1 Introduction
Hardy–Schrödinger operators on manifolds are of the form Δg-V, where Δg is the Laplace–Beltrami operator and V is a potential that has a quadratic singularity at some point of the manifold. For hyperbolic spaces, Carron [7] showed that, just like in the Euclidean case and with the same best constant, the following inequality holds on any Cartan–Hadamard manifold M,
(
n
-
2
)
2
4
∫
M
u
2
d
g
(
o
,
x
)
2
𝑑
v
g
≤
∫
M
|
∇
g
u
|
2
𝑑
v
g
for all
u
∈
C
c
∞
(
M
)
,
where dg(o,x) denotes the geodesic distance to a fixed point o∈M. There are many other works identifying suitable Hardy potentials, their relationship with the elliptic operator on hand, as well as corresponding energy inequalities [2, 8, 9, 13, 14, 16, 20]. In the Euclidean case, the Hardy potential V(x)=1|x|2 is distinguished by the fact that u2|x|2 has the same homogeneity as |∇u|2, but also u2*(s)|x|s, where 2*(s)=2(n-s)n-2 and 0≤s<2. In other words, the integrals
∫
ℝ
n
u
2
|
x
|
2
𝑑
x
,
∫
ℝ
n
|
∇
u
|
2
𝑑
x
and
∫
ℝ
n
u
2
*
(
s
)
|
x
|
s
𝑑
x
are invariant under the scaling u(x)↦λn-22u(λx), λ>0, which makes the corresponding minimization problem non-compact, hence giving rise to interesting concentration phenomena. In [1], Adimurthi and Sekar use the fundamental solution of a general second-order elliptic operator to generate natural candidates and derive Hardy-type inequalities. They also extended their arguments to Riemannian manifolds using the fundamental solution of the p-Laplacian. In [9], Devyver, Fraas and Pinchover study the case of a general linear second-order differential operator P on non-compact manifolds. They find a relation between positive super-solutions of the equation Pu=0, Hardy-type inequalities involving P and a weight W, as well as some properties of the spectrum of a corresponding weighted operator. See also [3]. Although not needed in this paper, we mention that the Hardy inequality is closely related to the inequality due to Caffarelli, Kohn and Nirenberg [6], via a power-type change of variable. For related parabolic problems, the readers may consult [17, 19].
In this paper, we shall focus on the Poincaré ball model of the hyperbolic space 𝔹n, n≥3, that is, the Euclidean unit ball B1(0):={x∈ℝn:|x|<1} endowed with the metric g𝔹n=(21-|x|2)2gEucl. This framework has the added feature of radial symmetry, which plays an important role and contributes to the richness of the structure. In this direction, Sandeep and Tintarev [18] recently came up with several integral inequalities involving weights on 𝔹n that are invariant under scaling, once restricted to the class of radial functions (see also Li and Wang [14]). As described below, this scaling is given in terms of the fundamental solution of the hyperbolic Laplacian Δ𝔹nu=div𝔹n(∇𝔹nu). Indeed, let
(1.1)
f
(
r
)
:=
(
1
-
r
2
)
n
-
2
r
n
-
1
and
G
(
r
)
:=
∫
r
1
f
(
t
)
𝑑
t
,
where r=∑i=1nxi2 denotes the Euclidean distance of a point x∈B1(0) to the origin. It is known that 1nωn-1G(r) is a fundamental solution of the hyperbolic Laplacian Δ𝔹n.
As usual, the Sobolev space H1(𝔹n) is defined as the completion of Cc∞(𝔹n) with respect to the norm
∥
u
∥
H
1
(
𝔹
n
)
2
=
∫
𝔹
n
|
∇
𝔹
n
u
|
2
𝑑
v
g
𝔹
n
.
We denote by Hr1(𝔹n) the subspace of radially symmetric functions. For functions u∈Hr1(𝔹n), we consider the scaling
(1.2)
u
λ
(
r
)
=
λ
-
1
2
u
(
G
-
1
(
λ
G
(
r
)
)
)
,
λ
>
0
.
In [18], Sandeep and Tintarev have noted that for any u∈Hr1(𝔹n) and p≥1, one has the following invariance property:
∫
𝔹
n
|
∇
𝔹
n
u
λ
|
2
𝑑
v
g
𝔹
n
=
∫
𝔹
n
|
∇
𝔹
n
u
|
2
𝑑
v
g
𝔹
n
and
∫
𝔹
n
V
p
|
u
λ
|
p
𝑑
v
g
𝔹
n
=
∫
𝔹
n
V
p
|
u
|
p
𝑑
v
g
𝔹
n
,
where
(1.3)
V
p
(
r
)
:=
f
(
r
)
2
(
1
-
r
2
)
2
4
(
n
-
2
)
2
G
(
r
)
p
+
2
2
.
In other words, the hyperbolic scaling r↦G-1(λG(r)) is quite analogous to the Euclidean scaling. Indeed, in that case, by taking G¯(ρ)=ρ2-n, we see that G¯-1(λG¯(ρ))=λ¯=λ12-n for ρ=|x| in ℝn. Also, note that G¯ is – up to a constant – the fundamental solution of the Euclidean Laplacian Δ in ℝn. The weights Vp have the following asymptotic behaviors, for n≥3 and p>1,
V
p
(
r
)
=
{
c
0
(
n
,
p
)
r
n
(
1
-
p
2
*
)
(
1
+
o
(
1
)
)
as
r
→
0
,
c
1
(
n
,
p
)
(
1
-
r
)
(
n
-
1
)
(
p
-
2
)
2
(
1
+
o
(
1
)
)
as
r
→
1
.
Here 2*=2nn-2.
In particular, for n≥3, the weight
V
2
(
r
)
=
1
4
(
n
-
2
)
2
(
f
(
r
)
(
1
-
r
2
)
G
(
r
)
)
2
∼
r
→
0
1
4
r
2
,
and at r=1 has a finite positive value. In other words, the weight V2 is qualitatively similar to the Euclidean Hardy weight, and Sandeep and Tintarev have indeed established the following Hardy inequality on the hyperbolic space 𝔹n (cf. [18, Theorem 3.4]) (see also [9] where they deal with similar Hardy weights):
(
n
-
2
)
2
4
∫
𝔹
n
V
2
|
u
|
2
𝑑
v
g
𝔹
n
≤
∫
𝔹
n
|
∇
𝔹
n
u
|
2
𝑑
v
g
𝔹
n
for any
u
∈
H
1
(
𝔹
n
)
.
They also show in the same paper the following Sobolev inequality: for some constant C>0,
(
∫
𝔹
n
V
2
*
|
u
|
2
*
𝑑
v
g
𝔹
n
)
2
2
*
≤
C
∫
𝔹
n
|
∇
𝔹
n
u
|
2
𝑑
v
g
𝔹
n
for any
u
∈
H
1
(
𝔹
n
)
.
By interpolating between these two inequalities taking 0≤s≤2, one easily obtain the following Hardy–Sobolev inequality.
Lemma 1.1.
If γ<(n-2)24, then there exists a constant C>0 such that, for any u∈H1(Bn),
C
(
∫
𝔹
n
V
2
*
(
s
)
|
u
|
2
*
(
s
)
𝑑
v
g
𝔹
n
)
2
2
*
(
s
)
≤
∫
𝔹
n
|
∇
𝔹
n
u
|
2
𝑑
v
g
𝔹
n
-
γ
∫
𝔹
n
V
2
|
u
|
2
𝑑
v
g
𝔹
n
,
where 2*(s):=2(n-s)(n-2).
Note that, up to a positive constant, we have V2*(s)∼r→01rs, adding to the analogy with the Euclidean case, where we have for any u∈H1(ℝn),
C
(
∫
ℝ
n
|
u
|
2
*
(
s
)
|
x
|
s
𝑑
x
)
2
2
*
(
s
)
≤
∫
ℝ
n
|
∇
u
|
2
𝑑
x
-
γ
∫
ℝ
n
|
u
|
2
|
x
|
2
𝑑
x
.
Motivated by the recent progress on the Euclidean Hardy–Schrödinger equation (see for example Ghoussoub and Robert [12, 11], and the references therein), we shall consider the problem of existence of extremals for the corresponding best constant, that is,
(1.4)
μ
γ
,
λ
(
Ω
)
:=
inf
u
∈
H
0
1
(
Ω
)
∖
{
0
}
∫
Ω
|
∇
𝔹
n
u
|
2
𝑑
v
g
𝔹
n
-
γ
∫
Ω
V
2
|
u
|
2
𝑑
v
g
𝔹
n
-
λ
∫
Ω
|
u
|
2
𝑑
v
g
𝔹
n
(
∫
Ω
V
2
*
(
s
)
|
u
|
2
*
(
s
)
𝑑
v
g
𝔹
n
)
2
2
*
(
s
)
,
where H01(Ω) is the completion of Cc∞(Ω) with respect to the norm
∥
u
∥
=
∫
Ω
|
∇
u
|
2
𝑑
v
g
𝔹
n
.
Similarly to the Euclidean case, and once restricted to radial functions, the general Hardy–Sobolev inequality for the hyperbolic Hardy–Schrödinger operator is invariant under hyperbolic scaling described in (1.2), This invariance makes the corresponding variational problem non-compact and the problem of existence of minimizers quite interesting.
In Proposition 3.1, we start by showing that the extremals for the minimization problem (1.4) in the class of radial functions Hr1(𝔹n) can be written explicitly as
U
(
r
)
=
c
(
G
(
r
)
-
2
-
s
n
-
2
α
-
(
γ
)
+
G
(
r
)
-
2
-
s
n
-
2
α
+
(
γ
)
)
-
n
-
2
2
-
s
,
where c is a positive constant and α±(γ) satisfy
α
±
(
γ
)
=
1
2
±
1
4
-
γ
(
n
-
2
)
2
.
In other words, we show that
(1.5)
μ
γ
,
0
rad
(
𝔹
n
)
:=
inf
u
∈
H
r
1
(
𝔹
n
)
∖
{
0
}
∫
𝔹
n
|
∇
𝔹
n
u
|
2
𝑑
v
g
𝔹
n
-
γ
∫
𝔹
n
V
2
|
u
|
2
𝑑
v
g
𝔹
n
(
∫
𝔹
n
V
2
*
(
s
)
|
u
|
2
*
(
s
)
𝑑
v
g
𝔹
n
)
2
2
*
(
s
)
is attained by U.
Note that the radial function Gα(r) is a solution of -Δ𝔹nu-γV2u=0 on 𝔹n∖{0} if and only if α=α±(γ). These solutions have the asymptotic behavior
G
(
r
)
α
±
(
γ
)
∼
c
(
n
,
γ
)
r
-
β
±
(
γ
)
as
r
→
0
,
where
β
±
(
γ
)
=
n
-
2
2
±
(
n
-
2
)
2
4
-
γ
.
These then yield positive solutions to the equation
-
Δ
𝔹
n
u
-
γ
V
2
u
=
V
2
*
(
s
)
u
2
*
(
s
)
-
1
in
𝔹
n
.
We point out the paper [15] (also see [4, 5, 10]), where the authors considered the counterpart of the Brezis–Nirenberg problem on 𝔹n (n≥3), and discuss issues of existence and non-existence for the equation
-
Δ
𝔹
n
u
-
λ
u
=
u
2
*
-
1
in
𝔹
n
,
in the absence of a Hardy potential.
Next, we consider the attainability of μγ,λ(Ω) in subdomains of 𝔹n without necessarily any symmetry. In other words, we will search for positive solutions for the equation
(1.6)
{
-
Δ
𝔹
n
u
-
γ
V
2
u
-
λ
u
=
V
2
*
(
s
)
u
2
*
(
s
)
-
1
in
Ω
,
u
≥
0
in
Ω
,
u
=
0
on
∂
Ω
,
where Ω is a compact smooth subdomain of 𝔹n such that 0∈Ω, but Ω¯ does not touch the boundary of 𝔹n and λ∈ℝ. Note that the metric is then smooth on such Ω, and the only singularity we will be dealing with will be coming from the Hardy-type potential V2 and the Hardy–Sobolev weight V2*(s), which behaves like 1r2 (resp., 1rs) at the origin. This is analogous to the Euclidean problem on bounded domains considered by Ghoussoub and Robert [12, 11]. We shall therefore rely heavily on their work, at least in dimensions n≥5. Actually, once we perform a conformal transformation, the equation above reduces to the study of the following type of problems on bounded domains in ℝn:
{
-
Δ
v
-
(
γ
|
x
|
2
+
h
γ
,
λ
(
x
)
)
v
=
b
(
x
)
v
2
*
(
s
)
-
1
|
x
|
s
in
Ω
,
v
≥
0
in
Ω
,
v
=
0
on
∂
Ω
,
where b(x) is a positive C0(Ω¯) function with
(1.7)
b
(
0
)
=
(
n
-
2
)
n
-
s
n
-
2
2
2
-
s
,
(1.8)
h
γ
,
λ
(
x
)
=
γ
a
(
x
)
+
4
λ
-
n
(
n
-
2
)
(
1
-
|
x
|
2
)
2
,
(1.9)
a
(
x
)
=
a
(
r
)
=
{
4
r
+
8
+
g
3
(
r
)
when
n
=
3
,
8
log
1
r
-
4
+
g
4
(
r
)
when
n
=
4
,
4
(
n
-
2
)
n
-
4
+
r
g
n
(
r
)
when
n
≥
5
.
with gn(0)=0, for all n≥3. Ghoussoub and Robert [12] have recently tackled such an equation, but in the case where h(x) and b(x) are constants. We shall explore here to which extent their techniques could be extended to this setting. To start with, the following regularity result will then follow immediately.
Theorem 1 (Regularity).
Let Ω⋐Bn, n≥3, and γ<(n-2)24. If u≢0 is a non-negative weak solution of equation (1.6) in the hyperbolic Sobolev space H1(Ω), then
lim
|
x
|
→
0
u
(
x
)
G
(
|
x
|
)
α
-
=
K
>
0
.
We also need to define a notion of mass of a domain associated to the operator -Δ𝔹n-γV2-λ. We therefore show the following.
Theorem 2 (Hyperbolic Hardy-Singular Mass of Ω⋐Bn).
Let 0∈Ω⋐Bn, n≥3, and γ<(n-2)24. Let λ∈R be such that the operator -ΔBn-γV2-λ is coercive. Then there exists a solution KΩ∈C∞(Ω¯∖{0}) to the linear problem
{
-
Δ
𝔹
n
K
Ω
-
γ
V
2
K
Ω
-
λ
K
Ω
=
0
in
Ω
,
K
Ω
≥
0
in
Ω
,
K
Ω
=
0
on
∂
Ω
,
such that KΩ(x)≃|x|→0cG(|x|)α+ for some positive constant c. Furthermore:
If
K
Ω
′
∈
C
∞
(
Ω
¯
∖
{
0
}
)
is another solution of the above linear equation, then there exists a constant
C
>
0
such that
K
Ω
′
=
C
K
Ω
.
If
γ
>
max
{
n
(
n
-
4
)
4
,
0
}
, then there exists
m
γ
,
λ
H
(
Ω
)
∈
ℝ
such that
K
Ω
(
x
)
=
G
(
|
x
|
)
α
+
+
m
γ
,
λ
H
(
Ω
)
G
(
|
x
|
)
α
-
+
o
(
G
(
|
x
|
)
α
-
)
as
x
→
0
.
The constant
m
γ
,
λ
H
(
Ω
)
will be referred to as the hyperbolic mass of the domain Ω associated with the operator -Δ𝔹n-γV2-λ.
Just like the Euclidean case, solutions exist in high dimensions, while the positivity of the “hyperbolic mass” will be needed for low dimensions. More precisely, we have the following theorem.
Theorem 3.
Let Ω⋐Bn(n≥3) be a smooth domain with 0∈Ω, 0≤γ<(n-2)24 and let λ∈R be such that the operator -ΔBn-γV2-λ is coercive. Then the best constant μγ,λ(Ω) is attained under the following conditions:
n
≥
5
, γ≤n(n-4)4 and λ>n-2n-4(n(n-4)4-γ);
n
≥
3
, max{n(n-4)4,0}<γ<(n-2)24 and mγ,λH(Ω)>0.
As mentioned above, Theorem 3 will be proved by using a conformal transformation that reduces the problem to the Euclidean case, already considered by Ghoussoub and Robert [12]. Actually, this leads to the following variation of the problem they considered, where the perturbation can be singular but not as much as the Hardy potential.
Theorem 4.
Let Ω be a bounded smooth domain in Rn, n≥3, with 0∈Ω and 0≤γ<(n-2)24. Let h∈C1(Ω¯∖{0}) be such that
(1.10)
h
(
x
)
=
-
𝒞
1
|
x
|
-
θ
log
|
x
|
+
h
~
(
x
)
,
where
lim
x
→
0
|
x
|
θ
h
~
(
x
)
=
𝒞
2
for some
0
≤
θ
<
2
and
𝒞
1
,
𝒞
2
∈
ℝ
,
and the operator -Δ-(γ|x|2+h(x)) is coercive. Also, assume that b(x) is a non-negative function in C0(Ω¯) with b(0)>0. Then the best constant
μ
γ
,
h
(
Ω
)
:=
inf
u
∈
H
0
1
(
Ω
)
∖
{
0
}
∫
Ω
(
|
∇
u
|
2
-
(
γ
|
x
|
2
+
h
(
x
)
)
u
2
)
𝑑
x
(
∫
Ω
b
(
x
)
|
u
|
2
*
(
s
)
|
x
|
s
𝑑
x
)
2
2
*
(
s
)
is attained if one of the following two conditions is satisfied:
γ
≤
(
n
-
2
)
2
4
-
(
2
-
θ
)
2
4
and, either
𝒞
1
>
0
or
{
𝒞
1
=
0
, 𝒞2>0}.
(
n
-
2
)
2
4
-
(
2
-
θ
)
2
4
<
γ
<
(
n
-
2
)
2
4
and
m
γ
,
h
(
Ω
)
>
0
, where
m
γ
,
h
(
Ω
)
is the mass of the domain Ω associated to the operator -Δ-(γ|x|2+h(x)).
The paper is organized as follows. In Section 2, we introduce the Hardy–Sobolev-type inequalities in hyperbolic space. In Section 3, we find the explicit solutions for Hardy–Sobolev equations corresponding to (1.5) on 𝔹n. In Section 4, we show that our main equation (1.6) can be transformed into the Hardy–Sobolev-type equations in Euclidean space under a conformal transformation. Section 5 is then devoted to establish the existence results for (1.6) on compact submanifolds of 𝔹n by studying the transformed equations in Euclidean space.
4 The Corresponding Perturbed Hardy–Schrödinger Operator on Euclidean Space
We shall see in the next section that after a conformal transformation, equation (1.6) is transformed into the Euclidean equation
(4.1)
{
-
Δ
u
-
(
γ
|
x
|
2
+
h
(
x
)
)
u
=
b
(
x
)
u
2
*
(
s
)
-
1
|
x
|
s
in
Ω
,
u
>
0
in
Ω
,
u
=
0
on
∂
Ω
,
where Ω is a bounded domain in ℝn, n≥3, h∈C1(Ω¯∖{0}) with lim|x|→0|x|2h(x)=0 is such that the operator -Δ-(γ|x|2+h(x)) is coercive and b(x)∈C0(Ω¯) is non-negative with b(0)>0. Equation (4.1) is the Euler–Lagrange equation for the following energy functional on D1,2(Ω):
J
γ
,
h
Ω
(
u
)
:=
∫
Ω
(
|
∇
u
|
2
-
(
γ
|
x
|
2
+
h
(
x
)
)
u
2
)
𝑑
x
(
∫
Ω
b
(
x
)
|
u
|
2
*
(
s
)
|
x
|
s
𝑑
x
)
2
2
*
(
s
)
.
Here D1,2(Ω) – or H01(Ω) if the domain is bounded – is the completion of Cc∞(Ω) with respect to the norm given by
∥
u
∥
2
=
∫
Ω
|
∇
u
|
2
𝑑
x
.
We let
μ
γ
,
h
(
Ω
)
:=
inf
u
∈
D
1
,
2
(
Ω
)
∖
{
0
}
J
γ
,
h
Ω
(
u
)
.
A standard approach to find minimizers is to compare μγ,h(Ω) with μγ,0(ℝn). It is known that μγ,0(ℝn) is attained when γ≥0, and minimizers are explicit and take the form
U
ε
(
x
)
:=
c
γ
,
s
(
n
)
⋅
ε
-
n
-
2
2
U
(
x
ε
)
=
c
γ
,
s
(
n
)
⋅
(
ε
2
-
s
n
-
2
⋅
β
+
(
γ
)
-
β
-
(
γ
)
2
ε
2
-
s
n
-
2
⋅
(
β
+
(
γ
)
-
β
-
(
γ
)
)
|
x
|
(
2
-
s
)
β
-
(
γ
)
n
-
2
+
|
x
|
(
2
-
s
)
β
+
(
γ
)
n
-
2
)
n
-
2
2
-
s
for x∈ℝn∖{0}, where ε>0, cγ,s(n)>0, and β±(γ) are defined in (3.1), see [11]. In particular, there exists χ>0 such that
(4.2)
-
Δ
U
ε
-
γ
|
x
|
2
U
ε
=
χ
U
ε
2
*
(
s
)
-
1
|
x
|
s
in
ℝ
n
∖
{
0
}
.
We shall start by analyzing the singular solutions and then define the mass of a domain associated to the operator -Δ-(γ|x|2+h(x)).
Proposition 4.1.
Let Ω be a smooth bounded domain in Rn such that 0∈Ω and γ<(n-2)24. Let h∈C1(Ω¯∖{0}) be such that lim|x|→0|x|τh(x) exists and is finite, for some 0≤τ<2, and that the operator -Δ-γ|x|2-h(x) is coercive. Then the following hold:
There exists a solution
K
∈
C
∞
(
Ω
¯
∖
{
0
}
)
for the linear problem
{
-
Δ
K
-
(
γ
|
x
|
2
+
h
(
x
)
)
K
=
0
in
Ω
∖
{
0
}
,
K
>
0
in
Ω
∖
{
0
}
,
K
=
0
on
∂
Ω
,
such that for some
c
>
0
,
K
(
x
)
≃
x
→
0
c
|
x
|
β
+
(
γ
)
.
Moreover, if
K
′
∈
C
∞
(
Ω
¯
∖
{
0
}
)
is another solution for the above equation, then there exists a constant
λ
>
0
such that
K
′
=
λ
K
.
Let
θ
=
inf
{
θ
′
∈
[
0
,
2
)
:
lim
|
x
|
→
0
|
x
|
θ
′
h
(
x
)
exists and is finite
}
. If
γ
>
(
n
-
2
)
2
4
-
(
2
-
θ
)
2
4
, then there exist constants
c
1
,
c
2
∈
ℝ
with
c
1
>
0
such that
(4.3)
K
(
x
)
=
c
1
|
x
|
β
+
(
γ
)
+
c
2
|
x
|
β
-
(
γ
)
+
o
(
1
|
x
|
β
-
(
γ
)
)
as
x
→
0
.
The ratio
c
2
c
1
is independent of the choice of
K
.
We can therefore define the mass of Ω with respect to the operator -Δ-(γ|x|2+h(x)) as mγ,h(Ω):=c2c1.
The mass
m
γ
,
h
(
Ω
)
satisfies the following properties:
If
h
≤
h
′
and
h
≢
h
′
, then
m
γ
,
h
(
Ω
)
<
m
γ
,
h
′
(
Ω
)
.
If
Ω
′
⊂
Ω
, then
m
γ
,
h
(
Ω
′
)
<
m
γ
,
h
(
Ω
)
.
Proof.
The proof of (1) and (3) is similar to [12, Propositions 2 and 4] with only a minor change that accounts for the singularity of h. To illustrate the role of this extra singularity, we prove (2). For that, we let η∈Cc∞(Ω) be such that η(x)≡1 around 0. Our first objective is to write
K
(
x
)
:=
η
(
x
)
|
x
|
β
+
(
γ
)
+
f
(
x
)
for some
f
∈
H
0
1
(
Ω
)
.
Note that
γ
>
(
n
-
2
)
2
4
-
(
2
-
θ
)
2
4
⇔
β
+
-
β
-
<
2
-
θ
⇔
2
β
+
<
n
-
θ
.
Fix θ′ such that θ<θ′<min{2+θ2,2-(β+(γ)-β-(γ))}. Then lim|x|→0|x|θ′h(x) exists and is finite.
Consider the function
g
(
x
)
=
-
(
-
Δ
-
(
γ
|
x
|
2
+
h
(
x
)
)
)
(
η
|
x
|
-
β
+
(
γ
)
)
in
Ω
∖
{
0
}
.
Since η(x)≡1 around 0, we have
(4.4)
|
g
(
x
)
|
≤
|
h
(
x
)
|
x
|
β
+
(
γ
)
|
≤
C
|
x
|
-
(
β
+
(
γ
)
+
θ
′
)
as
x
→
0
.
Therefore g∈L2nn+2(Ω) if 2β+(γ)+2θ′<n+2, and this holds since by our assumption 2β+<n-θ and 2θ′<2+θ. Since L2nn+2(Ω)=L2nn-2(Ω)′⊂H01(Ω)′, there exists f∈H01(Ω) such that
-
Δ
f
-
(
γ
|
x
|
2
+
h
(
x
)
)
f
=
g
in
H
0
1
(
Ω
)
.
By regularity theory, we have f∈C2(Ω¯∖{0}). We now show that
(4.5)
|
x
|
β
-
(
γ
)
f
(
x
)
has a finite limit as
x
→
0
.
Define K(x)=η(x)|x|β+(γ)+f(x) for all x∈Ω¯∖{0}, and note that K∈C2(Ω¯∖{0}) and is a solution to
-
Δ
K
-
(
γ
|
x
|
2
+
h
(
x
)
)
K
=
0
.
Write g+(x):=max{g(x),0} and g-(x):=max{-g(x),0} so that g=g+-g-, and let f1,f2∈H01(Ω) be weak solutions to
(4.6)
-
Δ
f
1
-
(
γ
|
x
|
2
+
h
(
x
)
)
f
1
=
g
+
and
-
Δ
f
2
-
(
γ
|
x
|
2
+
h
(
x
)
)
f
2
=
g
-
in
H
0
1
(
Ω
)
.
In particular, uniqueness, coercivity and the maximum principle yield f=f1-f2 and f1,f2≥0. Assume that f1≢0 so that f1>0 in Ω∖{0}, fix α>β+(γ) and μ>0. Define u-(x):=|x|-β-(γ)+μ|x|-α for all x≠0. We then get that there exists a small δ>0 such that
(
-
Δ
-
(
γ
|
x
|
2
+
h
(
x
)
)
)
u
-
(
x
)
=
μ
(
-
Δ
-
γ
|
x
|
2
)
|
x
|
-
α
-
μ
h
(
x
)
|
x
|
-
α
-
h
(
x
)
|
x
|
-
β
-
(
γ
)
=
-
μ
(
α
-
β
+
(
γ
)
)
(
α
-
β
-
(
γ
)
)
-
|
x
|
2
h
(
x
)
(
|
x
|
α
-
β
-
(
γ
)
+
μ
)
|
x
|
α
+
2
<
0
for
x
∈
B
δ
(
0
)
∖
{
0
}
.
This implies that u-(x) is a sub-solution on Bδ(0)∖{0}. Let C>0 be such that f1≥Cu- on ∂Bδ(0). Since f1 and Cu-∈H01(Ω) are respectively super-solutions and sub-solutions to (-Δ-(γ|x|2+h(x)))u(x)=0, it follows from the comparison principle (via coercivity) that f1>Cu->C|x|-β-(γ) on Bδ(0)∖{0}. It then follows from (4.4) that
g
+
(
x
)
≤
|
g
(
x
)
|
≤
C
|
x
|
-
(
β
+
(
γ
)
+
θ
′
)
≤
C
1
|
x
|
(
2
-
θ
′
)
-
(
β
+
(
γ
)
-
β
-
(
γ
)
)
f
1
|
x
|
2
.
Then rewriting (4.6) as
-
Δ
f
1
-
(
γ
|
x
|
2
+
h
(
x
)
+
g
+
f
1
)
f
1
=
0
yields
-
Δ
f
1
-
(
γ
+
O
(
|
x
|
(
2
-
θ
′
)
-
(
β
+
(
γ
)
-
β
-
(
γ
)
)
)
|
x
|
2
)
f
1
=
0
.
With our choice of θ′ we can then conclude by the optimal regularity result in [12, Theorem 8] that |x|β-(γ)f1 has a finite limit as x→0. Similarly one also obtains that |x|β-(γ)f2 has a finite limit as x→0, and therefore (4.5) is verified.
It follows that there exists c2∈ℝ such that
K
(
x
)
=
1
|
x
|
β
+
(
γ
)
+
c
2
|
x
|
β
-
(
γ
)
+
o
(
1
|
x
|
β
-
(
γ
)
)
as
x
→
0
,
which proves the existence of a solution K to the problem with the relevant asymptotic behavior. The uniqueness result yields the conclusion. ∎
We now proceed with the proof of the existence results, following again [12]. We shall use the following standard sufficient condition for attainability.
Lemma 4.1.
Under the assumptions of Theorem 4, if
μ
γ
,
h
(
Ω
)
:=
inf
u
∈
H
0
1
(
Ω
)
∖
{
0
}
∫
Ω
(
|
∇
u
|
2
-
(
γ
|
x
|
2
+
h
(
x
)
)
u
2
)
𝑑
x
(
∫
Ω
b
(
x
)
|
u
|
2
*
(
s
)
|
x
|
s
𝑑
x
)
2
2
*
(
s
)
<
μ
γ
,
0
(
ℝ
n
)
b
(
0
)
2
2
*
(
s
)
,
then the infimum μγ,s(Ω) is achieved and equation (4.1) has a solution.
Proof of Theorem 4.
We will construct a minimizing sequence uε in H01(Ω)∖{0} for the functional Jγ,hΩ in such a way that μγ,h(Ω)<b(0)-22*(s)μγ,0(ℝn). As mentioned above, when γ≥0 the infimum μγ,0(ℝn) is achieved, up to a constant, by the function
U
(
x
)
:=
1
(
|
x
|
(
2
-
s
)
β
-
(
γ
)
n
-
2
+
|
x
|
(
2
-
s
)
β
+
(
γ
)
n
-
2
)
n
-
2
2
-
s
for
x
∈
ℝ
n
∖
{
0
}
.
In particular, there exists χ>0 such that
-
Δ
U
-
γ
|
x
|
2
U
=
χ
U
2
*
(
s
)
-
1
|
x
|
s
in
ℝ
n
∖
{
0
}
.
Define a scaled version of U by
(4.7)
U
ε
(
x
)
:=
ε
-
n
-
2
2
U
(
x
ε
)
=
(
ε
2
-
s
n
-
2
⋅
β
+
(
γ
)
-
β
-
(
γ
)
2
ε
2
-
s
n
-
2
⋅
(
β
+
(
γ
)
-
β
-
(
γ
)
)
|
x
|
(
2
-
s
)
β
-
(
γ
)
n
-
2
+
|
x
|
(
2
-
s
)
β
+
(
γ
)
n
-
2
)
n
-
2
2
-
s
for
x
∈
ℝ
n
∖
{
0
}
,
where β±(γ) are defined in (3.1). In the sequel, we write β+:=β+(γ) and β-:=β-(γ). Consider a cut-off function η∈Cc∞(Ω) such that η(x)≡1 in a neighborhood of 0 contained in Ω.
Case 1: Test-functions for the case when γ≤(n-2)24-(2-θ)24. For ε>0, we consider the test functions uε∈D1,2(Ω) defined by
u
ε
(
x
)
:=
η
(
x
)
U
ε
(
x
)
for x∈Ω¯∖{0}. To estimate Jγ,hΩ(uε), we use the bounds on Uε to obtain
∫
Ω
b
(
x
)
u
ε
2
*
(
s
)
|
x
|
s
𝑑
x
=
∫
B
δ
(
0
)
b
(
x
)
U
ε
2
*
(
s
)
|
x
|
s
𝑑
x
+
∫
Ω
∖
B
δ
(
0
)
b
(
x
)
u
ε
2
*
(
s
)
|
x
|
s
𝑑
x
=
∫
B
ε
-
1
δ
(
0
)
b
(
ε
x
)
U
2
*
(
s
)
|
x
|
s
𝑑
x
+
∫
B
ε
-
1
δ
(
0
)
b
(
ε
x
)
η
(
ε
x
)
2
*
(
s
)
U
2
*
(
s
)
|
x
|
s
𝑑
x
=
b
(
0
)
∫
ℝ
n
U
2
*
(
s
)
|
x
|
s
𝑑
x
+
O
(
ε
2
*
(
s
)
2
(
β
+
-
β
-
)
)
.
Similarly, one also has
∫
Ω
(
|
∇
u
ε
|
2
-
γ
|
x
|
2
u
ε
2
)
𝑑
x
=
∫
B
δ
(
0
)
(
|
∇
U
ε
|
2
-
γ
|
x
|
2
U
ε
2
)
𝑑
x
+
∫
Ω
∖
B
δ
(
0
)
(
|
∇
u
ε
|
2
-
γ
|
x
|
2
u
ε
2
)
𝑑
x
=
∫
B
ε
-
1
δ
(
0
)
(
|
∇
U
|
2
-
γ
|
x
|
2
U
2
)
𝑑
x
+
O
(
ε
β
+
-
β
-
)
=
∫
ℝ
n
(
|
∇
U
|
2
-
γ
|
x
|
2
U
2
)
𝑑
x
+
O
(
ε
β
+
-
β
-
)
=
χ
∫
ℝ
n
U
2
*
(
s
)
|
x
|
s
𝑑
x
+
O
(
ε
β
+
-
β
-
)
.
Estimating the lower-order terms as ε→0 gives
∫
Ω
h
~
(
x
)
u
ε
2
𝑑
x
=
{
ε
2
-
θ
[
𝒞
2
∫
ℝ
n
U
2
|
x
|
θ
𝑑
x
+
o
(
1
)
]
if
β
+
-
β
-
>
2
-
θ
,
ε
2
-
θ
log
(
1
ε
)
[
𝒞
2
ω
n
-
1
+
o
(
1
)
]
if
β
+
-
β
-
=
2
-
θ
,
O
(
ε
β
+
-
β
-
)
if
β
+
-
β
-
<
2
-
θ
,
and
-
𝒞
1
∫
Ω
log
|
x
|
|
x
|
θ
u
ε
2
𝑑
x
=
{
𝒞
1
ε
2
-
θ
log
(
1
ε
)
[
∫
ℝ
n
U
2
|
x
|
θ
𝑑
x
+
o
(
1
)
]
if
β
+
-
β
-
>
2
-
θ
,
𝒞
1
ε
2
-
θ
(
log
(
1
ε
)
)
2
[
ω
n
-
1
2
+
o
(
1
)
]
if
β
+
-
β
-
=
2
-
θ
,
O
(
ε
β
+
-
β
-
)
if
β
+
-
β
-
<
2
-
θ
.
Note that β+-β-≥2-θ if and only if γ≤(n-2)24-(2-θ)24. Therefore,
∫
Ω
h
(
x
)
u
ε
2
𝑑
x
=
{
ε
2
-
θ
∫
ℝ
n
U
2
|
x
|
θ
𝑑
x
[
𝒞
1
log
(
1
ε
)
(
1
+
o
(
1
)
)
+
𝒞
2
+
o
(
1
)
]
if
γ
<
(
n
-
2
)
2
4
-
(
2
-
θ
)
2
4
,
ε
2
-
θ
log
(
1
ε
)
ω
n
-
1
2
[
𝒞
1
log
(
1
ε
)
(
1
+
o
(
1
)
)
+
2
𝒞
2
+
o
(
1
)
]
if
γ
=
(
n
-
2
)
2
4
-
(
2
-
θ
)
2
4
.
Combining the above estimates, we obtain as ε→0,
J
γ
,
h
Ω
(
u
ε
)
=
∫
Ω
(
|
∇
u
ε
|
2
-
γ
u
ε
2
|
x
|
2
-
h
(
x
)
u
ε
2
)
𝑑
x
(
∫
Ω
b
(
x
)
|
u
ε
|
2
*
(
s
)
|
x
|
s
𝑑
x
)
2
2
*
(
s
)
=
μ
γ
,
0
(
ℝ
n
)
b
(
0
)
2
2
*
(
s
)
-
{
∫
ℝ
n
U
2
|
x
|
θ
𝑑
x
(
b
(
0
)
∫
ℝ
n
U
2
*
(
s
)
|
x
|
s
𝑑
x
)
2
2
*
(
s
)
ε
2
-
θ
[
𝒞
1
log
(
1
ε
)
(
1
+
o
(
1
)
)
+
𝒞
2
+
o
(
1
)
]
,
γ
<
(
n
-
2
)
2
4
-
(
2
-
θ
)
2
4
,
ω
n
-
1
2
(
b
(
0
)
∫
ℝ
n
U
2
*
(
s
)
|
x
|
s
𝑑
x
)
2
2
*
(
s
)
ε
2
-
θ
log
(
1
ε
)
[
𝒞
1
log
(
1
ε
)
(
1
+
o
(
1
)
)
+
2
𝒞
2
+
o
(
1
)
]
,
γ
=
(
n
-
2
)
2
4
-
(
2
-
θ
)
2
4
,
as long as β+-β-≥2-θ. Thus, for ε sufficiently small, the assumption that either 𝒞1>0 or 𝒞1=0, 𝒞2>0 guarantees that
μ
γ
,
h
(
Ω
)
≤
J
γ
,
h
Ω
(
u
ε
)
<
μ
γ
,
0
(
ℝ
n
)
b
(
0
)
2
2
*
(
s
)
.
It then follows from Lemma 4.1 that μγ,h(Ω) is attained.
Case 2: Test-functions for the case when (n-2)24-(2-θ)24<γ<(n-2)24. Here h(x) and θ given by (1.10) satisfy the hypothesis of Proposition 4.1. Since γ>(n-2)24-(2-θ)24, it follows from (4.3) that there exists β∈D1,2(Ω) such that
β
(
x
)
≃
x
→
0
m
γ
,
h
(
Ω
)
|
x
|
β
-
.
The function K(x):=η(x)|x|β++β(x) for x∈Ω∖{0} satisfies the equation:
{
-
Δ
K
-
(
γ
|
x
|
2
+
h
(
x
)
)
K
=
0
in
Ω
∖
{
0
}
,
K
>
0
in
Ω
∖
{
0
}
,
K
=
0
on
∂
Ω
.
Define the test functions
u
ε
(
x
)
:=
η
(
x
)
U
ε
+
ε
β
+
-
β
-
2
β
(
x
)
for
x
∈
Ω
¯
∖
{
0
}
The functions uε∈D1,2(Ω) for all ε>0. We estimate Jγ,hΩ(uε).
Step 1: Estimates for ∫Ω(|∇uε|2-(γ|x|2+h(x))uε2)𝑑x. Take δ>0 small enough such that η(x)=1 in Bδ(0)⊂Ω. We decompose the integral as
∫
Ω
(
|
∇
u
ε
|
2
-
(
γ
|
x
|
2
+
h
(
x
)
)
u
ε
2
)
𝑑
x
=
∫
B
δ
(
0
)
(
|
∇
u
ε
|
2
-
(
γ
|
x
|
2
+
h
(
x
)
)
u
ε
2
)
𝑑
x
+
∫
Ω
∖
B
δ
(
0
)
(
|
∇
u
ε
|
2
-
(
γ
|
x
|
2
+
h
(
x
)
)
u
ε
2
)
𝑑
x
.
By standard elliptic estimates, it follows that
lim
ε
→
0
u
ε
ε
β
+
-
β
-
2
=
K
in
C
loc
2
(
Ω
¯
∖
{
0
}
)
.
Hence
lim
ε
→
0
∫
Ω
∖
B
δ
(
0
)
(
|
∇
u
ε
|
2
-
(
γ
|
x
|
2
+
h
(
x
)
)
u
ε
2
)
𝑑
x
ε
β
+
-
β
-
=
∫
Ω
∖
B
δ
(
0
)
(
|
∇
K
|
2
-
(
γ
|
x
|
2
+
h
(
x
)
)
K
2
)
𝑑
x
=
∫
Ω
∖
B
δ
(
0
)
(
-
Δ
K
-
(
γ
|
x
|
2
+
h
(
x
)
)
K
)
K
𝑑
x
+
∫
∂
(
Ω
∖
B
δ
(
0
)
)
K
∂
ν
K
d
σ
=
∫
∂
(
Ω
∖
B
δ
(
0
)
)
K
∂
ν
K
d
σ
=
-
∫
∂
B
δ
(
0
)
K
∂
ν
K
d
σ
.
Since β++β-=n-2, using elliptic estimates, and the definition of K gives us
K
∂
ν
K
=
-
β
+
|
x
|
1
+
2
β
+
-
(
n
-
2
)
m
γ
,
h
(
Ω
)
|
x
|
n
-
1
+
o
(
1
|
x
|
n
-
1
)
as
x
→
0
.
Therefore,
∫
Ω
∖
B
δ
(
0
)
(
|
∇
u
ε
|
2
-
(
γ
|
x
|
2
+
h
(
x
)
)
u
ε
2
)
𝑑
x
=
ε
β
+
-
β
-
ω
n
-
1
(
β
+
δ
β
+
-
β
-
+
(
n
-
2
)
m
γ
,
h
(
Ω
)
+
o
δ
(
1
)
)
.
Now, we estimate the term
∫
B
δ
(
0
)
(
|
∇
u
ε
|
2
-
(
γ
|
x
|
2
+
h
(
x
)
)
u
ε
2
)
𝑑
x
.
First,
u
ε
(
x
)
=
U
ε
(
x
)
+
ε
β
+
-
β
-
2
β
(
x
)
for
x
∈
B
δ
(
0
)
,
therefore after integration by parts, we obtain
∫
B
δ
(
0
)
(
|
∇
u
ε
|
2
-
(
γ
|
x
|
2
+
h
(
x
)
)
u
ε
2
)
𝑑
x
=
∫
B
δ
(
0
)
(
|
∇
U
ε
|
2
-
(
γ
|
x
|
2
+
h
(
x
)
)
U
ε
2
)
𝑑
x
+
2
ε
β
+
-
β
-
2
∫
B
δ
(
0
)
(
∇
U
ε
⋅
∇
β
-
(
γ
|
x
|
2
+
h
(
x
)
)
U
ε
β
)
𝑑
x
+
ε
β
+
-
β
-
∫
B
δ
(
0
)
(
|
∇
β
|
2
-
(
γ
|
x
|
2
+
h
(
x
)
)
β
2
)
𝑑
x
=
∫
B
δ
(
0
)
(
-
Δ
U
ε
-
γ
|
x
|
2
U
ε
)
U
ε
𝑑
x
+
∫
∂
B
δ
(
0
)
U
ε
∂
ν
U
ε
d
σ
-
∫
B
δ
(
0
)
h
(
x
)
U
ε
2
𝑑
x
+
2
ε
β
+
-
β
-
2
∫
B
δ
(
0
)
(
-
Δ
U
ε
d
x
-
γ
|
x
|
2
U
ε
)
β
𝑑
x
-
2
ε
β
+
-
β
-
2
∫
B
δ
(
0
)
h
(
x
)
U
ε
β
𝑑
x
+
2
ε
β
+
-
β
-
2
∫
∂
B
δ
(
0
)
β
∂
ν
U
ε
d
σ
+
ε
β
+
-
β
-
∫
B
δ
(
0
)
(
|
∇
β
|
2
-
(
γ
|
x
|
2
+
h
(
x
)
)
β
2
)
𝑑
x
.
We now estimate each of the above terms. First, using equation (4.2) and the expression for Uε defined as in (4.7), we obtain
∫
B
δ
(
0
)
(
-
Δ
U
ε
-
γ
|
x
|
2
U
ε
)
U
ε
𝑑
x
=
χ
∫
B
δ
(
0
)
U
ε
2
*
(
s
)
|
x
|
s
𝑑
x
=
χ
∫
ℝ
n
U
2
*
(
s
)
|
x
|
s
𝑑
x
+
O
(
ε
2
*
(
s
)
2
(
β
+
-
β
-
)
)
and
∫
∂
B
δ
(
0
)
U
ε
∂
ν
U
ε
d
σ
=
-
β
+
ω
n
-
1
ε
β
+
-
β
-
δ
β
+
-
β
-
+
o
δ
(
ε
β
+
-
β
-
)
as
ε
→
0
.
Note that
β
+
-
β
-
<
2
-
θ
⇔
γ
>
(
n
-
2
)
2
4
-
(
2
-
θ
)
2
4
⟹
2
β
+
+
θ
<
n
.
Therefore,
∫
B
δ
(
0
)
h
(
x
)
U
ε
2
𝑑
x
=
O
(
ε
β
+
-
β
-
∫
B
δ
(
0
)
1
|
x
|
2
β
+
+
θ
𝑑
x
)
=
o
δ
(
ε
β
+
-
β
-
)
as
ε
→
0
.
Again from equation (4.2) and the expression for U and β, we get
∫
B
δ
(
0
)
(
-
Δ
U
ε
d
x
-
γ
|
x
|
2
U
ε
)
β
𝑑
x
=
ε
β
+
+
β
-
2
∫
B
ε
-
1
δ
(
0
)
(
-
Δ
U
d
x
-
γ
|
x
|
2
U
)
β
(
ε
x
)
𝑑
x
=
m
γ
,
h
(
Ω
)
ε
β
+
-
β
-
2
∫
B
ε
-
1
δ
(
0
)
(
-
Δ
U
d
x
-
γ
|
x
|
2
U
)
|
x
|
-
β
-
𝑑
x
+
o
δ
(
ε
β
+
-
β
-
2
)
=
m
γ
,
h
(
Ω
)
ε
β
+
-
β
-
2
∫
B
ε
-
1
δ
(
0
)
(
-
Δ
|
x
|
-
β
-
d
x
-
γ
|
x
|
2
|
x
|
-
β
-
)
U
𝑑
x
-
m
γ
,
h
(
Ω
)
ε
β
+
-
β
-
2
∫
∂
B
ε
-
1
δ
(
0
)
∂
ν
U
|
x
|
β
-
𝑑
σ
+
o
δ
(
ε
β
+
-
β
-
2
)
=
β
+
m
γ
,
h
(
Ω
)
ω
n
-
1
ε
β
+
-
β
-
2
+
o
δ
(
ε
β
+
-
β
-
2
)
.
Similarly,
∫
∂
B
δ
(
0
)
β
∂
ν
U
ε
d
σ
=
-
β
+
m
γ
,
h
(
Ω
)
ω
n
-
1
ε
β
+
-
β
-
2
+
o
δ
(
ε
β
+
-
β
-
2
)
.
Since β++β-+θ=n-(2-θ)<n, we have
∫
B
δ
(
0
)
h
(
x
)
U
ε
β
𝑑
x
=
O
(
ε
β
+
-
β
-
2
∫
B
δ
(
0
)
1
|
x
|
β
+
+
β
-
+
θ
𝑑
x
)
=
o
δ
(
ε
β
+
-
β
-
2
)
.
Finally,
ε
β
+
-
β
-
∫
B
δ
(
0
)
(
|
∇
β
|
2
-
(
γ
|
x
|
2
+
h
(
x
)
)
β
2
)
𝑑
x
=
o
δ
(
ε
β
+
-
β
-
)
.
Combining all the estimates, we get
∫
B
δ
(
0
)
(
|
∇
u
ε
|
2
-
(
γ
|
x
|
2
+
h
(
x
)
)
u
ε
2
)
𝑑
x
=
χ
∫
ℝ
n
U
2
*
(
s
)
|
x
|
s
𝑑
x
-
β
+
ω
n
-
1
ε
β
+
-
β
-
δ
β
+
-
β
-
+
o
δ
(
ε
β
+
-
β
-
)
.
So,
∫
Ω
(
|
∇
u
ε
|
2
-
(
γ
|
x
|
2
+
h
(
x
)
)
u
ε
2
)
𝑑
x
=
χ
∫
ℝ
n
U
2
*
(
s
)
|
x
|
s
𝑑
x
+
ω
n
-
1
(
n
-
2
)
m
γ
,
h
(
Ω
)
ε
β
+
-
β
-
+
o
δ
(
ε
β
+
-
β
-
)
.
Step 2: Estimating ∫Ωb(x)uε2*(s)|x|s𝑑x. One has for δ>0 small,
∫
Ω
b
(
x
)
u
ε
2
*
(
s
)
|
x
|
s
𝑑
x
=
∫
B
δ
(
0
)
b
(
x
)
u
ε
2
*
(
s
)
|
x
|
s
𝑑
x
+
∫
Ω
∖
B
δ
(
0
)
b
(
x
)
u
ε
2
*
(
s
)
|
x
|
s
𝑑
x
=
∫
B
δ
(
0
)
b
(
x
)
(
U
ε
(
x
)
+
ε
β
+
-
β
-
2
β
(
x
)
)
2
*
(
s
)
|
x
|
s
𝑑
x
+
o
(
ε
β
+
-
β
-
)
=
∫
B
δ
(
0
)
b
(
x
)
U
ε
2
*
(
s
)
|
x
|
s
𝑑
x
+
ε
β
+
-
β
-
2
2
*
(
s
)
∫
B
δ
(
0
)
b
(
x
)
U
ε
2
*
(
s
)
-
1
|
x
|
s
β
𝑑
x
+
o
(
ε
β
+
-
β
-
)
=
∫
B
δ
(
0
)
b
(
x
)
U
ε
2
*
(
s
)
|
x
|
s
𝑑
x
+
ε
β
+
-
β
-
2
2
*
(
s
)
χ
∫
B
δ
(
0
)
b
(
x
)
(
-
Δ
U
ε
d
x
-
γ
|
x
|
2
U
ε
)
β
𝑑
x
+
o
(
ε
β
+
-
β
-
)
=
b
(
0
)
∫
ℝ
n
U
2
*
(
s
)
|
x
|
s
𝑑
x
+
2
*
(
s
)
χ
b
(
0
)
β
+
m
γ
,
λ
,
a
(
Ω
)
ω
n
-
1
ε
β
+
-
β
-
+
o
(
ε
β
+
-
β
-
)
.
So, we obtain
J
γ
,
λ
,
a
Ω
(
u
ε
)
=
∫
Ω
(
|
∇
u
ε
|
2
-
γ
u
ε
2
|
x
|
2
-
h
(
x
)
u
ε
2
)
𝑑
x
(
∫
Ω
b
(
x
)
|
u
ε
|
2
*
(
s
)
|
x
|
s
𝑑
x
)
2
2
*
(
s
)
=
μ
γ
,
0
(
ℝ
n
)
b
(
0
)
2
2
*
(
s
)
-
m
γ
,
h
(
Ω
)
ω
n
-
1
(
β
+
-
β
-
)
(
b
(
0
)
∫
ℝ
n
U
2
*
(
s
)
|
x
|
s
𝑑
x
)
2
2
*
(
s
)
ε
β
+
-
β
-
+
o
(
ε
β
+
-
β
-
)
.
Therefore, if mγ,h(Ω)>0, we get for ε sufficiently small,
μ
γ
,
h
(
Ω
)
≤
J
γ
,
h
Ω
(
u
ε
)
<
μ
γ
,
0
(
ℝ
n
)
b
(
0
)
2
2
*
(
s
)
.
Then, from Lemma 4.1 it follows that μγ,h(Ω) is attained. ∎
Remark 4.1.
Assume for simplicity that h(x)=λ|x|-θ, where 0≤θ<2. There is a threshold λ*(Ω)≥0 beyond which the infimum μγ,λ(Ω) is achieved, and below which, it is not. In fact,
λ
*
(
Ω
)
:=
sup
{
λ
:
μ
γ
,
λ
(
Ω
)
=
μ
γ
,
0
(
ℝ
n
)
}
.
Performing a blow-up analysis like in [12], one can obtain the following sharp results:
In high dimensions, that is, for γ≤(n-2)24-(2-θ)24, one has λ*(Ω)=0 and the infimum μγ,λ(Ω) is achieved if and only if λ>λ*(Ω).
In low dimensions, that is, for (n-2)24-(2-θ)24<γ, one has λ*(Ω)>0 and μγ,λ(Ω) is not achieved for λ<λ*(Ω) and μγ,λ(Ω) is achieved for λ>λ*(Ω). Moreover, under the assumption μγ,λ*(Ω) is not achieved, we have mγ,λ*(Ω)=0, and λ*(Ω)=sup{λ:mγ,λ(Ω)≤0}.
5 Existence Results for Compact Submanifolds of 𝔹n
Consider the following Dirichlet boundary value problem in hyperbolic space. Let Ω⋐𝔹n (n≥3) be a bounded smooth domain such that 0∈Ω. We consider the Dirichlet boundary value problem:
(5.1)
{
-
Δ
𝔹
n
u
-
γ
V
2
u
-
λ
u
=
V
2
*
(
s
)
u
2
*
(
s
)
-
1
in
Ω
,
u
≥
0
in
Ω
,
u
=
0
on
∂
Ω
,
where λ∈ℝ, 0<s<2 and γ<γH:=(n-2)24.
We shall use the conformal transformation g𝔹n=φ4n-2gEucl, where φ=(21-r2)n-22, to map the problem into ℝn. We start by considering the general equation
(5.2)
-
Δ
𝔹
n
u
-
γ
V
2
u
-
λ
u
=
F
(
x
,
u
)
in
Ω
⋐
𝔹
n
,
where F(x,u) is a Carathéodory function such that
|
F
(
x
,
u
)
|
≤
C
|
u
|
(
1
+
|
u
|
2
*
(
s
)
-
2
r
s
)
for all
x
∈
Ω
.
If u satisfies (5.2), then v:=φu satisfies the equation
-
Δ
v
-
γ
(
2
1
-
r
2
)
2
V
2
v
-
[
λ
-
n
(
n
-
2
)
4
]
(
2
1
-
r
2
)
2
v
=
φ
n
+
2
n
-
2
f
(
x
,
v
φ
)
in
Ω
.
On the other hand, we have the following expansion for (21-r2)2V2:
(
2
1
-
r
2
)
2
V
2
(
x
)
=
1
(
n
-
2
)
2
(
f
(
r
)
G
(
r
)
)
2
where f(r) and G(r) are given by (1.1). We then obtain that
(
2
1
-
r
2
)
2
V
2
(
x
)
=
{
1
r
2
+
4
r
+
8
+
g
3
(
r
)
when
n
=
3
,
1
r
2
+
8
log
1
r
-
4
+
g
4
(
r
)
when
n
=
4
,
1
r
2
+
4
(
n
-
2
)
n
-
4
+
r
g
n
(
r
)
when
n
≥
5
,
where for all n≥3, gn(0)=0 and gn is C0([0,δ]) for δ<1.
This implies that v:=φu is a solution to
-
Δ
v
-
γ
r
2
v
-
[
γ
a
(
x
)
+
(
λ
-
n
(
n
-
2
)
4
)
(
2
1
-
r
2
)
2
]
v
=
φ
n
+
2
n
-
2
f
(
x
,
v
φ
)
,
where a(x) is defined in (1.9). We can therefore state the following lemma:
Lemma 5.1.
A non-negative function u∈H01(Ω) solves (5.1) if and only if v:=φu∈H01(Ω) satisfies
(5.3)
{
-
Δ
v
-
(
γ
|
x
|
2
+
h
γ
,
λ
(
x
)
)
v
=
b
(
x
)
v
2
*
(
s
)
-
1
|
x
|
s
in
Ω
,
v
≥
0
in
Ω
,
v
=
0
on
∂
Ω
,
where
h
γ
,
λ
(
x
)
=
γ
a
(
x
)
+
4
λ
-
n
(
n
-
2
)
(
1
-
|
x
|
2
)
2
,
a
(
x
)
is defined in (1.9), and b(x) is a positive function in C0(Ω¯) with
b
(
0
)
=
(
n
-
2
)
n
-
s
n
-
2
2
2
-
s
.
Moreover, the hyperbolic operator LγBn:=-ΔBn-γV2-λ is coercive if and only if the corresponding Euclidean operator Lγ,hRn:=-Δ-(γ|x|2+hγ,λ(x)) is coercive.
Proof.
Note that one has in particular
(5.4)
h
γ
,
λ
(
x
)
=
h
γ
,
λ
(
r
)
=
{
[
4
γ
r
+
8
γ
+
4
λ
-
3
]
+
γ
g
3
(
r
)
+
(
4
λ
-
3
)
r
2
(
2
-
r
2
)
(
1
-
r
2
)
2
when
n
=
3
,
[
8
γ
log
1
r
-
4
γ
+
4
λ
-
8
]
+
γ
g
4
(
r
)
+
(
4
λ
-
8
)
r
2
(
2
-
r
2
)
(
1
-
r
2
)
2
when
n
=
4
,
4
(
n
-
2
)
n
-
4
[
n
-
4
n
-
2
λ
+
γ
-
n
(
n
-
4
)
4
]
+
γ
r
g
n
(
r
)
+
(
4
λ
-
n
(
n
-
2
)
)
r
2
(
2
-
r
2
)
(
1
-
r
2
)
2
when
n
≥
5
,
with gn(0)=0 and gn is C0([0,δ]) for δ<1, for all n≥3. Let F(x,u)=V2*(s)u2*(s)-1 in (5.2). The above remarks show that v:=φu is a solution to (5.3).
For the second part, we first note that the following identities hold:
∫
Ω
(
|
∇
𝔹
n
u
|
2
-
n
(
n
-
2
)
4
u
2
)
𝑑
v
g
𝔹
n
=
∫
Ω
|
∇
v
|
2
𝑑
x
and
∫
Ω
u
2
𝑑
v
g
𝔹
n
=
∫
Ω
v
2
(
2
1
-
r
2
)
2
𝑑
x
.
If the operator Lγ𝔹n is coercive, then for any u∈C∞(Ω), we have
〈
L
γ
𝔹
n
u
,
u
〉
≥
C
∥
u
∥
H
0
1
(
Ω
)
2
,
which means
∫
Ω
(
|
∇
𝔹
n
u
|
2
-
γ
V
2
u
2
)
𝑑
v
g
𝔹
n
≥
C
∫
Ω
(
|
∇
𝔹
n
u
|
2
+
u
2
)
𝑑
v
g
𝔹
n
.
The latter then holds if and only if
〈
L
γ
,
φ
ℝ
n
u
,
u
〉
=
∫
Ω
(
|
∇
v
|
2
-
(
2
1
-
r
2
)
2
(
γ
V
2
-
n
(
n
-
2
)
4
)
v
2
)
𝑑
x
≥
C
∫
Ω
(
|
∇
v
|
2
+
(
2
1
-
r
2
)
2
(
n
(
n
-
2
)
4
+
1
)
v
2
)
𝑑
x
≥
C
′
∫
Ω
(
|
∇
v
|
2
+
v
2
)
𝑑
x
≥
c
∥
u
∥
H
0
1
(
Ω
)
2
,
where v=φu is in C∞(Ω). This completes the proof. ∎
At this point, the proofs of Theorems 1 and 2 follow verbatim as in the Euclidean case.
One can then use the results obtained in the last section to prove Theorem 3 stated in the introduction for the hyperbolic space. Indeed, it suffices to consider equation (5.3), where b is a positive function in C1(Ω¯) satisfying (1.7) and hγ,λ is given by (5.4).
If n≥5, then
lim
|
x
|
→
0
h
γ
,
λ
(
x
)
=
4
(
n
-
2
)
n
-
4
[
n
-
4
n
-
2
λ
+
γ
-
n
(
n
-
4
)
4
]
,
which is positive provided
λ
>
n
-
2
n
-
4
(
n
(
n
-
4
)
4
-
γ
)
.
Moreover, since in this case θ=0, the first alternative in Theorem 4 holds when γ≤(n-2)24-1=n(n-4)4. For (n-2)24-1<γ<(n-2)24, the existence of the extremal is guaranteed by the positivity of the hyperbolic mass mγ,λH(Ω) associated to the operator Lγ𝔹n, which is a positive multiple of the mass of the corresponding Euclidean operator.
When n=4, we can use the first option in Theorem 4 using the logarithmic perturbation if
lim
|
x
|
→
0
(
log
1
|
x
|
)
-
1
h
γ
,
λ
(
x
)
=
8
γ
>
0
and, since θ=0,
γ
≤
(
4
-
2
)
2
4
-
1
=
0
.
This is impossible. In the absence of the dominating term with log1|x|, i.e., when γ=0, we get existence of the extremal if λ>4(4-2)4=2. Otherwise, we require the positivity of the hyperbolic mass mγ,λH.
Similarly, if n=3, the threshold for γ with the singular perturbation 1|x| (i.e., θ=1) is γ≤(3-2)24-14=0. In order to use the first option in Theorem 4, we have to resort to the next term 4λ-3, which is positive when λ>34, in the case γ=0. When γ>0 or λ≤34, one needs that mγ,λH>0.
,
Saikat Mazumdar
