) ds
−R−2σh−N+2
∫ R
0
sN−1+σh
2σh+N−2 q(s θ)u(s θ) ds
)
ψh1 (θ) dV (θ)
Remarks:
• the term at the right hand side is independent of R
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.11/23
A Cauchy’s integral type formula ∀R > 0 such that B(0, R) ⊂ Ω
lim|x|→0
u(x)
|x|σhψh1(
x|x|
) =
∫
SN−1
(
R−σhu(Rθ)+
∫ R
0
s1−σh
2σh+N−2 q(s θ)u(s θ) ds
−R−2σh−N+2
∫ R
0
sN−1+σh
2σh+N−2 q(s θ)u(s θ) ds
)
ψh1 (θ) dV (θ)
Remarks:
• the term at the right hand side is independent of R• in the case of a radial perturbation q, an analogous formula
holds also for changing sign solutions
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.11/23
A Cauchy’s integral type formula ∀R > 0 such that B(0, R) ⊂ Ω
lim|x|→0
u(x)
|x|σhψh1(
x|x|
) =
∫
SN−1
(
R−σhu(Rθ)+
∫ R
0
s1−σh
2σh+N−2 q(s θ)u(s θ) ds
−R−2σh−N+2
∫ R
0
sN−1+σh
2σh+N−2 q(s θ)u(s θ) ds
)
ψh1 (θ) dV (θ)
Remarks:
• the term at the right hand side is independent of R• in the case of a radial perturbation q, an analogous formula
holds also for changing sign solutions• an analogous result holds for semilinear equations with at
most critical growth
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.11/23
Positivity We consider the problem of positivity for Schrodinger operatorswith multiple locally anisotropic inverse-square singularities:
a necessary condition for positivity of the quadratic form
∫
RN
|∇u|2−k
∑
i=1
∫
B(ai,ri)
hi(
x−ai
|x−ai|
)
|x− ai|2u2−
∫
RN\B(0,R)
h∞(
x|x|
)
|x|2u2 −
∫
RN
W (x)u2
with hi, h∞ ∈ L∞(
SN−1
)
, W ∈ LN/2(RN ) ∩ L∞(RN ), is
µ1(hi) > −(N − 2)2
4for any i = 1, . . . , k,∞ (∗)
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.12/23
Positivity We consider the problem of positivity for Schrodinger operatorswith multiple locally anisotropic inverse-square singularities:
a necessary condition for positivity of the quadratic form
∫
RN
|∇u|2−k
∑
i=1
∫
B(ai,ri)
hi(
x−ai
|x−ai|
)
|x− ai|2u2−
∫
RN\B(0,R)
h∞(
x|x|
)
|x|2u2 −
∫
RN
W (x)u2
with hi, h∞ ∈ L∞(
SN−1
)
, W ∈ LN/2(RN ) ∩ L∞(RN ), is
µ1(hi) > −(N − 2)2
4for any i = 1, . . . , k,∞ (∗)
If all hi, i = 1, . . . , k,∞ are constant (isotropic case) then (∗) isalso sufficient [F.-Marchini-Terracini , J. Funct. Analysis (2007)]
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.12/23
The class V We frame the analysis of coercivity of Schrodinger dipole-type operators in the class
V :=
V (x) =
kX
i=1
χB(ai,ri)
(x)hi
` x−ai
|x−ai|
´
|x− ai|2+ χ
RN\B(0,R)(x)h∞
`
x|x|
´
|x|2+W (x) :
k ∈ N, ri, R > 0, ai ∈ RN , ai 6= aj for i 6= j, W ∈ LN/2(RN ) ∩ L∞(RN ),
hi ∈ L∞`
SN−1´
, µ1(hi) > −(N − 2)2/4 for any i = 1, . . . , k,∞
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.13/23
The class V We frame the analysis of coercivity of Schrodinger dipole-type operators in the class
V :=
V (x) =
kX
i=1
χB(ai,ri)
(x)hi
` x−ai
|x−ai|
´
|x− ai|2+ χ
RN\B(0,R)(x)h∞
`
x|x|
´
|x|2+W (x) :
k ∈ N, ri, R > 0, ai ∈ RN , ai 6= aj for i 6= j, W ∈ LN/2(RN ) ∩ L∞(RN ),
hi ∈ L∞`
SN−1´
, µ1(hi) > −(N − 2)2/4 for any i = 1, . . . , k,∞
Hardy’s and Sobolev’s inequalities =⇒ ∀V ∈ V
µ(V ) = infu∈D1,2
u 6≡0
∫
RN
(
|∇u|2 − V u2)
∫
RN |∇u|2> −∞.
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.13/23
An Allegretto-Piepenbrink type criterion in V
Positivity criterion in V . Let V ∈ V. Then
µ(V ) > 0
m
∃ ε > 0 and ϕ ∈ D1,2(RN ), ϕ positive
and continuous in RN \ a1, . . . , ak, such that
−∆ϕ− V ϕ ≥ ε V ϕ in (D1,2(RN ))⋆.
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.14/23
An Allegretto-Piepenbrink type criterion in V
Positivity criterion in V . Let V ∈ V. Then
µ(V ) > 0
m
∃ ε > 0 and ϕ ∈ D1,2(RN ), ϕ positive
and continuous in RN \ a1, . . . , ak, such that
−∆ϕ− V ϕ ≥ ε V ϕ in (D1,2(RN ))⋆.
Can we obtain coercive operators by summing up multisingular potentialsgiving rise to positive quadratic forms, after pushing them very far away fromeach other to weaken the interactions among poles?
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.14/23
Localization of binding
Sigal and Ouchinnokov, 1979 : If −∆ − V1 and −∆ − V2 are positiveoperators, is −∆ − V1 − V2(· − y) positive for |y| large?
• Simon, 1980 : yes for potentials with compact support• Pinchover, 1995 : yes for potentials in the Kato class
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.15/23
Localization of binding
Sigal and Ouchinnokov, 1979 : If −∆ − V1 and −∆ − V2 are positiveoperators, is −∆ − V1 − V2(· − y) positive for |y| large?
• Simon, 1980 : yes for potentials with compact support• Pinchover, 1995 : yes for potentials in the Kato class
If, for j = 1, 2,
Vj =
kjX
i=1
χB(a
ji,r
ji)
hji ((x− aji )/|x− aji |)
|x− aji |2
+ χBc
Rj
hj∞(x/|x|)
|x|2+Wj ∈ V,
then a necessary condition for positivity of −∆ − V1 − V2(· − y) for some y is that
µ1(h1∞ + h2
∞) > −
„
N − 2
2
«2
(∗∗)
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.15/23
Localization of binding
Sigal and Ouchinnokov, 1979 : If −∆ − V1 and −∆ − V2 are positiveoperators, is −∆ − V1 − V2(· − y) positive for |y| large?
• Simon, 1980 : yes for potentials with compact support• Pinchover, 1995 : yes for potentials in the Kato class
If, for j = 1, 2,
Vj =
kjX
i=1
χB(a
ji,r
ji)
hji ((x− aji )/|x− aji |)
|x− aji |2
+ χBc
Rj
hj∞(x/|x|)
|x|2+Wj ∈ V,
then a necessary condition for positivity of −∆ − V1 − V2(· − y) for some y is that
µ1(h1∞ + h2
∞) > −
„
N − 2
2
«2
(∗∗)
[F.-Marchini-Terracini , J. Funct. Analysis (2007)] (∗∗) is also sufficientfor localization of binding for locally isotropic singularities.
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.15/23
Localization of binding A lack of isotropy could produce the failure of localiza-tion of binding even under assumption (∗∗)
N ≥ 4, y = (0, . . . , 0, 1) ∈ RN , x = (x′, xN ) ∈ RN−1 × R
[Secchi-Smets-Willem (2003)] ; there exists ψ ∈ C∞c
`
(RN−1 \ 0) × R´
such that
R
RN |∇ψ(x)|2 dxR
RNψ2(x)
|x′|2dx
∼
„
N − 3
2
«2
and suppψ ⊂ Q
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.16/23
Localization of binding A lack of isotropy could produce the failure of localiza-tion of binding even under assumption (∗∗)
N ≥ 4, y = (0, . . . , 0, 1) ∈ RN , x = (x′, xN ) ∈ RN−1 × R
[Secchi-Smets-Willem (2003)] ; there exists ψ ∈ C∞c
`
(RN−1 \ 0) × R´
such that
R
RN |∇ψ(x)|2 dxR
RNψ2(x)
|x′|2dx
∼
„
N − 3
2
«2
and suppψ ⊂ Q
xN
x′C+
xN
x′
Q
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.16/23
Localization of binding A lack of isotropy could produce the failure of localiza-tion of binding even under assumption (∗∗)
N ≥ 4, y = (0, . . . , 0, 1) ∈ RN , x = (x′, xN ) ∈ RN−1 × R
[Secchi-Smets-Willem (2003)] ; there exists ψ ∈ C∞c
`
(RN−1 \ 0) × R´
such that
R
RN |∇ψ(x)|2 dxR
RNψ2(x)
|x′|2dx
∼
„
N − 3
2
«2
and suppψ ⊂ Q
xN
x′C+
xN
x′
Q
Q = C+ ∩ (y + C−), C− = −C+
V1(x) =λχC+
|x′|2, V2(x) =
λχC−
|x′|2,
12
`
N−32
´2< λ <
`
N−32
´2
[Badiale-Tarantello, (2002)] ; µ(V1), µ(V2) > 0, µ(V1 + V2) > 0, and thus (∗∗) holds. But
µ`
V1 + V2(· − µ y)´
< 0 for all µ > 0.
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.16/23
Localization of binding
Theorem 2 [F.-Marchini-Terracini, Indiana Univ. Math. J., to appear ]Let, for j = 1, 2,
Vj =
kj∑
i=1
χB(aj
i ,rj
i )
hji ((x− aji )/|x− aji |)
|x− aji |2
+ χBcRj
hj∞(x/|x|)
|x|2+Wj ∈ V.
If µ(V1), µ(V2) > 0, and ‖(h∞1 )+‖L∞ + ‖(h∞2 )+‖L∞ < (N − 2)2/4
⇓
µ(V1 + V2(· − y)) > 0 for |y| large
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.17/23
Localization of binding
Theorem 2 [F.-Marchini-Terracini, Indiana Univ. Math. J., to appear ]Let, for j = 1, 2,
Vj =
kj∑
i=1
χB(aj
i ,rj
i )
hji ((x− aji )/|x− aji |)
|x− aji |2
+ χBcRj
hj∞(x/|x|)
|x|2+Wj ∈ V.
If µ(V1), µ(V2) > 0, and ‖(h∞1 )+‖L∞ + ‖(h∞2 )+‖L∞ < (N − 2)2/4
⇓
µ(V1 + V2(· − y)) > 0 for |y| large
Idea of the proof. To the positive operators −∆ − V1 and −∆ − V2 correspond positivesupersolutions φ1 e φ2. The function φ1 + φ2(· − y) provides the positive supersolutionto the equation with potential V1 + V2(· − y) we are looking for. If |y| is large, then theinteraction between potentials is negligible.
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.17/23
Localization of binding
Theorem 2 [F.-Marchini-Terracini, Indiana Univ. Math. J., to appear ]Let, for j = 1, 2,
Vj =
kj∑
i=1
χB(aj
i ,rj
i )
hji ((x− aji )/|x− aji |)
|x− aji |2
+ χBcRj
hj∞(x/|x|)
|x|2+Wj ∈ V.
If µ(V1), µ(V2) > 0, and ‖(h∞1 )+‖L∞ + ‖(h∞2 )+‖L∞ < (N − 2)2/4
⇓
µ(V1 + V2(· − y)) > 0 for |y| large
Idea of the proof. To the positive operators −∆ − V1 and −∆ − V2 correspond positivesupersolutions φ1 e φ2. The function φ1 + φ2(· − y) provides the positive supersolutionto the equation with potential V1 + V2(· − y) we are looking for. If |y| is large, then theinteraction between potentials is negligible.
The control of such interaction is based on the asymptotics provided by Theorem 1.
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.17/23
Essential self-adjointness
For V ∈ V, let us consider the operator
−∆ − V, D(−∆ − V ) = C∞c (RN \ a1, . . . , ak).
If µ1(hi) ≥ − (N−2)2
4 + 1, i = 1, . . . , k, then −∆ − V has a unique
self-adjoint extension
[Kalf-Schmincke-Walter-Wust] for 1 isotropic pole[F.-Marchini-Terracini] for many anisotropic poles
which is the Friedrichs extension:
(−∆ − V )F : u 7→ −∆u− V u
D(
(−∆ − V )F)
= u ∈ H1 : −∆u− V u ∈ L2.
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.18/23
Essential self-adjointness
For V ∈ V, let us consider the operator
−∆ − V, D(−∆ − V ) = C∞c (RN \ a1, . . . , ak).
If µ1(hi) ≥ − (N−2)2
4 + 1, i = 1, . . . , k, then −∆ − V has a unique
self-adjoint extension
[Kalf-Schmincke-Walter-Wust] for 1 isotropic pole[F.-Marchini-Terracini] for many anisotropic poles
which is the Friedrichs extension:
(−∆ − V )F : u 7→ −∆u− V u
D(
(−∆ − V )F)
= u ∈ H1 : −∆u− V u ∈ L2.
If µ1(hi) < − (N−2)2
4 + 1 for some i, then −∆ − V admits many
self-adjoint extensions, among which (−∆ − V )F is the only one
whose domain is included in H1.
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.18/23
A multi-center nonlinear elliptic problem
(NL) − ∆v −k
∑
i=1
hi(
x−ai
|x−ai|
)
|x− ai|2v = v2∗−1, v > 0 in R
N \ a1, . . . , ak,
2∗ = 2NN−2 , hi ∈ C1(SN−1), ai ∈ R
N , ai 6= aj for i 6= j.
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.19/23
A multi-center nonlinear elliptic problem
(NL) − ∆v −k
∑
i=1
hi(
x−ai
|x−ai|
)
|x− ai|2v = v2∗−1, v > 0 in R
N \ a1, . . . , ak,
2∗ = 2NN−2 , hi ∈ C1(SN−1), ai ∈ R
N , ai 6= aj for i 6= j.
Look for solutions with the smallest energy (ground states), i.e. minimizing
Rayleigh quotient:
S(h1, h2, . . . , hk)= infu∈D1,2
u 6≡0
∫
RN|∇u|2dx −
∑ki=1
∫
RN
hi
(
x−ai|x−ai|
)
|x−ai|2u2(x) dx
( ∫
RN|u|2∗dx
)2/2∗
Minimizers in S(h1, h2, . . . , hk) ; solutions to (NL) (up to Lagrange multipliers)
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.19/23
References
• Terracini [Advances in Differential Equations (1996)]:If h ∈ C1(SN ) satisfies and
(H) µ1(h) > −(N − 2
2
)2and
maxSN−1 h > 0, if N ≥ 4,∫
SN−1 h ≥ 0, if N = 3,
then S(h) < S (= Sobolev const) and S(h) is achieved.
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.20/23
References
• Terracini [Advances in Differential Equations (1996)]:If h ∈ C1(SN ) satisfies and
(H) µ1(h) > −(N − 2
2
)2and
maxSN−1 h > 0, if N ≥ 4,∫
SN−1 h ≥ 0, if N = 3,
then S(h) < S (= Sobolev const) and S(h) is achieved.
• F.-Terracini [Comm. Partial Differential Equations (2006)]:isotropic case (hi constant).
• F.-Terracini [Calc. Var. PDE’s (2006)]: isotropic singularitieslocated on the vertices of regular polygons.
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.20/23
Minimization of the Rayleigh quotient
Difficulties arise from the non-compact embedding D1,2(RN ) → L2∗ (RN ).
What are possible reasons for lack of compactness of minimizing sequences (and hencefor non existence of minimizers)?
ր
non-singular points
concentration of mass at → singularities
ց
infinity
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.21/23
Minimization of the Rayleigh quotient
Difficulties arise from the non-compact embedding D1,2(RN ) → L2∗ (RN ).
What are possible reasons for lack of compactness of minimizing sequences (and hencefor non existence of minimizers)?
ր
non-singular points
concentration of mass at → singularities
ց
infinity
P. L. Lions Concentration–Compactness ; a minimizing sequence can diverge only
րS (concentration at non-singular points)
S(h1, . . . , hk) = → S(hi) (concentration at ai)
ցS
`Pki=1 hi
´
(concentration at infinity)
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.21/23
Minimization of the Rayleigh quotient
Difficulties arise from the non-compact embedding D1,2(RN ) → L2∗ (RN ).
What are possible reasons for lack of compactness of minimizing sequences (and hencefor non existence of minimizers)?
ր
non-singular points
concentration of mass at → singularities
ց
infinity
P. L. Lions Concentration–Compactness ; a minimizing sequence can diverge only
րS (concentration at non-singular points)
S(h1, . . . , hk) = → S(hi) (concentration at ai)
ցS
`Pki=1 hi
´
(concentration at infinity)
Below these energy thresh-
olds, minimizing sequences
satisfy the Palais-Smale con-
dition.
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.21/23
Existence theorem [F.-Terracini, Comm. Partial Differential Equations (2006)]: isotropic case
[F., Preprint (2008) ]: extension to anisotropic case
Theorem 3 If hi ∈ C1(SN ), Q is positive definite, and
(A) S(hk) = minS(hj) : j = 1, . . . , k, hk satisfies (H),
(B)
8
>
>
>
>
>
<
>
>
>
>
>
:
k−1X
i=1
hi` ak−ai
|ak−ai|
´
|ak − ai|2> 0, if µ1(hk) ≥ −
`
N−22
´2+ 1,
k−1X
i=1
Z
RN
hi`
x|x|
´
h
ψhk1
` x+ai−ak
|x+ai−ak|
´
i2
|x|2|x+ ai − ak|2(σhk
+N−2)> 0, if −
`
N−22
´2< µ1(hk) < −
`
N−22
´2+ 1,
(C) S(hk) ≤ S“
Pki=1hi
”
,
then S(h1, h2, . . . , hk) is achieved and problem (NL) admits a solution in D1,2(RN ).
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.22/23
Example: let us consider the case of two dipoles k = 2
hi(θ) = λi θ · di, i = 1, 2, λi > 0 and di ∈ RN , |di| = 1.
Assume that 0 < λ1 ≤ λ2, λ2 is small, and N is large in such away that the associated quadratic form is positive definite and
µ1(h2) ≥ −(
N−22
)2+ 1.
assumption (B) ; (a2 − a1) · d1 > 0
assumption (C) ; d1 · d2 < −λ1
2λ2.
If the first dipole λ1d1 is fixed at point a1, (B) gives a constraint
on the location of the second dipole while (C) gives a condition
on its orientation.
“7th AIMS International Conference on Dyn. Systems, Diff. Equations and Applications”, Arlington, May 20, 2008 – p.23/23