On Schrödinger equations with multisingular inverse-square ...felli/talks/arlington.pdfOn Schrödinger equations with multisingular inverse-square anisotropic potentials Veronica

On Schrödinger equations with multisingular inverse-squareanisotropic potentials

Veronica Felli

Dipartimento di Matematica ed Applicazioni

University of Milano–Bicocca

[email protected]

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)].


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



Lλ,d := −∆ −λ (x · d)

|x|3, x ∈ R

N , N ≥ 3,

λ = 2me~



Plan of the talk: asymptotics near the singularity of solutionsto equations associated to Lλ,d



Lλ,d := −∆ −λ (x · d)

|x|3, x ∈ R

N , N ≥ 3,

λ = 2me~




ւ

positivity properties(localization of binding)



Lλ,d := −∆ −λ (x · d)

|x|3, x ∈ R

N , N ≥ 3,

λ = 2me~




ւ ↓


essentialself-adjointness



Lλ,d := −∆ −λ (x · d)

|x|3, x ∈ R

N , N ≥ 3,

λ = 2me~




ւ ↓ ց


essentialself-adjointness

study of nonlinearSchrodinger equations

with multi-singularpotentials


Schrodinger operators with dipole-type potentials

Dipole potentials have the sameorder of homogeneity as inverse

square potentials 1/|x|2




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).







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 .


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, ...


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.


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.


References


• F.-Marchini-Terracini [Indiana Univ. Math. J., to appear]positivity, essential self-adjointness, and spectral propertiesof Schrödinger operators with multiple locally anisotropicinverse-square singularities.


References


• 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.


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


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


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




Qh(u) :=

∫

RN

|∇u(x)|2dx−

∫

RN

h(x/|x|)u2(x)

|x|2dx.



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).




Qh(u) :=

∫

RN

|∇u(x)|2dx−

∫

RN

h(x/|x|)u2(x)

|x|2dx.



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).


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.


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)




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.


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


)

ψh1 (θ) dV (θ)



lim|x|→0

u(x)

|x|σhψh1(

x|x|

) =

∫

SN−1

(

R−σhu(Rθ)+

∫ R

0

s1−σh


−R−2σh−N+2

∫ R

0

sN−1+σh


)

ψh1 (θ) dV (θ)

Remarks:

• the term at the right hand side is independent of R



lim|x|→0

u(x)

|x|σhψh1(

x|x|

) =

∫

SN−1

(

R−σhu(Rθ)+

∫ R

0

s1−σh


−R−2σh−N+2

∫ R

0

sN−1+σh


)

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



lim|x|→0

u(x)

|x|σhψh1(

x|x|

) =

∫

SN−1

(

R−σhu(Rθ)+

∫ R

0

s1−σh


−R−2σh−N+2

∫ R

0

sN−1+σh


)

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


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,∞ (∗)


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)]


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,∞


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> −∞.


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 ))⋆.


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?


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

(∗∗)





If, for j = 1, 2,

Vj =

kjX

i=1

χB(a

ji,r

ji)


|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.


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



N ≥ 4, y = (0, . . . , 0, 1) ∈ RN , x = (x′, xN ) ∈ RN−1 × R


`

(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



N ≥ 4, y = (0, . . . , 0, 1) ∈ RN , x = (x′, xN ) ∈ RN−1 × R


`

(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.



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 )


|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




Vj =

kj∑

i=1

χB(aj

i ,rj

i )


|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.




Vj =

kj∑

i=1

χB(aj

i ,rj

i )


|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.


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.


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.


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.


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)


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.


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.


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





ր

non-singular points


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)





ր

non-singular points


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.


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 ).


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.


Documents

On Schrödinger equations with multisingular inverse-square ...felli/talks/arlington.pdfOn Schrödinger equations with multisingular inverse-square anisotropic potentials Veronica