ELLIPTIC EQUATIONS WITH MULTI-SINGULAR INVERSE-SQUARE
POTENTIALS AND CRITICAL NONLINEARITY
VERONICA FELLI AND SUSANNA TERRACINI
Abstract. This paper deals with a class of nonlinear elliptic equations involving a critical
power-nonlinearity as well as a potential featuring multiple inverse square singularities. We show
that existence of solutions heavily depends on the strength and the location of the singularities.
We associate to the problem the corresponding Rayleigh quotient and give both sufficient and
necessary conditions on masses and location of singularities for the minimum to be achieved.
Both the cases of whole RN and bounded domains are taken into account.
1. Introduction and statement of the main results
This paper deals with a class of nonlinear elliptic equations involving a critical power-nonlinearity
as well as a potential featuring multiple inverse square singularities:

k
X

λi
∗

−∆v −
v = v 2 −1 ,
2
|x − ai |
(1)
i=1



v > 0 in RN \ {a1 , . . . , ak },
where N ≥ 3, k ∈ N, (λ1 , λ2 , . . . , λk ) ∈ Rk , (a1 , a2 , . . . , ak ) ∈ RkN , and 2∗ = N2N
−2 . Among all
possible solutions of the problem, we are interested in those having the smallest energy, termed
ground states. These solutions minimize the Rayleigh quotient associated with problem (1):
Z
(2)
S(λ1 , λ2 , . . . , λk ) =
inf
|∇u|2 dx −
RN
u∈D 1,2 (RN )\{0}
k
X
Z
λi
i=1
Z
RN
u2 (x)
dx
|x − ai |2
2/2∗
|u| dx
,
2∗
RN
where D1,2 (RN ) denotes the closure space of C0∞ (RN ) with respect to the norm
Z
1/2
2
kukD1,2 (RN ) :=
|∇u| dx
.
RN
∗
While the exponent 2∗ appears in the inclusion of the Sobolev space D1,2 (RN ) into L2 (RN ),
inverse-square potentials are related to the Hardy inequality (see for instance [18, 15]), which
Supported by Italy MIUR, national project “Variational Methods and Nonlinear Differential Equations”.
2000 Mathematics Subject Classification. 35J60, 35J20, 35B33.
1
2
VERONICA FELLI AND SUSANNA TERRACINI
ensures the inclusion of D1,2 (RN ) into the weighted space L2 (RN , |x|−2 dx) and
Z
Z
|u|2
(3)
dx
≤
C
|∇u|2 dx,
N
2
RN |x|
RN
2
where CN = N 2−2 is optimal and not attained.
Problem (1) with only one singularity has been first studied in [27] where it is completely solved.
More precisely it is shown that if λ ∈ (0, (N − 2)2 /4) then problem

−∆u = λ u + u2∗ −1 , x ∈ RN ,

|x|2
(4)

u > 0 in RN \ {0}, and u ∈ D1,2 (RN ),
has exactly a one-dimensional C 2 manifold of positive solutions given by
λ
− N 2−2 (λ) x
(5)
Zλ = wµ (x) = µ
w
, µ>0 ,
µ
where we denote
N −2
1/2
N (N − 2)νλ2 4
4λ
(λ)
(6)
w (x) =
,
and
ν
=
1
−
.
λ
N −2
(N − 2)2
(|x|1−νλ (1 + |x|2νλ )) 2
As a matter of facts, all solutions of (4) minimize the associated Rayleigh quotient and the minimum
can be computed as:
NN−1
Qλ (wµλ )
4λ
Qλ (u)
(7) S(λ) :=
inf
S,
∗ = R
∗ = 1 − (N − 2)2
R
u∈D 1,2 (RN )\{0}
2∗ dx 2/2
λ |2∗ dx 2/2
|u|
|w
N
N
µ
R
R
R
R
u2
where we denoted the quadratic form Qλ (u) = RN |∇u|2 dx − λ RN |x|
2 dx, see [27], and S is the
best constant in the Sobolev inequality
Skuk2L2∗ (RN ) ≤ kuk2D1,2 (RN ) .
It turns out that the solvability of (1) is strongly related to the positivity of the quadratic form
associated with the singular potential.
Proposition 1.1. A necessary condition for the solvability of problem (1) is that the quadratic
form
Z
Z
k
X
u2 (x)
2
Q(u) = Qλ1 ,...,λk ,a1 ,...,ak (u) :=
|∇u| dx −
λi
dx
2
RN
RN |x − ai |
i=1
is positive semidefinite, i.e
Q(u) ≥ 0
for all u ∈ D1,2 (RN ).
To prove existence we shall actually require that Q is positive definite, i.e. there exists a positive
constant ε = ε(λ1 , . . . , λk , a1 , . . . , ak ) such that
Z
(8)
Q(u) ≥ ε(λ1 , . . . , λk , a1 , . . . , ak )
|∇u|2 dx.
RN
In such a case, from Sobolev’s inequality
S(λ1 , λ2 , . . . , λk ) ≥ ε(λ1 , . . . , λk , a1 , . . . , ak ) S > 0.
MULTI-SINGULAR ELLIPTIC EQUATIONS
3
In general the positivity of Q depends on the strength and the location of the singularities.
Proposition 1.2. A sufficient condition for Q to be positive definite for any choice of a1 , a2 , . . . , ak
is that
X
(N − 2)2
λi <
.
4
i=1,...,k
λi >0
Conversely, if
X
λi >
i=1,...,k
λi >0
(N − 2)2
.
4
then there exists points a1 , a2 , . . . , ak such that Q is not positive definite.
The minimization of the Rayleigh quotients is a non trivial issue, as the embedding D1,2 (RN ) ,→
L (RN ) is not compact. In the case of only one singularity, such a difficulty was overcome in [27]
by exploiting the invariances of the problem. This argument is no longer available in the multi–
singular case; indeed, while problem (4) is invariant by the rescaling µ−(N −2)/µ u(·/µ) and by the
Kelvin transform, problem (1) is not, though it is locally almost-invariant by scaling close to each
singularity. This can cause the nonexistence of a minimizer in some circumstances:
2
Pk
. Then the infimum
Theorem 1.3. Assume that λi > 0 for all i = 1, . . . , k and i=1 λi < (N −2)
4
in (2) is not achieved.
2∗
To go further in the analysis, we have deepened the study of the behavior of minimizing sequences, with the aid of P. L. Lions Concentration-Compactness [22, 23]. There are three possible
reasons for this lack of compactness: there might be concentration of mass at some non-singular
point, at one of the singularities or at infinity. Hence, a minimizing sequence can diverge only
when S(λ1 , . . . , λk ) takes one of the values S (concentration at a non-singular point), S(λi ) (conPk centration at the singular point ai ) or S
i λi (concentration at infinity). Next result provides
sufficient conditions for the infimum S(λ1 , . . . , λk ) to stay below all the energy thresholds at which
the compactness can be lost.
Theorem 1.4. Assume that (λ1 , λ2 , . . . , λk ) ∈ Rk and (a1 , a2 , . . . , ak ) ∈ RkN satisfy (8) and the
following conditions
(9)
(10)
λ 1 ≤ λ2 ≤ · · · ≤ λk <
X
λi


> 0,


|ak − ai |2

i6=k
X





i6=k
(11)
(N − 2)2
,
4
X
λ
√i
> 0,
2
|ai − ak | (N −2) −4λk
N (N − 4)
,
4
if
0 < λk ≤
if
N (N − 4)
(N − 2)2
< λk <
,
4
4
λi ≤ 0.
i6=k
Then the infimum in (2) is achieved. Therefore equation (1) admits a solution in D1,2 (RN ).
4
VERONICA FELLI AND SUSANNA TERRACINI
Hence the existence of a minimizer over the whole RN heavily depends on the strength and the
location of the singularities. The result above applies, for example, when the point carrying the
largest positive mass is surrounded by other positive singularities, while negative singularities do
appear, but far away. Surprisingly enough, and in contrast with the case of only one singularity,
the phenomenon of loss of compactness becomes less dramatic when working on bounded domains:
indeed the infimum can be achieved also in case of all positive masses, as the following existence
result shows.
Theorem 1.5. Assume that Ω is a bounded smooth domain, {a1 , a2 , . . . , ak } ⊂ Ω,
X
λi
N (N − 4)
> 0,
, and
λ1 ≤ λ2 ≤ · · · ≤ λk ≤
4
|ak − ai |2
i6=k
and the quadratic form
Z
(12)
QΩ (u) =:=
2
|∇u| dx −
Ω
k
X
Z
λi
i=1
Ω
u2 (x)
dx
|x − ai |2
is positive definite.
Then the infimum in
Z
(13)
SΩ (λ1 , λ2 , . . . , λk ) =
inf
u∈H01 (Ω)\{0}
|∇u|2 dx −
Ω
k
X
i=1
Z
Z
λi
Ω
u2 (x)
dx
|x − ai |2
2/2∗
∗
|u|2 dx
,
Ω
is achieved. Therefore equation

k
X

λi
∗

−∆u −
v = v 2 −1 ,
2
|x
−
a
|
i
(14)
i=1



v > 0 in Ω \ {a1 , . . . , ak }, v = 0
admits a solution in
on ∂Ω,
H01 (Ω).
The presence of the bound λk ≤ N (N − 4)/4 does not come unexpected, as it is related to
some peculiar phenomena occurring in concentration and cutting-off of our test functions at the
singularities as observed also in [19, 16, 14]. This restriction can be removed letting Ω containing
a sufficiently large ball. Let B(0, R) denote the ball {x ∈ RN : |x| < R}.
Theorem 1.6. Assume that λ1 ≤ · · · ≤ λk , (12) and (10) hold. Then there exists R > 0 such
that if Ω ⊃ B(0, R), the infimum in (13) is achieved. Therefore equation (14) admits a solution in
H01 (Ω).
Singular potentials appear in several fields of applications and have been the object of a wide
recent mathematical research. Besides the already mentioned papers [14, 15, 19, 27], we quote,
among others, [1, 9, 10, 12, 25, 26]. Equation (1) can deserve as a model for many problems coming
from Quantum Mechanics, Chemistry, Cosmology, Astrophysics and Differential Geometry.
Equation (1) is characterized by the presence of inverse-square multi-singular potentials, whose
physical relevance is briefly described below. Potentials of the type 1/|x|2 arise in many fields,
such as quantum mechanics, nuclear physics, molecular physics, and quantum cosmology. The
MULTI-SINGULAR ELLIPTIC EQUATIONS
5
relevance of singular potentials in nonrelativistic quantum mechanics is highlighted in [17], where
a classification of spherically symmetric potentials V (|x|) is given by considering the limit
(15)
lim r2 V (r).
r→0
The potential V (r) is said to be regular at 0 if the limit in (15) is 0 and singular if such a limit
is ±∞. When the limit in (15) is finite and different from 0 (as in the case of inverse square
potentials) the potential is said to be a transition potential. Moreover we say that the potential V
is attractively (respectively repulsively) singular when the limit in (15) is −∞ (respectively +∞).
A simple argument by Landau and Lifshitz (see [17] and[20]) explains why 1/r2 can be regarded
as the transition threshold in the classification of singular potentials in a nonrelativistic context.
Let us consider a particle near the origin in the presence of a potential 1/rm . From the Uncertainty
Principle, its kinetic energy scales like r−2 , so that the energy is approximatively given by r−2 +
λr−m . For λ < 0 and m > 2 (attractively singular potential), the energy is not lower-bounded and
the particle “falls” to the center. On the other hand, if m < 2 the discrete spectrum has a lower
bound.
The potential 1/r2 also arises in point-dipole interactions in molecular physics (see [21]), where
the interactions between the charge of the electron and the dipole moment of the molecule gives rise
to long-distance forces and to the presence of an inverse-square potential in the Schrödinger equation for the wave function of the electron. We also mention that inverse-square singular potentials
appear in the linearization of standard combustion models leading to blow-up phenomenon (see
[3, 15, 28]) and in quantum cosmological models such as the Wheeler-de-Witt equation (see [4]).
In Quantum Chemistry, multi-singular potentials arise for example when considering molecular
systems consisting of k nuclei of unit charge located at a finite number of points a1 , . . . , ak and
of k electrons. This type of systems are described by the Hartree-Fock model, where Coulomb
multi-singular potentials arise in correspondence to the interactions between the electrons and the
fixed nuclei, see [8, 24].
We also mention that Schödinger operators with multipolar inverse square singular potentials
are studied in [11], where estimates on resolvant truncated at high frequencies are proved.
The presence of the nonlinear term in equation (1) is motivated by the fact that in some physical
problems interaction phenomena (e.g. the presence of many particles interacting in quantum
physics or the possible joining or splitting of different universes in quantum cosmology) lead to
nonlinear terms, which are power-type in a first approximation. As far as the meaning of the
critical exponent is concerned, see [5].
Let us finally remark that (1) has also a geometric motivation as it is related to the Yamabe
problem on the sphere SN . Indeed, if we identify RN with SN through the stereographic projection
and endow SN with a metric whose scalar curvature is singular at the north pole and at a finite
number of points, then the problem of finding a conformal metric with prescribed scalar curvature
1 leads to solve equation (1), where the unknown v has the meaning of a conformal factor (see [2]).
The paper is organized as follows. In section 2 we prove that the Palais-Smale
condition is
Pk
satisfied below some critical threshold involving S, S(λi ), and S
i−1 λi . Section 3 contains
some interaction estimates and the proof of Theorem 1.4. Sections 4, respectively 5, are devoted
to the proof of Proposition 1.1, respectively Theorem 1.3. Section 6 deals with multi-singular
problems in bounded domains. Finally in the Appendix we prove some technical estimates stated
in Section 3.
6
VERONICA FELLI AND SUSANNA TERRACINI
2. The Palais-Smale condition
Let us introduce the functional
Z
Z
Z
k
X
∗
1
S(λ1 , λ2 , . . . , λk )
u2 (x)
λi
(16)
J(u) =
dx
−
|∇u|2 dx −
|u|2 dx.
2
∗
2 RN
2
|x
−
a
|
2
N
N
i
R
R
i=1
∗
If u is a critical points of J in D1,2 (RN ), u > 0, then v = S(λ1 , λ2 , . . . , λk )1/(2 −2) u is a solution
to equation (1). The following theorem provides a local Palais-Smale condition for J below some
critical threshold.
Theorem 2.1. Assume (8). Let {un }n∈N ⊂ D1,2 (RN ) be a Palais-Smale sequence for J, namely
lim J(un ) = c < ∞ in R
n→∞
lim J 0 (un ) = 0 in the dual space (D1,2 (RN ))? .
and
n→∞
If
(17)
c < c∗ =
Xk
N/2
N
1
S(λ1 , λ2 , . . . , λk )1− 2 min S, S(λ1 ), . . . , S(λk ), S
,
λj
j=1
N
then {un }n∈N has a converging subsequence.
Proof. Let {un } be a Palais-Smale sequence for J, then from (8) there exists some positive
constant c1 such that
Z
Z
k
X
N −2 0
u2 (x)
dx = N J(un ) −
hJ (un ), un i
c1 kun k2D1,2 (RN ) ≤
|∇u|2 dx −
λi
2
|x
−
a
|
2
N
N
i
R
R
i=1
= N c + o(kun kD1,2 (RN ) ) + o(1)
hence {un } is a bounded sequence in D1,2 (RN ). Then, up to a subsequence, we have
un * u0 in D1,2 (RN ),
∗
un → u0 almost everywhere, and un → u0 in Lα
loc for any α ∈ [1, 2 ).
Therefore, from the Concentration Compactness Principle by P. L. Lions, (see [22] and [23]), we
deduce the existence of a subsequence, still denoted by {un }, for which there exist an at most
countable set J , xj ∈ RN \ {a1 , . . . , ak }, real numbers µxj , νxj , j ∈ J , and µai , νai , γi , i = 1, . . . , k
such that the following convergences hold in the sense of measures
(18)
|∇un |2 * dµ ≥ |∇u0 |2 +
k
X
µai δai +
i=1
(19)
∗
∗
|un |2 * dν = |u0 |2 +
k
X
i=1
(20)
λi
νai δai +
X
µxj δxj ,
j∈J
X
νxj δxj ,
j∈J
u20
u2n
* dγai = λi
+ γi δai ,
2
|x − ai |
|x − ai |2
for any
i = 1, . . . , k.
From Sobolev’s inequality it follows that
2
(21)
Sνx2j∗ ≤ µxj for all j ∈ J
2
and Sνa2i∗ ≤ µai for all i = 1, . . . , k.
MULTI-SINGULAR ELLIPTIC EQUATIONS
7
To study the concentration at infinity of the sequence we also introduce the following quantities
Z
Z
2∗
ν∞ = lim lim sup
|un | dx, µ∞ = lim lim sup
|∇un |2 dx
R→∞ n→∞
R→∞ n→∞
|x|>R
|x|>R
and
k
X
Z
γ∞ = lim lim sup
R→∞ n→∞
|x|>R
λi
i=1
u2
n
dx.
|x|2
Claim 1. We claim that
J
(22)
is finite and for j ∈ J either νxj = 0 or νxj ≥
N/2
S
.
S(λ1 , λ2 , . . . , λk )
For ε > 0, let φj be a smooth cut-off function centered at xj , 0 ≤ φj (x) ≤ 1 such that
if |x − xj | ≤
φj (x) = 1
ε
,
2
if |x − xj | ≥ ε,
φj (x) = 0
and |∇φj | ≤
4
.
ε
Testing J 0 (un ) with un φj we obtain
0 = lim hJ 0 (un ), un φj i
n→∞
Z
Z
|∇un | φj +
= lim
n→∞
Z
2
RN
un ∇un · ∇φj −
RN
RN
Z
k
X
λi u2n φj
2∗
− S(λ1 , λ2 , . . . , λk )
φj |un | .
|x − ai |2
RN
i=1
From (18–20), and since ai 6∈ supp(φj ) for all i = 1, . . . , k provided ε is sufficiently small, we find
that
Z
Z
Z
Z
∗
|un |2 φj =
|∇un |2 φj =
φj dµ,
lim
φj dν,
lim
n→∞
RN
n→∞
RN
RN
RN
and
Z
k
k
X
X
λi u2n φj
λi u20 φj
=
.
2
2
Bε (xj ) i=1 |x − ai |
Bε (xj ) i=1 |x − ai |
Z
lim
n→∞
Taking limits as ε → 0 we obtain
Z
lim lim ε→0 n→∞ RN
un ∇un ∇φj → 0.
Hence
0 = lim lim hJ 0 (un ), un φi ≥ µxj − S(λ1 , λ2 , . . . , λk )νxj .
ε→0 n→∞
2
2∗
xj
By (21) we have that Sν ≤ µxj , then we obtain that either νxj = 0 or νxj ≥
which implies that J is finite. Claim 1 is proved.
N/2
S
,
S(λ1 ,λ2 ,...,λk )
Claim 2. We claim that
(23)
for each i = 1, 2, . . . , k
either νai = 0
or
νai ≥
N/2
S(λi )
.
S(λ1 , λ2 , . . . , λk )
In order to prove claim 2, for each i = 1, 2, . . . , k we consider a smooth cut-off function ψi satisfying
0 ≤ ψi (x) ≤ 1,
ψi (x) = 1
if |x − ai | ≤
ε
,
2
ψi (x) = 0
if |x − ai | ≥ ε,
and |∇ψi | ≤
4
.
ε
8
VERONICA FELLI AND SUSANNA TERRACINI
From (7) we obtain that
R
RN
(24)
R
ψi2 u2n
|∇(un ψi )|2 dx − λi RN |x−a
2 dx
i|
≥ S(λi )
R
2/2∗
2∗
|ψ
u
|
i n
RN
hence
Z
RN
ψi2 |∇un |2 dx +
Z
RN
Z
≥ λi
RN
u2n |∇ψi |2 dx + 2
Z
un ψi ∇un · ∇ψi dx
RN
Z
2/2∗
ψi2 u2n
2∗
dx + S(λi )
|ψi un |
.
|x − ai |2
RN
It is easy to verify that
Z
lim lim sup
ε→0 n→∞
RN
u2n |∇ψi |2 dx + 2
Z
un ψi ∇un · ∇ψ dx = 0.
RN
Then from (18–20) we obtain
∗
µai ≥ γi + S(λi )νa2/2
.
i
(25)
Testing J 0 (un ) with un ψi we infer
0 = lim hJ 0 (un ), un ψi i
n→∞
Z
= lim
n→∞
|∇un |2 ψi +
RN
Z
Z
un ∇un · ∇ψi −
RN
RN
Z
k
X
λj u2n ψi
2∗
−
S(λ
,
λ
,
.
.
.
,
λ
)
ψ
|u
|
.
1
2
k
i n
|x − aj |2
RN
j=1
Hence from (18), (19) and the following fact
(
Z
0
λj u2n ψi
lim lim sup
dx =
2
ε→0 n→∞
|x
−
a
|
N
γi
j
R
if i 6= j
if i = j
(which easily follows from (20)), we deduce that
µai − γi ≤ S(λ1 , λ2 , . . . , λk )νai .
(26)
From (25) and (26) we conclude that either νai = 0 or νai ≥
proved.
S(λi )
S(λ1 ,λ2 ,...,λk )
N
2
. Claim 2 is thereby
Claim 3. We claim that
(27)
either ν∞ = 0 or
ν∞ ≥
Pk
N/2
S( i=1 λi )
.
S(λ1 , λ2 , . . . , λk )
In order to prove claim 3, we study the possibility of concentration at ∞. Let ψ be a regular cut-off
function such that
2
1, if |x| > 2R
0 ≤ ψ(x) ≤ 1, ψ(x) =
and |∇ψ| ≤ .
0, if |x| < R,
R
MULTI-SINGULAR ELLIPTIC EQUATIONS
From (7) we obtain that
P
R
R
k
|∇(un ψ)|2 dx −
i=1 λi
RN
RN
(28)
R
2/2∗
|ψun |2∗
RN
ψ 2 u2n
|x|2 dx
≥S
9
P
k
i=1
λi .
Therefore we have
Z
Z
ψ 2 |∇un |2 dx +
(29)
Z
u2n |∇ψ|2 dx + 2
un ψ∇un · ∇ψdx
RN
RN
P
Z
2/2∗
Z ψ 2 u2
P
k
k
n
2∗
dx
+
S
λ
|ψu
|
.
λ
≥
i
n
i
i=1
i=1
2
RN
RN |x|
RN
We claim that
Z
lim lim sup
R→∞ n→∞
RN
u2n |∇ψ|2 dx
Indeed using Hölder inequality we obtain
Z
Z
|un |ψ|∇un ||∇ψ|dx ≤
un ψ∇un · ∇ψ dx = 0.
Z
+2
RN
1/2 Z
|un | |∇ψ| dx
2
RN
R<|x|<2R
Z
Z
1/2
|∇un | dx
.
2
2
R<|x|<2R
Hence
|un |ψ|∇un ||∇ψ| dx ≤ const
lim sup
n→∞
1/2
|u0 | |∇ψ| dx
2
RN
2
R<|x|<2R
Z
2∗
≤ const
|u0 |
1/2∗ Z
dx
R<|x|<2R
N
1/N
|∇ψ| dx
R<|x|<2R
2/2∗
|u0 | dx
.
Z
2∗
≤ const
R<|x|<2R
Therefore we conclude that
Z
Z
lim lim sup
|un |ψ|∇un ||∇ψ| dx ≤ const lim
R→∞ n→∞
R→∞
RN
∗
|u0 |2 dx
2/2∗
= 0.
R<|x|<2R
Using the same argument we can prove that
Z
lim lim sup
R→∞ n→∞
RN
u2n |∇ψ|2 = 0.
Then from (29) we infer
µ∞ − γ∞ ≥ S
(30)
P
k
i=1
2/2∗
.
λ i ν∞
Testing J 0 (un ) with un ψ we obtain
0 = lim hJ 0 (un ), un ψi
n→∞
Z
= lim
n→∞
RN
|∇un |2 ψ +
Z
Z
un ∇un · ∇ψ −
RN
RN
Z
k
X
λj u2n ψ
2∗
−
S(λ
,
λ
,
.
.
.
,
λ
)
ψ|u
|
.
1
2
k
n
|x − aj |2
RN
j=1
10
VERONICA FELLI AND SUSANNA TERRACINI
Since
u2n ψ
u2n ψ u2n ψ |x|2 − |x − aj |2 u2n ψ
=
−
≤
c̃
|x − aj |2
|x|2 |x|2
|x − aj |2
|x|3
for some constant c̃ independent of R, and by Hölder’s inequality
Z
2/2∗ Z
2/N
Z
u2n ψ
2∗
− 32 N
dx
≤
u
|x|
= O(R−1 ),
n
3
|x|
N
|x|>R
R
|x|>R
we deduce that
Z
(31)
lim lim sup
R→+∞ n→∞
RN
Z X
k
k
X
λj u2n ψ
λj u2n ψ
=
lim
lim
sup
= γ∞ .
2
R→+∞ n→∞
|x − aj |2
RN j=1 |x|
j=1
Hence from (18), (19) and (31) we deduce that
µ∞ − γ∞ ≤ S(λ1 , λ2 , . . . , λk )ν∞ .
(32)
From (30) and (32) we conclude that either ν∞ = 0 or ν∞ ≥
Claim 3.
P
N2
S( k
i=1 λi )
S(λ1 ,λ2 ,...,λk )
, thus proving
As a conclusion we obtain
(33)
1
c = J(un ) − hJ 0 (un ), un i + o(1)
2
Z
1
|un |2∗ dx + o(1)
= S(λ1 , λ2 , . . . , λk )
N
N
R
Z
k
X
X
S(λ1 , λ2 , . . . , λk )
=
|u0 |2∗ dx +
νai + ν∞ +
νxj .
N
RN
i=1
j∈J
From (17), (33), (22), (23), and (27), we deduce that νxj = 0 for any j ∈ J , νai = 0 for any
i = 1, . . . , k, and ν∞ = 0. Then up to a subsequence un → u0 in D1,2 (RN ).
3. Interaction estimates and proof of Theorem 1.4
In order to prove Theorem 1.4 some interaction estimates are needed. To begin with, we prove a
continuity lemma related to Hardy’s inequality.
Proposition 3.1. For any φ ∈ D1,2 (RN ), there holds
Z
Z
φ2 (x)
φ2 (x)
(34)
lim
dx
=
dx.
2
|ξ|→0 RN |x + ξ|2
RN |x|
Proof. For any u ∈ D1,2 (RN ), u ≥ 0 a.e., we consider the Schwarz symmetrization of u defined as
(35)
u∗ (x) := inf t > 0 : {y ∈ RN : u(y) > t} ≤ ωN |x|N
where | · | denotes the Lebesgue measure of RN and ωN is the volume of the standard unit N -ball.
From [29, Theorem 21.8], it follows that for any measurable set Ω and any φ ∈ D1,2 (RN )
∗ 2
Z
Z
1
φ2
∗
2
dx
≤
(|φ|
(x))
dx,
2
|x + ξ|
Ω∗
Ω |x + ξ|
MULTI-SINGULAR ELLIPTIC EQUATIONS
11
where Ω∗ = B 0, (|Ω|/ωN )1/N and |φ|∗ denoted the Schwarz symmetrization of |φ|, see (35). A
direct calculation shows that
∗
1
1
=
|x + ξ|
|x|
hence
Z
Z
(|φ|∗ )2
φ2
dx
≤
dx.
(36)
2
|x|2
B 0,(|Ω|/ωN )1/N
Ω |x + ξ|
Let φ ∈ D1,2 (RN ). It is easy to see that
φ2
|x + ξ|2
(37)
converges to
φ2
|x|2
a.e. in RN as |ξ| → 0.
Moreover from (36) it follows that
for any ε > 0 there exists δ > 0 such that for any measurable set Ω with |Ω| < δ
R
φ2
N
there holds Ω |x+ξ|
.
2 dx < ε for any ξ ∈ R
(38)
On the other hand for any R > 2 and |ξ| < 1 we have that
Z
Z
φ2
φ2
dx
≤
4
dx
2
2
RN \BR (0) |x + ξ|
RN \BR (0) |x|
hence
φ2
dx = 0 uniformly with respect to ξ, |ξ| < 1.
R→∞ RN \B (0) |x + ξ|2
R
φ2
Hence the family of functions |x+ξ|
is uniformly integrable. In view of (37–39), the
2 : |ξ| < 1
Vitali’s convergence Theorem allows to conclude.
Z
(39)
lim
As minimizers of problem (7), we consider
zµλ (x)
wµλ (x)
− N 2−2
− N2−2
x 1−νλ x 1+νλ
+ µ
µ
1/2∗ = αλ,N µ
|wµλ |2∗ dx
N −2
where αλ,N = N (N − 2)νλ2 4 kw(λ) k−1
is a positive constant depending only on λ and N . A
L2∗
direct consequence of Proposition 3.1 is the following corollary.
(40)
=
R
RN
Corollary 3.2. For any ξ ∈ RN and λ ∈ (−∞, (N − 2)2 /4) there holds
Z
Z
|zµλ (x)|2
|zµλ (x)|2
(41)
dx
=
dx + o(1) as µ → +∞.
2
|x|2
RN |x + ξ|
RN
Proof. By the change of variable x = µy we have
Z
Z
Z
|zµλ (x)|2
|z1λ (x/µ)|2
−(N −2)
dx
=
µ
dx
=
2
2
RN |x + ξ|
RN |x + ξ|
RN
Since
ξ
µ
|z1λ (y)|2
dy.
y + ξ 2
µ
→ 0 as µ → +∞, the corollary follows from Proposition 3.1.
Pk
As a consequence of the above corollary, we can bound S(λ1 , . . . , λk ) from above by S( i=1 λi ).
12
VERONICA FELLI AND SUSANNA TERRACINI
Corollary 3.3. Assume that (a1 , a2 , . . . , ak ) ∈ RkN and (λ1 , λ2 , . . . , λk ) ∈ Rk satisfy
(0, (N − 2)2 /4). Then
Pk
S(λ1 , . . . , λk ) ≤ S( i=1 λi ).
Proof. Let us set λ̃ =
Pk
i=1
Z
S(λ1 , . . . , λk ) ≤
RN
Z
=
RN
Pk
i=1
λi ∈
λi . From Corollary 3.2 it follows that
|∇zµλ̃ |2 dx −
|∇zµλ̃ |2
dx −
k
X
|zµλ̃ |2
dx
|x − ai |2
Z
λi
i=1
RN
k
X
Z
λi
RN
i=1
|zµλ̃ |2
Pk
dx + o(1) = S( i=1 λi ) + o(1)
2
|x|
as µ → ∞. Letting µ → ∞, we obtain the desired estimate.
The following lemma describes the behavior of
technical and is contained in the appendix.
R
|x + ξ|−2 |zµλ |2 as µ → 0. The proof is quite
Lemma 3.4. For any ξ ∈ RN there holds
 2 R
µ
λ 2
2


|ξ|2 RN |z1 | dx + o µ



Z

|zµλ |2
µ2 | ln µ|
2
dx
=
αλ,N
+ o(µ2 | ln µ|)
|ξ|2
2

RN |x + ξ|



√
√
√

2
2
α2 β
(N −2)2 −4λ
|ξ|− (N −2) −4λ + o µ (N −2) −4λ
λ,N λ,N µ
if λ <
N (N −4)
,
4
if λ =
N (N −4)
,
4
if λ >
N (N −4)
,
4
as µ → 0, where
Z
βλ,N =
RN
dx
|x|2 |x
N −2+
− e1 |
√
(N −2)2 −4λ
,
e1 = (1, 0, . . . , 0) ∈ RN .
From Lemma 3.4, we deduce the following corollary which provides sufficient conditions for S(λ1 , . . . , λk )
to stay below S(λj ).
Corollary 3.5. Let j ∈ {1, 2, . . . , k}. If λj > 0 and one of the following assumptions is satisfied
(42)
0 < λj ≤
N (N − 4)
4
and
X
and
X
i6=j
2
(43)
(N − 2)
N (N − 4)
< λj <
4
4
i6=j
λi
> 0,
|aj − ai |2
λ
√i
> 0,
2
|aj − ai | (N −2) −4λj
then
S(λ1 , . . . , λk ) < S(λj ).
MULTI-SINGULAR ELLIPTIC EQUATIONS
13
Proof. Under assumption (42), from Lemma 3.4 we obtain
Z
Z
λ
X Z |zµλj (x − aj )|2
|zµ j (x − aj )|2
dx
−
λi
dx
S(λ1 , . . . , λk ) ≤
|∇zµλj (x − aj )|2 dx − λj
|x − aj |2
|x − ai |2
RN
RN
RN
i6=j
Z
Z
λ
λ
X Z
|zµ j (x)|2
|zµ j (x)|2
=
dx
−
λ
dx
|∇zµλj (x)|2 dx − λj
i
2
|x|2
RN
RN |x + aj − ai |
RN
i6=j
λ
X Z
|zµ j (x)|2
= S(λj ) −
λi
dx
2
RN |x + aj − ai |
i6=j
 R
λj 2 P
λi
2

µ
|
|z
+
o(1)
if λj < N (N4−4)

1
i6=j |aj −ai |2
RN
= S(λj ) −

λi
α2 µ2 | ln µ| P
if λj = N (N4−4) ,
λj ,N
i6=j |aj −ai |2 + o(1)
as µ → 0. Taking µ sufficiently small we obtain that S(λ1 , . . . , λk ) < S(λj ). Under assumption (43), from Lemma 3.4 we obtain
X
√
λ
2
√i
S(λ1 , . . . , λk ) ≤ S(λj ) − αλ2 j ,N βλj ,N µ (N −2) −4λ
+
o(1)
.
(N −2)2 −4λ
i6=j |ai − aj |
Again, taking µ sufficiently small we obtain that S(λ1 , . . . , λk ) < S(λj ).
Proof of Theorem 1.4. By (8) the quadratic form Q is positive definite and provides an
equivalent norm on D1,2 (RN ). Let {un }n ⊂ D1,2 (RN ) be a minimizing sequence for (2). From the
homogeneity of the quotient there is no restriction requiring kun kL2∗ (RN ) = 1, while from Ekeland’s
variational principle we can assume that the sequence has the Palais-Smale property, i.e for any
v ∈ D1,2 (RN )
Z
Z
Z
k
X
∗
un (x)v(x)
dx
−
S(λ
λ
,
.
.
.
,
λ
)
|un |2 −2 un v dx = o kvkD1,2 (RN ) .
λi
∇un · ∇v −
1
2
k
2
RN |x − ai |
RN
RN
i=1
Hence J 0 (un ) → 0 in (D1,2 (RN ))? and
1
1
1
J(un ) →
− ∗ S(λ1 λ2 , . . . , λk ) = S(λ1 λ2 , . . . , λk ).
2 2
N
From assumption (10) and Corollary 3.5, we infer that
(44)
S(λ1 , . . . , λk ) < S(λk ).
Since the function λ 7→ S(λ) is decreasing, from assumption (9) we deduce that
for all i = 1, . . . , k − 1 and S(λk ) < S(0) = S,
Pk
and from assumption (11) we deduce that i=1 λi ≤ λk hence (again by monotonicity of λ 7→ S(λ))
we obtain
k
X
(46)
S(λk ) ≤ S
λi .
(45)
S(λk ) ≤ S(λi )
i=1
14
VERONICA FELLI AND SUSANNA TERRACINI
Gathering (44), (45), and (46), we finally have
Xk
S(λ1 , . . . , λk ) < min S, S(λ1 ), . . . , S(λk ), S
j=1
λj
and hence
Xk
N/2
N
1
1
λj
.
S(λ1 λ2 , . . . , λk ) < S(λ1 , λ2 , . . . , λk )1− 2 min S, S(λ1 ), . . . , S(λk ), S
j=1
N
N
Therefore Theorem 2.1 applies and we can conclude that {un }n∈N has a subsequence strongly
converging to some u0 ∈ D1,2 (RN ), which satisfies J(u0 ) = N1 S(λ1 , λ2 , . . . , λk ). Hence u0 achieves
the infimum in (2). Since J(u0 ) = J(|u0 |), we have that also |u0 | is a minimizer in (2) and then
∗
v0 = S(λ1 λ2 , . . . , λk )1/(2 −2) |u0 | is a nonnegative solution to equation (1). The maximum principle
N
in R \ {a1 , . . . , ak } implies that v0 > 0 in RN \ {a1 , . . . , ak }.
4. Proof of Proposition 1.1 and 1.2
∞
(RN \
Proof of Proposition 1.1. Let u be a positive solution to problem (1). Let φ ∈ CC
{a1 , . . . , ak }), i.e. let φ be a smooth function with a compact support A which is disjoint from
the set of the singularities. By classical regularity results, u is smooth in RN \ {a1 , . . . , ak }; in
particular u is bounded from below by a positive constant on the set A. Hence we can test equation
2
(1) with the function φu thus obtaining
Z
Z
Z
Z
k
X
∗
φ2
φ2
φ
2
∇φ · ∇u dx −
|∇u| dx −
λi
dx −
φ2 u2 −2 dx = 0.
2
2
2
u
u
|x
−
a
|
N
N
N
N
i
R
R
R
R
i=1
Since
2
we deduce
Z
Z
k
X
λi
|∇φ|2 −
RN
Since
i=1
RN
φ
φ2
∇φ · ∇u − 2 |∇u|2 ≤ |∇φ|2
u
u
φ2
dx ≥
|x − ai |2
Z
φ2 u2
∗
−2
dx ≥ 0
RN
∞
for all φ ∈ CC
(RN \ {a1 , . . . , ak }).
∞
(RN
CC
\ {a1 , . . . , ak }) is dense in D1,2 (RN ) (see [7, Lemma 2.1]), we obtain
Z
Z
k
X
φ2
2
λi
Q(φ) =
|∇φ| −
dx ≥ 0 for all φ ∈ D1,2 (RN ),
2
N |x − ai |
R
RN
i=1
namely Q is positive definite.
Proof of Proposition 1.2.
(47)
From Hardy’s inequality, it follows that
X Z
4
Q(u) ≥ 1 −
λi
|∇u|2 dx.
(N − 2)2
RN
λi >0
Hence a sufficient condition for Q to be positive definite is that
X
(N − 2)2
λi <
.
4
i=1,...,k
λi >0
MULTI-SINGULAR ELLIPTIC EQUATIONS
15
P
Assume now that λi >0 λi > (N − 2)2 /4. From optimality of the constant (N − 2)2 /4 in Hardy’s
inequality, we have that there exists some function φ ∈ C0∞ (RN ) such that
Z
X Z
φ2
2
dx < 0.
|∇φ| −
λi
2
RN
RN |x|
λi >0
N −2
For any µ > 0, consider the function φµ (x) = µ− 2 φ(x/µ). A change of variable yields
Z
Z
X Z
X Z
φ2µ
φ2
2
dx
=
|∇φ|
−
λi
|∇φµ |2 −
λi
dx for all µ > 0.
2
a 2
RN
RN
RN x − i RN |x − ai |
λi >0
λi >0
µ
Letting µ → ∞, Proposition 3.1 yields
Z
Z
X Z
X Z
φ2µ
φ2
2
2
|∇φµ | −
λi
dx
−→
|∇φ|
dx < 0
−
λ
i
2
2
RN
RN |x − ai |
RN
RN |x|
λi >0
λi >0
therefore there exists some large µ̄ such that the function ψ = φµ̄ satisfies
Z
X Z
ψ2
dx < 0.
|∇ψ|2 −
λi
2
RN
RN |x − ai |
λi >0
We now notice that since ψ has compact support, i.e. supp ψ ⊂ B(0, R),
Z
Z
ψ2
1
ψ 2 dx −→ 0 as |a| → ∞,
dx ≤
2
(|a| − R)2 B(0,R)
RN |x − a|
hence it is possible to locate the poles carrying negative masses far away from supp ψ in order to
get
Z
Z
k
X
ψ2
|∇ψ|2 −
dx < 0,
λi
2
RN
RN |x − ai |
i=1
thus proving the second part of Proposition 1.2.
5. Proof of Theorem 1.3
We start by showing that if all masses λi are positive, then the inequality of Corollary 3.3 is
indeed an equality.
2
Pk
Proposition 5.1. Assume that λi > 0 for all i = 1, . . . , k and i=1 λi < (N −2)
. Then
4
Pk
S(λ1 , . . . , λk ) = S( i=1 λi ).
Proof. For any u ∈ D1,2 (RN ), u ≥ 0 a.e., we consider the Schwarz symmetrization u∗ of u,
see (35). From [29, Theorem 21.8], it follows that
∗ 2
Z
Z
1
u2
∗
2
dx ≤
(u (x))
.
2
|x − ai |
RN
RN |x − ai |
∗
1
1
Since |x−a
= |x|
, it follows
i|
Z
Z
u2
(u∗ (x))2
(48)
dx
≤
dx for any u ≥ 0 a.e, u ∈ D1,2 (RN ).
2
2
|x
−
a
|
|x|
N
N
i
R
R
16
VERONICA FELLI AND SUSANNA TERRACINI
Moreover
Z
|u∗ |p =
(49)
Z
RN
|u|p ,
RN
see for example [29, Corollary 21.7], and, by the Pólya-Szegö inequality
Z
Z
(50)
|∇u∗ |2 ≤
|∇u|2 ,
RN
RN
which together with (48) imply that for all u ∈ D1,2 (RN ), u ≥ 0 a.e.
Z
(51)
|∇u|2 dx −
RN
k
X
Z
λi
RN
i=1
Z
u2 (x)
dx
|x − ai |2
Z
RN
≥
2/2∗
∗
|u|2 dx
|∇u∗ |2 dx −
k
X
Z
λi
RN
i=1
Z
RN
∗
|u∗ |2 dx
|u∗ (x)|2
dx
|x|2
2/2∗
RN
≥S
X k
i=1
λi .
Since the Rayleigh quotient above remains unchanged when replacing u with |u|, we have that
Z
S(λ1 , λ2 , . . . , λk ) =
|∇u|2 dx −
RN
inf
k
X
Z
λi
i=1
u∈D 1,2 (RN )\{0}, u≥0
Z
RN
u2 (x)
dx
|x − ai |2
2/2∗
∗
|u|2 dx
RN
Pk
hence passing to the infimum in (51), we obtain S(λ1 , λ2 , . . . , λk ) ≥ S( i=1 λi ), which together
Pk
with the estimate of Corollary 3.3 gives equality S(λ1 , λ2 , . . . , λk ) = S( i=1 λi ).
We are now in position to prove Theorem 1.3.
Proof of Theorem 1.3. We argue by contradiction. Assume that the infimum in (2) is attained
by some function u0 ∈ D1,2 (RN ) \ {0}. We may assume that u0 ≥ 0 a.e. in RN , otherwise we take
|u0 | which also is a minimizer in (2). Hence we can consider the Schwarz symmetrization u∗0 , see
(35). From (51), we have that
Z
2
|∇u0 | dx −
(52)
S(λ1 , λ2 , . . . , λk ) =
RN
k
X
Z
λi
RN
i=1
Z
u20 (x)
dx
|x − ai |2
2/2∗
|u0 | dx
2∗
RN
Z
≥
RN
|∇u∗0 |2 dx −
k
X
i=1
Z
RN
Z
λi
RN
|u∗0 (x)|2
dx
|x|2
2/2
∗
|u∗0 |2 dx
∗
≥S
X k
i=1
λi .
MULTI-SINGULAR ELLIPTIC EQUATIONS
17
From (52) and Proposition 5.1, we deduce that all inequalities in (52) are indeed equalities. In
particular
Z
Z
Z
Z
k
k
X
X
|u∗0 (x)|2
u20 (x)
∗ 2
dx
|∇u
|
dx
−
dx
|∇u0 |2 dx −
λ
λi
i
0
2
|x|2
RN
RN
RN
RN |x − ai |
i=1
i=1
(53)
=
Z
2/2∗
Z
2/2∗
∗
∗
|u0 |2 dx
RN
RN
=S
X k
i=1
|u∗0 |2 dx
λi .
From (53) and (49) it follows that
Z
2
|∇u0 | dx −
RN
k
X
Z
λi
RN
i=1
u20 (x)
dx =
|x − ai |2
Z
RN
|∇u∗0 |2 dx
−
k
X
i=1
Z
λi
RN
|u∗0 (x)|2
dx
|x|2
hence in view of (48) and (50) we obtain
Z
0≤
|∇u0 |2 dx −
RN
Z
RN
|∇u∗0 |2 dx =
k
X
i=1
Z
k
X
u20 (x)
|u∗0 (x)|2
λ
dx
−
dx ≤ 0.
i
|x − ai |2
|x|2
RN
i=1
Z
λi
RN
Therefore
Z
|∇u0 |2 dx =
(54)
RN
Z
RN
|∇u∗0 |2 dx.
On the other hand from (53), it follows that u∗0 is a minimizer of (7) and solves equation (4) with
Pk
λ = i=1 λi . Hence in view of the classification of solutions to (4) given in [27], u∗0 must be equal
Pk
to wµλ for some µ > 0 where λ = i=1 λi and wµλ is defined in (5). In particular u∗0 (|x|) is strictly
decreasing and hence
{x ∈ RN : ∇u∗0 (x) = 0} = 0.
(55)
(54) and (55) allow to use [6, Theorem 1.1] to conclude that there exists some point x0 ∈ RN
such that u0 = u∗0 (· − x0 ), namely u0 is spherically symmetric with respect to x0 . Since u0 is a
∗
minimizer in (2), then v0 = S(λ1 , λ2 , . . . , λk )1/(2 −2) u0 is a solution to equation (1). Consequently
Pk
λi
i=1 |x−ai |2 must be spherically symmetric with respect to x0 , which gives a contradiction. Hence
the infimum in (2) cannot be attained.
6. The problem on bounded domains
Let Ω be a bounded smooth domain in RN , N ≥ 3. In this section we study equation (14) and the
associated minimization problem (13). The corresponding functional is given by
(56)
1
JΩ (u) =
2
Z
2
|∇u| dx −
Ω
Z
k
X
λi
i=1
2
Ω
u2 (x)
SΩ (λ1 λ2 , . . . , λk )
dx −
|x − ai |2
2∗
The following theorem contains a local Palais-Smale condition.
Z
Ω
∗
|u|2 dx.
18
VERONICA FELLI AND SUSANNA TERRACINI
Theorem 6.1. Assume that (12) holds. Let {un }n∈N ⊂ H01 (Ω) be a Palais-Smale sequence for
JΩ , namely
lim JΩ (un ) = c < ∞ in R
n→∞
lim JΩ0 (un ) = 0 in the dual space (H01 )? .
and
n→∞
If
c < c∗Ω =
N/2
N
1
SΩ (λ1 , λ2 , . . . , λk )1− 2 min S, S(λ1 ), . . . , S(λk )
,
N
then {un }n∈N has a converging subsequence.
The proof of the above theorem is analogous to the proof of Theorem 2.1. In this case, due
to boundedness
of the domain, there is no possibility of loss of mass at infinity, so that the term
Pk
λ
S
is
not
involved in the level at which Palais-Smale condition fails.
j=1 j
Lemma 6.2. Let j ∈ {1, 2, . . . , k}. There holds
(57)
SΩ (λ1 , . . . , λk ) ≤ S(λj ) + O(µνλj (N −2) )
P
 R
λ
λi

µ2 RN |z1 j |2
2 + o(1)

i6
=
j
|a
−a
|
j
i




P
 2
λi
2
+
o(1)
α
µ
|
ln
µ|
2
λj ,N
i6=j |aj −ai |
−




P


αλ2 ,N βλ,N µνλj (N −2)
√λi
+
o(1)
i6=j
j
(N −2)2 −4λ
|ai −aj |
if λj <
N (N −4)
4
if λj =
N (N −4)
4
if λj >
N (N −4)
.
4
Moreover if and
0 < λj ≤
N (N − 4)
,
4
and
X
i6=j
λi
>0
|aj − ai |2
then
SΩ (λ1 , . . . , λk ) < S(λj ).
Proof. Let ω be an open set such that ω ⊂ Ω and aj ∈ ω and let ψ be a smooth cut-off function
such that
0 ≤ ψ(x) ≤ 1,
ψ≡0
in RN \ Ω,
ψ≡1
in ω.
λ
Then ψ(x)zµ j (x − aj ) ∈ H01 (Ω). (57) follows from
SΩ (λ1 , . . . , λk )
R
R |ψ(x)zµλj (x−aj )|2
R
P
λj
2
|∇(ψ(x)z
dx − i6=j λi RN
(x
−
a
))|
dx
−
λ
µ
j
j
N
|x−aj |2
R
RN
≤
2/2∗
R
λj
2∗ dx
|ψ(x)z
(x
−
a
)|
µ
j
RN
λ
|ψ(x)zµ j (x−aj )|2
|x−ai |2
dx
MULTI-SINGULAR ELLIPTIC EQUATIONS
and the following estimates
Z
Z
|∇(ψ(x)zµλj (x − aj ))|2 dx =
(58)
RN
RN
λ
Z
(59)
RN
Z
(61)
RN
Z
λ
|ψ(x)zµ j (x
Z
|∇zµλj (x)|2 dx + O(µνλj (N −2) )
λ
|zµ j (x)|2
dx + O(µνλj (N −2) )
|x|2
RN
λ
− aj )|2
|zµ j (x)|2
dx
=
dx + O(µνλj (N −2) )
2
2
|x
−
a
|
|x
+
(a
−
a
)|
N
N
i
j
i
R
R
2/2∗
∗
|ψ(x)zµλj (x − aj )|2 dx
= 1 + O(µνλj (N −2) ).
Z
(60)
|ψ(x)zµ j (x − aj )|2
dx =
|x − aj |2
19
Let us prove (58). We have that
Z
Z
λj
2
|∇(ψ(x)zµ (x − aj ))| dx =
ψ(x)2 |∇zµλj (x − aj )|2 dx
(62)
RN
RN
Z
Z
+
|zµλj (x − aj )|2 |∇ψ(x)|2 dx + 2
ψ(x)zµλj (x − aj )∇ψ(x) · ∇zµλj (x − aj ) dx.
RN
RN
In view of (40) we have
Z
Z
(63) ψ(x)2 |∇zµλj (x − aj )|2 dx −
|∇zµλj (x − aj )|2 dx
RN
RN
Z
λ x − aj 2
= µ−N
(1 − ψ 2 (x))∇z1 j
dx
µ
RN \ω
Z
λ
=
(1 − ψ 2 (µy + aj ))|∇z1 j (y)|2 dy
µ−1 ((RN \ω)−aj )
Z +∞
−1−νλj N +2νλj
ds = const µνλj (N −2) ,
Z
λ
2
2
λj
|z1 j (y)|2 µ2 |∇ψ(µy + aj )|2 dy
|zµ (x − aj )| |∇ψ(x)| dx ≤ const
≤ const
s
µ−1 r
Z
(64)
RN
µ−1 ((Ω\ω)−aj )
Z
≤ const
µ−1 ((Ω\ω)−aj )
Z
µ−1 R
≤ const
∗
λ
|z1 j (y)|2
−1−νλj N
s
2/2∗
dy
Z
N
2/N
|∇ψ(µy + aj )| dy
µ
µ−1 ((Ω\ω)−aj )
2/2∗
ds
2
= const µνλj (N −2) ,
µ−1 r
and
Z
(65)
RN
ψ(x)zµλj (x − aj )∇ψ(x) · ∇zµλj (x − aj ) dx
Z
µ−1 R
≤ const µ
s−νλj (N −2) ds = const µνλj (N −2) .
µ−1 r
Estimate (58) follows from (63–65). The proof of (59–61) is analogous. To show that SΩ (λ1 , . . . , λk ) <
S(λj ), it is enough to observe that if λj < N (N4−4) then νλj (N − 2) > 2 and hence O(µνλj (N −2) ) =
20
VERONICA FELLI AND SUSANNA TERRACINI
o(µ2 ) as µ → 0+ , while if λj = N (N4−4) then νλj (N − 2) = 2 and hence O(µνλj (N −2) ) = o(µ2 | ln µ|).
Taking µ sufficiently small, we obtain SΩ (λ1 , . . . , λk ) < S(λj ).
Proof of Theorem 1.5.
of Theorem 1.4.
It follows from Theorem 6.1 and Lemma 6.2, arguing as in the proof
Proof of Theorem 1.6.
Using a density argument it is easy to prove that
for any ε > 0 there exists R̄ > 0 such that if Ω ⊃ B(0, R)
(66)
then SΩ (λ1 , . . . , λk ) < S(λ1 , . . . , λk ) + ε.
Theorem 1.6 follows from Theorem 6.1, Corollary 3.5, and (66).
Proof of Lemma 3.4.
z1λ
2
7. Appendix
q
Set γλ = 1 − 1 − (N4λ
−2)2 . For λ <
N (N −4)
4
we have that γλ <
N −4
N −2
and
N
∈ L (R ). Note that
Z
Z
Z
|zµλ |2
|z1λ |2
|z1λ |2
2
2
(67)
dx
=
µ
dx
=
µ
dx
2
2
2
|ξ|
RN |x + ξ|
RN |µx + ξ|
|x|< 2µ |µx + ξ|
−(N −2)
Z
µ2(1−γλ ) + |x − ξ|2(1−γλ )
2
(1−γλ )(N −2)
+ αλ,N µ
dx.
|ξ|
|x|2 |x − ξ|γλ (N −2)
|x−ξ|≥ 2
Since
Z
|ξ|
|z1λ (x)|2
|x|< 2µ
1
1
− 2 dx
2
|µx + ξ|
|ξ|
−µ2 |x|2 − 2µξ · x =
|x|
|x|
+ |x|
|ξ|
|ξ|2 |µx + ξ|2 |x|< 2µ
−(N −2) 2
−(N −2)
Z
Z
2
2
αλ,N
µ2
2αλ,N
µ
|x|1−γλ + |x|γλ −1
|x|1−γλ + |x|γλ −1
|x|
|x|
≤
+
N −2 |µx + ξ|2
N −2 |µx + ξ|2
|ξ|
|ξ|
|ξ|2
|x|
|ξ|
|x|
|x|< 2µ
|x|< 2µ
|ξ|
Z
2
2 Z |ξ|
2
2µ
4αλ,N µ
8αλ,N µ 2µ
r3
r2
≤
N −2 dr +
N −2 dr
4
3
|ξ|
|ξ|
0
0
r1−γλ + rγλ −1
r1−γλ + rγλ −1
2
αλ,N
Z
−(N −2)
γλ −1 −(N −2)
1−γλ
and
Z
|ξ|
2µ
0
Z
0
r
r1−γλ
|ξ|
2µ
+ rγλ −1
r
r1−γλ
3
N −2
2
+ rγλ −1
N −2


O(1)



dr = O(| log µ|)



O(µ(1−γλ )(N −2)−4 )
if γλ <
N −6
N −2
if γλ =
N −6
N −2
if γλ >
N −6
N −2


O(1)
if γλ <



dr = O(| log µ|)
if γλ =



O(µN −5−γλ (N −2) ) if γ >
λ
N −5
N −2
N −5
N −2
N −5
N −2
MULTI-SINGULAR ELLIPTIC EQUATIONS
21
we deduce that
Z
|ξ|
|z1λ (x)|2
|x|< 2µ
1
1
− 2 dx = o(1)
|µx + ξ|2
|ξ|
as µ → 0 and hence, since z1λ ∈ L2 (RN ),
Z
Z
Z
1
dx
1
λ
2
|z1λ (x)|2
=
|z
(68)
(x)|
dx
+
o(1)
=
|z1λ (x)|2 dx + o(1).
1
2
|ξ|
|µx + ξ|2
|ξ|2 |x|< |ξ|
|ξ|
N
R
|x|< 2µ
2µ
On the other hand we have
Z
(69)
|x−ξ|≥
|ξ|
2
−(N −2)
µ2(1−γλ ) + |x − ξ|2(1−γλ )
dx
|x|2 |x − ξ|γλ (N −2)
Z
≤
≤
|x−ξ|≥|ξ|/2
|x|<2|ξ|
2(N −2)(2−γλ ) dx
+
|ξ|(N −2)(2−γλ ) |x|2
2(N −2)(2−γλ )
|ξ|(N −2)(2−γλ )
Z
Z
|x−ξ|≥|ξ|/2
|x|>2|ξ|
2|ξ|
rN −3 dr + 4(N −2)(2−γλ )
0
4(N −2)(2−γλ ) dx
|x|(N −2)(2−γλ ) |x|2
Z
∞
2|ξ|
dr
rN −1−γλ (N −2)
= O(1).
From (67), (68), and (69) we deduce that
Z
RN
For λ =
N (N −4)
4
Z
(70)
RN
Since
Z
we have that γλ =
|zµλ |2
dx = µ2
|x + ξ|2
N −4
N −2
Z
RN
|z1λ (x)|2 dx + o(µ2 ).
and
Z
|z1λ |2
|z1λ |2
2
dx
=
µ
dx
2
2
|ξ|
RN |µx + ξ|
|x|< 2µ |µx + ξ|
−(N −2)
Z
µ4/(N −2) + |x − ξ|4/(N −2)
2
2
+ αλ,N µ
dx.
|ξ|
|x|2 |x − ξ|N −4
|x−ξ|≥ 2
Z
1
1
− 2 dx
2
|µx + ξ|
|ξ|
|ξ|
Z
Z |ξ|
2
2
4αλ,N
µ2 2µ
8αλ,N
µ 2µ
r3
r2
≤
dr +
dr = O(1),
2
2 N −2
2
2 N −2
4
3
|ξ|
|ξ|
0
0
r N −2 + r− N −2
r N −2 + r− N −2
|z1λ (x)|2
|ξ|
|x|< 2µ
|zµλ |2
µ2
dx
=
|x + ξ|2
|ξ|2
we deduce that
Z
(71)
|ξ|
|x|< 2µ
|z1λ (x)|2
dx
1
= 2
|µx + ξ|2
|ξ|
Z
|ξ|
|x|< 2µ
|z1λ (x)|2 dx + O(1).
22
VERONICA FELLI AND SUSANNA TERRACINI
On the other hand
Z
|ξ|
|z1λ (x)|2
dx =
|x|< 2µ
2
αλ,N
Z
|ξ|
2µ
r3 dr
4
(1 + r N −2 )N −2
Z |ξ|
Z 1
Z |ξ|
2µ dr
2µ
r3 dr
r3
1
2
= αλ,N
+
dr .
+
−
4
4
r
r
1
0 (1 + r N −2 )N −2
1
(1 + r N −2 )N −2
0
Since
r3
4
(1 + r N −2 )N −2
1 (N − 2)
− ∼
N +2
r
r N −2
as
r → +∞,
we can conclude that
Z
(72)
|ξ|
2
|z1λ (x)|2 dx = αλ,N
| ln µ| + O(1).
|x|< 2µ
From (71) and (72) we then deduce that
Z
(73)
|ξ|
|z1λ (x)|2
|x|< 2µ
2
αλ,N
dx
=
| ln µ| + O(1).
|µx + ξ|2
|ξ|2
As above (see (69))
−(N −2)
µ2(1−γλ ) + |x − ξ|2(1−γλ )
dx = O(1).
|x|2 |x − ξ|γλ (N −2)
Z
(74)
|x−ξ|≥
|ξ|
2
Gathering (70), (73), and (74) we deduce that
Z
RN
For λ >
N (N −4)
4
|zµλ |2
µ2 | ln µ|
2
dx
=
α
+ o(µ2 | ln µ|).
λ,N
|x + ξ|2
|ξ|2
we have that γλ >
N −4
N −2
and
(75)
Z
−(N −2)
Z
|zµλ |2
µ2(1−γλ ) + |x − ξ|2(1−γλ )
2
(1−γλ )(N −2)
dx = αλ,N µ
dx
2
|x|2 |x − ξ|γλ (N −2)
RN |x + ξ|
RN
Z
dx
2
= αλ,N
µ(1−γλ )(N −2)
2 |x − ξ|(N −2)(2−γλ )
|x|
N
R
−(N −2)
Z 2(1−γλ )
µ
+ |x − ξ|2(1−γλ )
1
2
+ αλ,N
µ(1−γλ )(N −2)
−
dx.
|x|2 |x − ξ|γλ (N −2)
|x|2 |x − ξ|(N −2)(2−γλ )
RN
From the elementary inequality
|(a + b)s − as | ≤ C(as−1 b + bs )
MULTI-SINGULAR ELLIPTIC EQUATIONS
23
which holds for some C = C(s) > 0 where s ≥ 1, and for any a, b ≥ 0, it follows that
−(N −2)
2(1−γλ )
µ
+ |x − ξ|2(1−γλ )
1
(76)
− 2
2
γ
(N
−2)
(N
−2)(2−γ
)
λ
λ
|x| |x − ξ|
|x| |x − ξ|
2(1−γλ ) 2(1−γλ )
−(N −2)
µ
µ
+ |x − ξ|2(1−γλ )
≤ const
|x|2 |x − ξ|2+γλ (N −4)
−(N −2) µ2(1−γλ )(N −2) µ2(1−γλ ) + |x − ξ|2(1−γλ )
+
.
|x|2 |x − ξ|(2−γλ )(N −2)
Since
−(N −2)
Z
µ2(1−γλ ) µ2(1−γλ ) + |x − ξ|2(1−γλ )
|x|2 |x − ξ|2+γλ (N −4)
RN
"
Z
Z
dx
dx
2(1−γλ )
≤ const µ2(1−γλ )
+
µ
|ξ|
|ξ|
2
N
|x−ξ|> 2 |x|
|x−ξ|> 2 |x| (2−γλ )
|x|<2|ξ|
|x|>2|ξ|
#
Z |ξ|
N
−1
2µ
r
dr
γλ (N −2)−N +4
+µ
r2+γλ (N −4) (1 + r2(1−γλ ) )N −2
0
= O(µ2(1−γλ ) ) + O(µγλ (N −2)−N +4 ) = o(1)
as
µ → 0+
and
−(N −2)
µ2(1−γλ )(N −2) µ2(1−γλ ) + |x − ξ|2(1−γλ )
|x|2 |x − ξ|(2−γλ )(N −2)
RN
"
Z
Z
dx
dx
2(1−γλ )(N −2)
+
µ
≤ const µ2(1−γλ )(N −2)
|ξ|
|ξ|
|x−ξ|> 2 |x|4N −6−3γλ (N −2)
|x−ξ|> 2 |x|2
|x|<2|ξ|
|x|>2|ξ|
#
Z |ξ|
N −1
2µ
r
dr
+ µγλ (N −2)−N +4
r(2−γλ )(N −2) (1 + r2(1−γλ ) )N −2
0
#
"
Z |ξ|
2µ
dr
2(1−γλ )(N −2)
γλ (N −2)−N +4
γλ (N −2)−N +4
≤ const µ
+µ
+µ
r−3γλ (N −2)+3N −7
1
≤ const µ2(1−γλ )(N −2) + µγλ (N −2)−N +4 + µ2(1−γλ )(N −2) = o(1) as µ → 0+
Z
from (75) and (76) we deduce that
Z
Z
|zµλ |2
dx
2
(1−γλ )(N −2)
(77)
dx = αλ,N µ
+ o µ(1−γλ )(N −2) .
2
2
(N
−2)(2−γ
)
λ
RN |x + ξ|
RN |x| |x − ξ|
Note that the function
Z
dx
ϕ(ξ) :=
2
(N −2)(2−γλ )
RN |x| |x − ξ|
is invariant by rotation and homogeneous, more precisely
√
2
ϕ(ηξ) = η − (N −2) −4λ ϕ(ξ),
24
hence
(78)
VERONICA FELLI AND SUSANNA TERRACINI
√
√
2
2
ϕ(ξ) = |ξ|− (N −2) −4λ ϕ(ξ/|ξ|) = |ξ|− (N −2) −4λ ϕ(e1 ).
(77) and (78) yield the required estimate for λ >
N (N −4)
.
4
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Università di Milano Bicocca, Dipartimento di Matematica e Applicazioni, Via Cozzi 53, 20125 Milano.
E-mail address: [email protected], [email protected].