Journal of Functional Analysis 281 (2021) 109092
Contents lists available at ScienceDirect
Journal of Functional Analysis
www.elsevier.com/locate/jfa
Uniqueness of ground state and minimal-mass
blow-up solutions for focusing NLS with Hardy
potential
Debangana Mukherjee a
, Phan Thành Nam b,c,∗
,
Phuoc-Tai Nguyen a
a
Department of Mathematics and Statistics, Masaryk University, Brno, Czech
Republic
b
Department of Mathematics, LMU Munich, Theresienstrasse 39, D-80333
Munich, Germany
c
Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4,
D-80799 Munich, Germany
a r t i c l e i n f o a b s t r a c t
Article history:
Received 1 January 2021
Accepted 17 April 2021
Available online 4 May 2021
Communicated by Benjamin Schlein
MSC:
35Q55
35B44
35A02
35A15
35Q40
Keywords:
Nonlinear Schrödinger equation
Inverse square potential
Hardy-Gagliardo-Nirenberg
inequality
Ground state solutions
We consider the focusing nonlinear Schrödinger equation with
the critical inverse square potential. We give the first proof of
the uniqueness of the ground state solution. Consequently,
we obtain a sharp Hardy-Gagliardo-Nirenberg interpolation
inequality. Moreover, we provide a complete characterization
for the minimal mass blow-up solutions to the time dependent
problem.
© 2021 Elsevier Inc. All rights reserved.
* Corresponding author.
E-mail addresses: mukherjeed@math.muni.cz (D. Mukherjee), nam@math.lmu.de (P.T. Nam),
ptnguyen@math.muni.cz (P.-T. Nguyen).
https://doi.org/10.1016/j.jfa.2021.109092
0022-1236/© 2021 Elsevier Inc. All rights reserved.
2 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2. Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1. Ground state problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2. Dynamical problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3. Hardy-Gagliardo-Nirenberg inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1. Existence of ground states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2. A-priori estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3. Uniqueness of ground state solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4. Basic properties of the NLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1. Local well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2. Global existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3. Finite time blowup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5. Minimal mass blowup solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.1. Compactness of minimizing sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2. Blowup profile at t → T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.3. Virial identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6. Extension to the case c|x|−2
with c < c∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1. Introduction
In this paper, we consider the Cauchy problem for the focusing nonlinear Schrödinger
(NLS) equation with the critical inverse square potential
i∂tu(t, x) = (−Δ − c∗|x|−2
)u(t, x) − |u(t, x)|p−2
u(t, x), x ∈ Rd
, t > 0,
u(0, x) = u0(x), x ∈ Rd
.
(1.1)
Here
d 3, 2 < p < 2∗
=
2d
d − 2
and c∗ =
(d − 2)2
4
.
Since c∗ is the best constant in Hardy’s inequality
Rd
|∇φ(x)|2
dx c∗
Rd
|φ(x)|2
|x|2
dx, ∀φ ∈ H1
(Rd
),
the term −c∗|x|−2
is thus called the Hardy potential.
The classical focusing NLS equation (without Hardy potential) is a huge subject; see
e.g. [6,35] for excellent textbooks. In the last decades, a considerable amount of work
has been devoted to the study of the NLS equation with an inverse square potential.
Since the singularity of |x|−2
is critical to the Laplacian (as we can see from Hardy’s
inequality), its effect cannot be simply understood by perturbation methods. Therefore,
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 3
several well-known results for the classical NLS are not standardly extended to this
case.
The scale-covariance operator Hc = Δ − c|x|−2
plays an important role in quantum
mechanics [14]. The heat equation associated with Hc has been first studied by
Vazquez-Zuazua [37]. In the subcritical case c < c∗, the Strichartz’s estimates for Hc
have been established by Burq-Planchon-Stalker-Tahvildar-Zadeh [5], using RodnianskiSchlag’s
approach [28] (originally developed for potentials decays like |x|−2−ε
for large
|x|) but bypassing some dispersive estimates thanks to Kato-Yajima smoothing and
Morawetz estimates. In particular, the key results in [5] ensure the local well-posedness
of the corresponding NLS from standard techniques. For further results in the subcritical
case c < c∗, see Zhang-Zheng [40], Killip-Murphy-Visan-Zheng [17] and Lu-Miao-Murphy
[20] for the long-time behavior of solutions; Killip-Miao-Visan-Zhang-Zheng [16] for the
global well-posedness and scattering in the energy-critical case; Csobo-Genoud [8] for
the classification of the mass-critical blow-up solutions; and Bensouilah-Dinh-Zhu [2] for
the stability and instability of standing waves.
The critical case c = c∗ is more interesting, but less understood. In this case, the
analysis is more subtle and intricate in several aspects, for example the energy space of
the NLS (1.1) is strictly larger than H1
(Rd
) and the endpoint Strichartz’s estimates fail
[5]. Nevertheless, the non-endpoint Strichartz’s estimates for (1.1) remain valid, as proved
by Suzuki [32], and hence the local well-posedness in the energy-subcritical p < 2∗
follows
from the abstract framework in Okazawa-Suzuki-Yokota [25,24]. In another development,
the existence and the asymptotic behavior of the standing waves of the NLS (1.1) have
been established by Trachanas-Zographopoulos [36].
In spite of the above remarkable works, there are still several open questions regarding
the NLS with an inverse square potential. In the present paper, we will prove the uniqueness
of the ground state solution, thus extending the fundamental results of Coffman [7]
and McLeod-Serrin [21] for the classical NLS. As a consequence, we also obtain the full
characterization for the minimal mass blow-up solution for (1.1), in the spirit of Merle
[22].
To simplify the representation, we will mostly focus on the critical case c = c∗, but
our results can be extended easily to the subcritical case c < c∗. The precise statements
of our results are represented in the next section.
Acknowledgments
Debangana Mukherjee and Phuoc-Tai Nguyen are supported by Czech Science Foundation,
project GJ19 – 14413Y. Phan Thành Nam is funded by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under Germany’s Excellence Strategy
– EXC-2111 – 390814868.
4 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
2. Main results
2.1. Ground state problem
Let us start by recalling some fundamental facts. By Hardy’s inequality
Rd
|∇u(x)|2
dx − c∗
Rd
|u(x)|2
|x|2
dx 0, ∀u ∈ H1
(Rd
)
and Friedrichs’ method, we may extend the operator
H = −Δ − c∗|x|−2
to be a non-negative self-adjoint operator on L2
(Rd
).
The energy space associated with (1.1) is the quadratic form domain Q of H, which
is the Hilbert space with norm
u Q =
√
Hu 2
L2(Rd) + u 2
L2(Rd)
1
2
.
Obviously
H1
(Rd
) ⊂ Q ⊂ L2
(Rd
).
Moreover, H1
(Rd
) is strictly included in Q. In fact, it is possible to construct a function
ϕ ∈ Q such that ϕ(x) ∼ |x|− d−2
2 as |x| → 0,1
and hence
Q ⊂ L2∗
(Rd
) (2.1)
while H1
(Rd
) ⊂ L2∗
(Rd
) by Sobolev’s embedding. On the other hand, a fundamental
result of Brezis-Vázquez [3, Theorem 4.1] ensures the continuous embedding
Q ⊂ Lp
(Rd
), ∀ p ∈ [2, 2∗
). (2.2)
Actually, there is a stronger result proved by Frank [10, Theorem 1.2] that Q can be
continuously embedded into fractional Sobolev spaces:
Q ⊂ Hs
(Rd
), ∀s ∈ (0, 1). (2.3)
See also Solovej-Sørensen-Spitzer [29] for a previous result on relativistic particles and
Vazquez-Zuazua [37] for a local version of (2.3). Consequently, for every 2 < p < 2∗
and
ϕ ∈ Q the following energy functional is well-defined
1
Note that the function ψ(x) = |x|− d−2
2 is exactly the “ground state” of Hardy’s inequality, namely
(−Δ − c∗|x|−2
)ψ = 0 for all x = 0, but ψ /∈ L2
(R2
) and hence Hardy’s inequality is strict.
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 5
E(ϕ) :=
1
2
√
Hϕ 2
L2(Rd) −
1
p
ϕ p
Lp(Rd)
.
The Euler-Lagrange equation associated to the minimizer of this energy functional
under the mass constraint (namely ϕ L2(Rd) is fixed) reads
HQ − |Q|p−2
Q − μQ = 0 (2.4)
with a constant μ ∈ R (the Lagrange multiplier or chemical potential). Note that any
solution of (2.4) gives a solution of the NLS (1.1) of the form
u(t, x) = e−iμt
Q(x).
By a standard scaling argument (i.e. replacing u(x) by au(bx) with constants a, b > 0)
from now on we will fix the Lagrange multiplier μ = −1 for simplicity.
Our first main result is the existence and uniqueness of a radial positive solution to
the ground state equation.
Theorem 1 (Existence and uniqueness of the ground state solution). Let d 3 and
2 < p < 2∗
. Then there exists a unique radial positive solution Q ∈ Q to the equation
HQ − Qp−1
+ Q = 0. (2.5)
As a consequence of Theorem 1, we obtain a sharp interpolation inequality.
Theorem 2 (Hardy-Gagliardo-Nirenberg inequality). Let d 3 and 2 < p < 2∗
. Then we
have
√
Hu θ
L2(Rd) u 1−θ
L2(Rd)
CHGN u Lp(Rd), θ =
d
2
−
d
p
, (2.6)
for all u ∈ Q, with the sharp constant
CHGN = Q
p−2
p
L2(Rd)
(1 − θ)
1
p
θ
1 − θ
d(p−2)
4p
. (2.7)
Moreover, any optimizer of the interpolation inequality (2.6) has the form
u(x) = zQ(λx)
with constants z ∈ C and λ > 0. Here Q is the unique solution in Theorem 1.
While the existence of solution to (2.5) is well-known (it essentially follows from
Weinstein’s method [38]), the uniqueness part has remained an open problem for a while.
6 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
Our result in Theorem 1 can be extended to the subcritical case 0 < c < c∗ where it is
also new (see Section 6 for further discussions).
For the classical NLS (without the inverse square potential), the uniqueness of the
ground state solutions goes back to Coffman [7] who treated the cubic NLS (p = 4) in 3D
and McLeod-Serrin [21] who handled the general case. It is worth noting that the radial
symmetry of the ground state solution generally follows from rearrangement inequalities
(see e.g. [19,4]), although the symmetry can be also derived from the equation itself by
the moving plane method (see Gidas-Ni-Nirenberg [11] and Kwong [18]). However, it
seems that the existing methods for the classical NLS do not apply easily to (2.5) due
to the strong effect of the critical Hardy potential.
Our ideas of proving the uniqueness in Theorem 1 are as follows. First, since the
solution behaves as |x|−(d−2)/2
close to the origin (similarly to the non-integrable ground
state of Hardy’s inequality), we introduce the function
v(x) = |x|
d−2
2 Q(x)
which is uniformly bounded and decays fast at infinity. The equation for v admits a
generalized Pohozaev type identity, following the spirit of the recent work of ShiojiWatanabe
[34] (although the uniqueness result in [34] does not apply in our case as the
condition (II) in [34, Theorem 1] is so restricted). Then the conclusion follows from a
careful implementation of the general shooting argument of Yanagida [39], taking the
specific scaling of equation (2.5) into account.
We will prove Theorems 1 and 2 in Section 3.
2.2. Dynamical problem
Next, we consider the NLS (1.1) with an initial datum u0 ∈ Q.
Definition 1 (Weak solutions). A function u is called a weak solution to (1.1) in (0, T)
with initial datum u0 ∈ Q if
u ∈ C([0, T); Q) ∩ C1
([0, T); Q∗
)
and it satisfies the Duhamel formula
u(t) = e−itH
u0 + i
t
0
e−i(t−s)H
|u(s)|p−2
u(s)ds ∀t ∈ (0, T). (2.8)
Here we ignore the x-dependence in the notation. If T = ∞ then u is called a global
solution.
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 7
For the reader’s convenience, we collect some basic properties of equation (1.1) with
initial datum in the energy space Q. Recall that Q is the unique ground state solution
of (2.5).
Theorem 3 (Basic properties of focusing NLS with Hardy potential). Let d 3 and
2 < p < 2∗
. For any u0 ∈ Q the followings hold true.
(i) (Local well-posedness). There exist a constant T = T( u0 Q) > 0 and a unique
weak solution u ∈ C([0, T); Q) ∩ C1
([0, T); Q∗
) of (1.1) in (0, T). The mass and
energy conservation laws hold, i.e.
u(t) L2(Rd) = u0 L2(Rd), E(u(t)) = E(u0), ∀t ∈ [0, T). (2.9)
Moreover, the above solution u admits a unique continuation up to a maximum time
T∗
such that either the solution is global, namely T∗
= ∞, or the solution blows up
at finite time, namely T∗
< ∞ and
lim
t T ∗
√
Hu L2(Rd) = ∞. (2.10)
(ii) (Sufficient conditions for global existence). The solution of (1.1) is global in one of
the following three cases: 2 < p < 2+4/d; p = 2+4/d and u0 L2(Rd) < Q L2(Rd);
2 + 4/d < p < 2∗
and
E(u0) u0
q
L2 < E(Q) Q q
L2 ,
√
Hu0
2
L2 u0
q
L2 <
√
HQ 2
L2 Q q
L2
with
q =
4d + 4p − 2pd
dp − 2d − 4
. (2.11)
Moreover, in any of the above cases,
√
Hu(t) L2(Rd) remains uniformly bounded
in t ∈ (0, ∞).
(iii) (Sufficient conditions for blowup). The solution of (1.1) blows up at finite time if
2 + 4/d p < 2∗
, |x|u0 ∈ L2
(Rd
) and either E(u0) < 0, or
E(u0) u0
q
L2 < E(Q) Q q
L2 ,
√
Hu0
2
L2 u0
q
L2 >
√
HQ 2
L2 Q q
L2
with q as in (2.11).
Most of the results in Theorem 3 are known or easily obtained by adapting the analysis
for the usual NLS. More precisely, the local well-posedness is due to Suzuki [32], based
on the abstract result of Okazawa-Suzuki-Yokota [25,24] (the uniqueness in (i) was not
addressed explicitly but it follows from the standard method in [6] using the Strichartz
8 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
estimates for Hardy potential by Suzuki [32, Proposition 4.8]; see also Mizutani [23,
Corollary 2.3]). In the global existence (ii), the first case (of subcritical nonlinearity) is
standard; the second case is analogous to Weinstein’s famous theorem for the usual NLS
[38]; the last case follows from the strategy in the usual NLS of Kenig-Merle [15] (we
will recall the representation of Holmer-Roudenko [13] for the energy subcritical case,
see also Ogawa-Tsutsumi [26] for an earlier work on the radial case). In the blow-up
conditions (iii), the first case (of negative energy) can be found in Suzuki [33, Theorem
4.1], while the other case follows from the analysis for the usual NLS [15,13].
For the sake of completeness, in Section 4 we will briefly recall standard arguments
in the proof of Theorem 3. Some key ingredients, e.g. Virial’s identities, will be useful
later.
Now we concentrate on the mass-critical case p = 2+4/d. Our new result is a complete
characterization of the minimal-mass blow-up solutions. Recall that Q is the unique
ground state solution of (2.5).
Theorem 4 (Minimal mass blow-up solutions). Assume d 3 and p = 2 + 4/d.
(i) (Existence). Let γ ∈ R, λ > 0 and T > 0 arbitrarily. Then
u(t, x) = eiγ
ei λ2
T −t e−i |x|2
4(T −t)
λ
T − t
d
2
Q
λx
T − t
∀x ∈ Rd
, t ∈ [0, T) (2.12)
solves the NLS (1.1) in (0, T) and blows up at the finite time T.
(ii) (Uniqueness). For any finite time T > 0, if u ∈ C([0, T); Q) ∩ C1
([0, T), Q∗
) is a
solution of (1.1) in (0, T) with u0 L2 = Q L2 and blows up at T, then u is given
in (2.12) for some constants γ ∈ R and λ > 0.
Theorem 4 is an extension of Merle’s celebrated result [22] (for the classical NLS) to
the case of Hardy potential. A similar result was obtained recently by Csobo-Genoud [8]
in the subcritical case c < c∗ (although the uniqueness result for the ground state was
not available in [8]).
A crucial ingredient of the proof of Theorem 4 is a compactness lemma on the minimizing
sequences of the Hardy-Gagliardo-Nirenberg inequality (2.6). In the classical NLS,
the compactness lemma follows standardly from the concentration-compactness method.
This method was also used in [8] for the subcritical case c < c∗, but the analysis cannot
be extended to the case c = c∗. In the present paper, we will prove the compactness result
for the critical case by a refined method based on geometric localization techniques.
This enables us to implement the approach of Hmidi-Keraani [12] to achieve the full
characterization for the mass-critical blow-up solutions.
Finally, let us remark that all the results in Theorems 1, 2, 3, 4 hold true in the
subcritical case, where the Hardy potential −c∗|x|−2
is replaced by −c|x|−2
with c < c∗.
In fact, the proof in the subcritical case is similar, even simpler, as will be explained in
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 9
Section 6. Moreover, if c < c∗, then some results in Theorems 3 and 4 can be extended
to the energy–critical case p = 2∗
; see [2,9] for further details. In the case c = c∗, the
restriction p < 2∗
is natural due to (2.1).
Organization of the paper. Section 3 is devoted to the proof of Theorem 1 and Theorem 2.
In Section 4, we present basic properties of problem (1.1). In Section 5, we demonstrate
the characterization of blow-up solutions with minimal mass. Finally, in Section 6, we
discuss the extension to the subcritical case 0 < c < c∗.
Notations.
• When there is no confusion, we will write u(t) instead of u(t, x), ||u||Lp instead of
||u||Lp(Rd).
• For u, v ∈ Q, we use the notation
u, Hv =
Rd
uHvdx =
Rd
∇u · ∇v −
c∗
|x|2
uv dx
and
√
Hu L2 = u, Hu
1
2 .
• The notation A B (resp. A B) means A C B (resp. A C B) where C is a
positive constant depending on some initial parameters.
• (z), (z) and ¯z denote the real part, the imaginary part and the complex conjugate
of z ∈ C respectively.
3. Hardy-Gagliardo-Nirenberg inequality
In this section we prove Theorems 1 and 2.
Following Weinstein’s strategy [38], we consider the Hardy-Gagliardo-Nirenberg interpolation
problem
CHGN = inf
u∈Q\{0}
√
Hu θ
L2 u 1−θ
L2
u Lp
, θ =
d
2
−
d
p
. (3.1)
Recall from (2.2) that
√
Hu L2 + u L2 u Lp .
By a standard scaling argument, i.e. changing u(x) → u(λx) and optimizing over λ > 0,
we find that
10 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
√
Hu θ
L2 u 1−θ
L2 u Lp , θ =
d
2
1 −
2
d
. (3.2)
Thus CHGN > 0.
3.1. Existence of ground states
This part follows from the standard direct method in the calculus of variations, using
the rearrangement inequalities and the compactness of radial functions (see [30,31], [36,
Lemma 3.1]). We will give a short proof for the reader’s convenience.
Let {un} be a minimizing sequence for CHGN. By the Hardy–Littlewood and the
Pólya–Szegö rearrangement inequalities (see [19, Theorem 3.4] and [27,4]) we can assume
that the functions un’s are non-negative and radially symmetric decreasing. By a scaling
argument we can also assume that
un L2 =
√
Hun L2 = 1, un Lp → C−1
HGN.
Since |x| → un(x) is decreasing, we have the pointwise estimate
|un(x)|2 1
B(0, |x|)
|y| |x|
|un(y)|2
dy C|x|−d
with a constant C > 0 independent of n. Therefore, by Helly’s selection theorem we find
that, up to a subsequence when n → ∞, we have the pointwise convergence
un(x) → u0(x) for all x = 0.
This implies that for any ε > 0,
1{|x| ε}(un − u0) → 0 strongly in Lp
(Rd
)
as n → ∞ by Dominated Convergence Theorem. On the other hand, from the uniform
bound un Lq C for p < q < 2∗
, we find that
1{|x|<ε}(un − u0) → 0 strongly in Lp
(Rd
)
as ε → 0, uniformly in n. Thus un → u0 strongly in Lp
(Rd
), and hence
u0 Lp = lim
n→∞
un Lp = C−1
HGN.
On the other hand, by the Banach-Alaoglu theorem, up to a subsequence as n → ∞
again, we can assume that
un u0,
√
Hun
√
Hu0 weakly in L2
(Rd
).
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 11
Hence, we have
u0 L2 lim inf
n→∞
un L2 = 1,
√
Hu0 L2 lim inf
n→∞
√
Hun L2 = 1.
In summary, we have proved that
√
Hu0
θ
L2 u0
1−θ
L2
u0 Lp
CHGN.
Thus u0 is a minimizer for the variational problem (2.6). From the above proof, since
the minimizing sequence {un} are nonnegative radially symmetric decreasing, the limit
u0 is also nonnegative radially symmetric decreasing.
Euler-Lagrange equation. By standard variational techniques, we can show that the
above minimizer u0 satisfies the Euler-Lagrange equation
θHu0 + (1 − θ)u0 − (CHGN)
p
2 up−1
0 = 0. (3.3)
Here the relevant coefficients come from the constraints
u0 L2 = 1 =
√
Hu0 L2 , u0 Lp = C−1
HGN.
If we define
u0(x) = αQ(βx), β =
1 − θ
θ
1
2
, α = (1 − θ)
1
p−2 (CHGN)− p
p−2 ,
then (3.3) becomes
HQ + Q − Qp−1
= 0. (3.4)
Moreover, Q is nonnegative radially symmetric decreasing and
Q L2 = β
d
2 α−1
u0 L2 =
1 − θ
θ
d
4
(1 − θ)− 1
p−2 (CHGN)
p
p−2 . (3.5)
3.2. A-priori estimates
The following a-priori estimates will be important for the proof of the uniqueness of
Q.
Lemma 5 (A-priori estimates). Let Q ∈ Q be a nonnegative radial solution to (3.4). Then
Q ∈ C2
(Rd
\ {0}) and Q is strictly positive. Moreover,
12 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
lim
|x|→0
|x|
d−2
2 Q(x) ∈ (0, ∞), (3.6)
and
lim
|x|→∞
|x|m
Q(x) = 0, ∀m > 0. (3.7)
Proof. Since Q ∈ Q, by a standard bootstrap argument, together with Sobolev embeddings,
one can show that Q ∈ C2
(Rd
\ {0}).
The equation (3.4) can be rewritten as
Q(x) = (I − Δ)−1
c∗|x|−2
Q(x) + Q(x)p−1
=
Rd
G(x − y) c∗|y|−2
Q(y) + Q(y)p−1
dy (3.8)
where G is the Green function of I −Δ (the Yukawa potential). Recall that [19, Theorem
6.23]
G(x) =
∞
0
(4πt)−d/2
exp −
|x|2
4t
− t dt
and
0 < G(x)
∞
0
(4πt)−d/2
exp −
|x|2
4t
dt =
cd
|x|d−2
, ∀x ∈ Rd
\ {0} (3.9)
and
lim
|x|→∞
−
log G(x)
|x|
= 1. (3.10)
In particular, since Q(x) is the convolution of G(x) > 0 and c∗|x|−2
Q(x)+Q(x)p−1
0, we find that Q(x) > 0 for all x ∈ Rd
.
The formula (3.6) has been proved by Trachanas-Zographopoulos [36, Theorem 1.2].
Now we consider the decay of Q at infinity. We take |x| large and decompose the
integral domain in (3.8) into |x − y| < |x|1/4
and |x − y| |x|1/4
.
In the region |x − y| < |x|1/4
, by the triangle inequality we have
|y| > |x| − |x|1/4
|x|/2 1
for |x| large. Combining with the upper bound (3.9) and Newton’s Theorem [19, Theorem
9.7] (here Q is radial) we can bound
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 13
|x−y|<|x|1/4
G(x − y) c∗|y|−2
Q(y) + Q(y)p−1
dy
|x−y|<|x|1/4
1
|x − y|d−2
Q(y) + Q(y)p−1
dy
|x−y|<|x|1/4
1
|x|d−2
Q(y) + Q(y)p−1
dy. (3.11)
On the other hand, since Q ∈ L2
(Rd
) ∩ Lp
(Rd
), by Hölder’s inequality and keeping in
mind that p > 2, we have
|x−y|<|x|1/4
Q(y)dy
⎛
⎜
⎝
|x−y|<|x|1/4
dy
⎞
⎟
⎠
1/2 ⎛
⎜
⎝
|x−y|<|x|1/4
|Q(y)|2
dy
⎞
⎟
⎠
1/2
|x|d/8
and
|x−y|<|x|1/4
Q(y)p−1
dy
⎛
⎜
⎝
|x−y|<|x|1/4
dy
⎞
⎟
⎠
1/p ⎛
⎜
⎝
|x−y|<|x|1/4
|Q(y)|p
dy
⎞
⎟
⎠
(p−1)/p
|x|d/8
.
Thus from (3.11) it follows that
|x−y|<|x|1/4
G(x − y) c∗|y|−2
Q(y) + Q(y)p−1
dy
|x|d/8
|x|d−2
|x|−5/8
. (3.12)
Here we have used d 3 in the last estimate.
In the region |x − y| |x|1/4
, using (3.10) we have
G(x − y) e−|x−y|/2
e−|x−y|/4
e−|x|1/4
/4
for |x| large. Hence,
|x−y| |x|1/4
G(x − y) c∗|y|−2
Q(y) + Q(y)p−1
dy
e−|x|1/4
/4
Rd
e−|x−y|/4
|y|−2
Q(y) + Q(y)p−1
dy. (3.13)
By Hölder’s inequality we can bound
14 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
Rd
e−|x−y|/4
Q(y)p−1
dy
⎛
⎝
Rd
e−|x−y|p/4
dy
⎞
⎠
1/p ⎛
⎝
Rd
|Q(y)|p
dy
⎞
⎠
(p−1)/p
1
and
Rd
e−|x−y|/4
|y|−2
Q(y)dy
|y| 1
|y|−2
Q(y)dy +
|y|>1
e−|x−y|/4
Q(y)dy
⎛
⎜
⎝
|y| 1
1
|y|d−1/2
dy
⎞
⎟
⎠
4/(2d−1) ⎛
⎜
⎝
|y| 1
|Q(y)|(2d−1)/(2d−5)
dy
⎞
⎟
⎠
(2d−5)/(2d−1)
+
⎛
⎜
⎝
|y|>1
e−|x−y|/2
dy
⎞
⎟
⎠
1/2 ⎛
⎜
⎝
|y|>1
|Q(y)|2
dy
⎞
⎟
⎠
1/2
1.
In the last estimate we have used that Q ∈ Lq
(Rd
) for all 2 q < 2∗
and that
2d − 1
2d − 5
<
2d
d − 2
= 2∗
for all d 3. Thus (3.13) gives us
|x−y| |x|1/4
G(x − y) c∗|y|−2
Q(y) + Q(y)p−1
dy e−|x|1/4
/4
(3.14)
for |x| large.
Putting (3.12) and (3.14) together, we obtain
Q(x) =
Rd
G(x − y) c∗|y|−2
Q(y) + Q(y)p−1
dy |x|−5/8
+ e−|x|1/4
/4
|x|−5/8
for |x| large.
Next, assume that we have proved that Q(x) |x|−m
for |x| large, with some constant
m > 0. Then for |x| large, in the region |x−y| |x|1/4
using again the triangle inequality
|y| > |x| − |x|1/4
|x|/2 we get
Q(y) + Q(y)p−1
|x|−m
.
Inserting this pointwise bound into (3.11) we find that
|x−y|<|x|1/4
G(x − y) c∗|y|−2
Q(y) + Q(y)p−1
dy
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 15
|x−y|<|x|1/4
1
|x|d−2
Q(y) + Q(y)p−1
dy
|x−y|<|x|1/4
1
|x|d−2
|x|−m
dy
|x|d/4
|x|d−2
|x|−m
|x|−m−1/4
(3.15)
for all d 3. Putting the latter bound together with (3.14) we obtain
Q(x) =
Rd
G(x − y) c∗|y|−2
Q(y) + Q(y)p−1
dy |x|−m−1/4
+ e−|x|1/4
/4
|x|−m−1/4
for |x| large.
Thus in summary, we have proved that if Q(x) |x|−m
for |x| large, then Q(x)
|x|−m−1/4
for |x| large. By induction, we conclude that for any constant m > 0, then
Q(x) |x|−m
for |x| large. Consequently, we get (3.7), namely
lim
|x|→∞
|x|m
Q(x) = 0, ∀m > 0.
The proof is complete.
3.3. Uniqueness of ground state solution
Now we study the uniqueness of positive radial solutions to equation (3.4). Thanks to
Lemma 5, it suffices to show that there exists at most one positive radial C2
solution to
⎧
⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
−HQ + Q − Qp−1
= 0 in Rd
\ {0},
lim
|x|→0
|x|
d−2
2 Q(x) ∈ (0, ∞),
lim
|x|→∞
|x|m
Q(x) ∈ [0, ∞),
(3.16)
for some m > d+2
2 . Here the first equation in (3.16) is understood in the classical sense
in Rd
\ {0}.
Note that if Q is radial solution of (3.16) then using the polar coordinate r = |x| we
can write
⎧
⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
d2
dr2
Q +
d − 1
r
d
dr
Q +
c∗
r2
Q − Q + Qp−1
= 0 in (0, ∞),
lim
r→0+
r
d−2
2 Q(r) ∈ (0, ∞),
lim
r→+∞
rm
Q(r) ∈ [0, ∞).
(3.17)
16 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
By putting
v(r) = r
d−2
2 Q(r), (3.18)
we deduce that v ∈ C2
((0, ∞)), v > 0 in (0, ∞) and v satisfies
⎧
⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎩
d2
dr2
v +
1
r
d
dr
v − v + r− (d−2)(p−2)
2 vp−1
= 0 in (0, ∞),
v(0) ∈ (0, ∞),
lim
r→+∞
rm− d−2
2 v(r) ∈ [0, ∞).
(3.19)
We will prove that there exists at most one positive solution v of (3.19). In the following
we will use the notation vr instead of d
dr v.
Pohozaev identity. By using the computation in Shioji-Watanabe [34] with
f(r) = r, g(r) = −1, h(r) = r− (d−2)(p−2)
2 ,
we obtain the generalized Pohozaev identity
d
dr
J(r, u) = G(r)u(r)2
∀r ∈ (0, ∞) (3.20)
where
a(r) := r
d(p−2)+4
p+2 , (3.21)
b(r) :=
d + 2 − (d − 2)(p − 1)
2(p + 2)
r
(d−1)(p−2)
p+2 , (3.22)
c(r) :=
[(d + 2 − (d − 2)(p − 1)]2
2(p + 2)2
r− d+2−(d−2)(p−1)
d+2 ,
G(r) := b(r) +
1
2
cr(r) −
1
2
ar(r)
= −
(d − 1)(p − 2)
p + 2
r
(d−1)(p−2)
p+2 −
[(d + 2 − (d − 2)(p − 1)]3
2(p + 2)3
r− d+5−(d−3)(p−1)
p+2 ,
J(r, v) :=
1
2
a(r)vr(r)2
+ b(r)vr(r)v(r) (3.23)
+
1
2
(c(r) − a(r))v(r)2
+
1
p
a(r)r− (d−2)(p−2)
2 v(r)p
.
We will follow the general strategy in [34], but we use the following result to relax the
condition (II) in [34, Theorem 1].
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 17
Lemma 6. Assume 2 < p < 2∗
and m > d+2
2 . Let v ∈ C2
((0, ∞)) be a positive solution
of (3.19). Then
vr(r) +
1
r
r
0
s(−v(s) + s− (d−2)(p−2)
2 v(s)p−1
)ds = 0 ∀r ∈ (0, ∞). (3.24)
Moreover,
lim
r→0+
rγ
vr(r) = 0 ∀ γ >
(d − 2)(p − 2)
2
− 1 (3.25)
and
lim
r→+∞
rν
vr(r) = 0 ∀ ν < m −
d
2
. (3.26)
Proof. From (3.19), we have, for any 0 < r < r,
rvr(r) − r vr(r ) +
r
r
s(−v(s) + s− (d−2)(p−2)
2 v(s)p−1
)ds = 0. (3.27)
Note that
r
0
s1− (d−2)(p−2)
2 ds < ∞
since
1 −
(d − 2)(p − 2)
2
> −1
as p < 2∗
. Consequently, from (3.27), we deduce that the limit
k = lim
r →0+
r vr(r )
exists as a real number. Letting r → 0+
in (3.27) yields
vr(r) −
k
r
+
1
r
r
0
s(−v(s) + s− (d−2)(p−2)
2 v(s)p−1
)ds = 0. (3.28)
Integrating this equation over (ε, r) with 0 < ε < r implies
v(r) − v(ε) − k ln
r
ε
+
r
ε
1
r
r
0
s(−v(s) + s− (d−2)(p−2)
2 v(s)p−1
)dsdr = 0. (3.29)
18 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
Using again p < 2∗
we get
r
0
1
r
r
0
s(−v(s) + s− (d−2)(p−2)
2 v(s)p−1
)dsdr < ∞.
By letting ε → 0 in (3.29) we find that k = 0. Thus (3.24) follows from (3.28).
Moreover, by (3.24) and since p < 2∗
and γ > (d−2)(p−2)
2 − 1,
lim
r→0+
rγ
vr(r) = − lim
r→0+
rγ−1
r
0
s(−v(s) + s− (d−2)(p−2)
2 v(s)p−1
)dsdr = 0.
Thus we obtain (3.25).
Next we prove (3.26). From (3.19), there exists r0 > 0 large enough such that
v(r) r
d−2
2 −m
∀r ∈ [r0, ∞). (3.30)
We use (3.27) for large numbers r0 < r < r. Since m > d+2
2 , we deduce from (3.30)
and the third equality in (3.19) that
lim
r→+∞
r
r
sv(s)ds lim
r→+∞
r
r
s
d−2
2 −m+1
ds =
2(r )
d+2−2m
2
2m − d − 2
(3.31)
and
lim
r→+∞
r
r
s1− (d−2)(p−2)
2 v(s)p−1
ds lim
r→+∞
r
r
s
d−2m(p−1)
2 ds =
2(r )
d+2−2m(p−1)
2
2m(p − 1) − d − 2
.
(3.32)
Therefore, we see from (3.27), using (3.31) and (3.32), that the limit
K = lim
r→+∞
rvr(r)
exists as a real number. Letting r → +∞ in (3.27) implies
K
r
− vr(r ) +
1
r
∞
r
s(−v(s) + s− (d−2)(p−2)
2 v(s)p−1
)ds = 0. (3.33)
By integrating over (r, R) with r0 < r < R, we find
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 19
K ln
R
r
− v(R) + v(r) +
R
r
1
r
∞
r
s(−v(s) + s− (d−2)(p−2)
2 v(s)p−1
)dsdr = 0.
(3.34)
Since m > d+2
2 , we derive
∞
r
1
r
∞
r
s(−v(s) + s− (d−2)(p−2)
2 v(s)p−1
)dsdr < ∞.
This, together with the fact that v decays at infinity and (3.34), implies K = 0. Consequently,
we infer from (3.33) that
(r )ν
vr(r ) = (r )ν−1
∞
r
s(−v(s) + s− (d−2)(p−2)
2 v(s)p−1
)ds. (3.35)
From (3.31) and (3.32) and the fact that p > 2, we deduce
(r )ν−1
∞
r
s(−v(s) + s− (d−2)(p−2)
2 v(s)p−1
)ds (r )ν−m+ d
2 . (3.36)
Consequently, as ν < m − d
2 , it follows
lim
r →+∞
(r )ν−1
∞
r
s(−v(s) + s− (d−2)(p−2)
2 v(s)p−1
)ds = 0. (3.37)
Combining (3.35) and (3.37) leads to (3.26).
Now we are ready to conclude
Proposition 7 (Uniqueness of (3.19)). Assume 2 < p < 2∗
and m > d+2
2 . Then problem
(3.19) admits at most one positive solution.
Proof. Let v and v be two positive solutions of (3.19). We will prove that v = v by using
a shooting argument.
Step 1. We show that if v(0) = v(0) then v = v in (0, ∞).
Proof. Let R > 0. From Lemma 6, we see that, for any 0 < r < R,
v(r) = v(0) −
r
0
1
σ
σ
0
s(−v(s) + s− (d−2)(p−2)
2 v(s)p−1
)dsdσ
20 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
= v(0) −
r
0
⎛
⎝
r
s
1
σ
dσ
⎞
⎠ s(−v(s) + s− (d−2)(p−2)
2 v(s)p−1
)ds
= v(0) −
r
0
ln
r
s
s(−v(s) + s− (d−2)(p−2)
2 v(s)p−1
)ds.
Similarly, we have
v(r) = v(0) −
r
0
ln
r
s
s(−v(s) + s− (d−2)(p−2)
2 v(s)p−1
)ds.
Keeping in mind that v(0) = v(0), v, v are bounded in [0, R] and p > 2, we deduce from
the above equalities that
|v(r) − v(r)|
r
0
ln
r
s
s(|v(s) − v(s)| + s− (d−2)(p−2)
2 |v(s)p−1
− v(s)p−1
|)ds
C(R)
r
0
ln
r
s
s(1 + s− (d−2)(p−2)
2 )|v(s) − v(s)|ds.
Since p < 2∗
we have
1 −
(d − 2)(p − 2)
2
> −1,
and hence
r
0
ln
r
s
s(1 + s− (d−2)(p−2)
2 )ds < ∞.
Therefore, by Gronwall’s inequality, we find that v = v in [0, R). Since R > 0 is arbitrary,
we deduce that v = v in [0, ∞).
Step 2. We show that
d
dr
v(r)
v(r)
=
1
rv(r)2
r
0
s1− (d−2)(p−2)
2 (v(s)p−2
− v(s)p−2
)v(s)v(s)ds ∀r ∈ (0, ∞).
(3.38)
Proof. Since v and v are two solutions of (3.19), we obtain
(rvr)r + r(−v + r− (d−2)(p−2)
2 vp−1
) = 0, (3.39)
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 21
(rvr)r + r(−v + r− (d−2)(p−2)
2 vp−1
) = 0. (3.40)
Multiplying (3.39) by v and multiplying (3.40) by v, then integrating over [r , r] with
0 < r < r, we obtain
r(vr(r)v(r) − v(r)vr(r)) − r (vr(r )v(r ) − v(r )vr(r ))
+
r
r
s1− (d−2)(p−2)
2 (v(s)p−2
− v(s)p−2
)v(s)v(s)ds = 0.
(3.41)
Since p < 2∗
, thanks to Lemma 6, limr →0+ (r vr(r )) = limr →0+ (r vr(r )) = 0. Therefore,
by letting r → 0 in (3.41), we obtain (3.38).
Step 3. We show that if v(0) < v(0) then
d
dr
v(r)
v(r)
> 0 ∀r ∈ (0, ∞). (3.42)
Proof. If v does not intersect v then v(r) > v(r) for any r ∈ (0, ∞) by the continuity. In
this case (3.42) follows immediately from (3.38).
Now we suppose that v intersects v at least one point. Denote by r1 ∈ (0, ∞) the first
intersection of v and v. Put w = v/v. By (3.38), wr(r) > 0 for any r ∈ (0, r1].
Suppose by contradiction that (3.42) does not hold. Then there exists r2 > r1 such
that wr(r) > 0 for any r ∈ (0, r2) and wr(r2) = 0. Hence we see that w(r2)v(r) > v(r)
for any r ∈ (0, r2) and
w(r2) > 1, w(r2)v(r2) = v(r2), w(r2)vr(r2) = vr(r2). (3.43)
From the generalized Pohozaev identity (3.20), for r < r2 we have
w(r2)2
J(r2, v) − J(r2, v)
=
r2
r
G(s)(w(r2)2
v(s)2
− v(s)2
)ds + w(r2)2
J(r, v) − J(r, v).
(3.44)
The left hand side of (3.44) can be estimated by (3.43) as
w(r2)2
J(r2, v) − J(r2, v) =
w(r2)2
− w(r2)p
p
a(r2)r
− (d−2)(p−2)
2
2 v(r2)p
< 0. (3.45)
For the right hand side of (3.44), since wr(r) > 0 for any r ∈ (0, r2), it follows that
0 < w(r) < w(r2) for any 0 < r < r2 and w(r)v(s) < v(s) for any s ∈ (r, r2). From
(3.20) and G(r) < 0 < v(r) it follows that
22 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
d
dr
J(r, v) = G(r)v(r)2
< 0, ∀r ∈ (0, ∞).
Consequently, J(r, v) is strictly decreasing with respect to r. Moreover, since v satisfies
the third limit in (3.19) with m > d+2
2 and the limit (3.26) with ν < m − d
2 , we have
lim
r→+∞
a(r)
1
2 vr(r) = lim
r→+∞
r
d(p−2)+4
2(p+2) vr(r) = 0,
lim
r→+∞
b(r)vr(r)v(r) lim
r→+∞
r
(d−1)(p−2)
p+2 + d−2
2 −m
vr(r) = 0,
lim
r→+∞
c(r)v(r)2
lim
r→+∞
r− d+2−(d−2)(p−1)
d+2 +d−2−2m
= 0,
lim
r→+∞
a(r)v(r)2
lim
r→+∞
r
d(p−2)+4
p+2 +d−2−2m
= 0,
lim
r→+∞
a(r)r− (d−2)(p−2)
2 v(r)p
lim
r→+∞
r
d(p−2)+4
p+2 − (d−2)(p−2)
2 + (d−2−2m)p
2 = 0.
(3.46)
Inserting these limits into the formula of J(r, v) we obtain
lim
r→+∞
J(r, v) = 0.
Similarly
lim
r→+∞
J(r, v) = 0.
Therefore
J(r, v) > 0 ∀r ∈ (0, ∞). (3.47)
This leads to
r2
r
G(s)v(s)2
ds + J(r, v) = J(r2, v) > 0.
Combining the above estimates, we can estimate the right hand side of (3.44) as follows
r2
r
G(s)(w(r2)2
v(s)2
− v(s)2
)ds + w(r2)2
J(r, v) − J(r, v)
= w(r2)2
⎛
⎝
r2
r
G(s)v(s)2
ds + J(r, v)
⎞
⎠ −
r2
r
G(s)v(s)2
ds − J(r, v)
w(r)2
⎛
⎝
r2
r
G(s)v(s)2
ds + J(r, v)
⎞
⎠ −
r2
r
G(s)v(s)2
ds − J(r, v) (3.48)
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 23
=
r2
r
G(s)(w(r)2
v(s)2
− v(s)2
)ds + w(r)2
J(r, v) − J(r, v)
w(r)2
J(r, v) − J(r, v)
for all r ∈ (0, r2). Here in the last estimate, we have used G(s) < 0 and w(r)v(s) < v(s)
for all s ∈ (r, r2).
Next, by (3.23), we have
w(r)2
J(r, v) − J(r, v) =
1
2
a(r)(w(r)2
vr(r)2
− vr(r)2
)
+ b(r)(w(r)2
vr(r)v(r) − vr(r)v(r))
+
1
p
a(r)r− (d−2)(p−2)
2 (w(r)2
v(r)p
− v(r)p
).
(3.49)
Since p < 2∗
it follows that
d(p − 2) + 4
2(p + 2)
>
(d − 2)(p − 2)
2
− 1,
and hence by (3.21) and (3.25) we deduce
lim
r→0+
a(r)
1
2 vr(r) = lim
r→0+
r
d(p−2)+4
2(p+2) vr(r) = 0.
Similarly
lim
r→0+
a(r)
1
2 vr(r) = lim
r→0+
r
d(p−2)+4
2(p+2) vr(r) = 0.
Moreover, p < 2∗
also implies that
(d − 1)(p − 2)
p + 2
>
(d − 2)(p − 2)
2
− 1,
and hence by (3.22) and (3.25) we deduce
lim
r→0+
b(r)vr(r) = lim
r→0+
d + 2 − (d − 2)(p − 1)
2(p + 2)
r
(d−1)(p−2)
p+2 vr(r) = 0.
Similarly
lim
r→0+
b(r)vr(r) = lim
r→0+
d + 2 − (d − 2)(p − 1)
2(p + 2)
r
(d−1)(p−2)
p+2 vr(r) = 0.
Since p < 2∗
, we have
24 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
d(p − 2) + 4
p + 2
−
(d − 2)(p − 2)
2
> 0,
which implies
lim
r→0+
a(r)r− (d−2)(p−2)
2 = lim
r→0+
r
d(p−2)+4
p+2 − (d−2)(p−2)
2 = 0.
Combining the above equalities and taking into account that v, v, w are bounded near
0, by letting r → 0+
in (3.49) we deduce
lim
r→0+
(w(r)2
J(r, v) − J(r, v)) = 0. (3.50)
From (3.44), (3.48) and (3.50), we deduce that
w(r2)2
J(r2, v) − J(r2, v) 0,
which contradicts (3.45). Therefore, we conclude (3.42).
Step 4: Conclusion. Suppose by contradiction that there exist two distinct solutions v
and v of (3.19). By Step 1, we know that v(0) = v(0). Without loss of generality, we may
assume that 0 < v(0) < v(0). We see from (3.47) that J(r, v) > 0 for any r ∈ (0, ∞).
Define
X(r) := J(r, v)
v(r)
v(r)
2
− J(r, v), r ∈ (0, ∞). (3.51)
By inserting (3.23) into (3.51), we deduce
X(r) =
1
2
a(r)
v(r)2
v(r)2
vr(r)2
− vr(r)2
+ b(r)v(r)
v(r)
v(r)
vr(r) − vr(r)
+
1
p
a(r)r− (d−2)(p−2)
2 v(r)2
(v(r)p−2
− v(r)p−2
).
By using the argument as in Step 3, we obtain
lim
r→0+
X(r) = 0. (3.52)
From (3.42), we find
v(r)
v(r)
<
v(r0)
v(r0)
∀r ∈ (r0, ∞).
By combining this and (3.46), we deduce
lim
r→+∞
X(r) = 0. (3.53)
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 25
On the other hand, we observe that
d
dr
X(r) = 2J(r, v)
v(r)
v(r)
d
dr
v(r)
v(r)
.
This, together with the conclusion in Step 3, yields d
dr X(r) < 0 for any r ∈ (0, ∞).
However, the latter fact contradicts (3.52) and (3.53). Thus we conclude that problem
(3.19) admits at most one solution. This ends the proof of Proposition 7.
Now let us conclude.
Proof of Theorem 1. The existence has been proved in subsection 3.1, in particular (3.4).
The uniqueness in the critical case c = c∗ has been derived from the transformation
(3.18), Lemma 5 and Proposition 7.
Remark 8. In the case 0 < c < c∗ we can replace (3.18) by
v(r) = rκ
Q(r), (3.54)
with κ is defined in (6.4), and then obtain the uniqueness by the same way.
Next, we prove Theorem 2.
Proof of Theorem 2. The existence of a minimizer for (2.6) has been proved in subsection
3.1. The sharp constant follows from (3.5). It remains to prove that any minimizer for
(2.6) has the form u(x) = zQ(λx). This part follows a standard technique, but let us
quickly explain for the reader’s convenience. If u is a minimizer for (2.6), then by the
convexity of gradients [19, Theorem 7.8] we know that |u| is also a minimizer. Moreover,
by the rearrangement inequality [19, Theorem 3.4] and the fact that |x|−2
is strictly
radially symmetric decreasing, we know that |u| must be radially symmetric decreasing.
Up to dilations, |u| is nonnegative radial solution to equation (2.5). Thus the uniqueness
result in Theorem 1 ensures that |u| equals Q up to dilations. In particular, we know
that |u| > 0.
Note that if |u| > 0 and
∇u L2 = ∇|u| L2
then u/|u| is a constant. This can be seen, e.g. as in [8, Proposition 3], using the identity
|∇u|2
= |∇(|u|w)|2
= |∇(|u|)w + |u|(∇w)|2
= |∇|u||2
+ |u|2
|∇w|2
,
with w = u/|u|. Here the cross term disappears since
2 (w)∇w = ∇|w|2
= 0
26 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
as |w|2
= 1. Thus from ∇u L2 = ∇|u| L2 we find that w is a constant, namely u/|u|
is a constant. Thus u(x) = zQ(λx).
4. Basic properties of the NLS
In this section we prove Theorem 3.
4.1. Local well-posedness
As previously explained, due to the sharpness of the Hardy potential H, the energy
space associated to (1.1) is Q which is strictly larger than H1
(Rd
). Therefore, the wellposedness
results in [6] do not apply directly to our case.
Nevertheless, the existence and uniqueness of a local weak solution
u ∈ C([0, T); Q) ∩ C1
([0, T); Q∗
)
of (1.1) in a small time interval (0, T) can be obtained by adapting the fixed point
argument in [6]. We refer to [25, Theorems 2.2, 2.3] (see also [32, Section 4.3]) for details.2
Moreover, the conservation laws (2.9) also follow from [25, Theorem 2.3].
Fortunately, the local existence can be derived by examining abstract assumptions
stated in [25]. The uniqueness is strongly based on the Strichartz estimates for H which
were recently established in [32,23]. The blowup alternative follows from a standard
argument.
Next, we show that the short-time solution obtained previously can be extended
uniquely to a maximum life time T∗
. This step is nontrivial as the fixed point argument
only works in short-time. Normally this requires a further argument using Strichartz
estimates, as explained carefully for the usual NLS in [6]. Note that the Strichartz estimates
with inverse square potentials are more subtle than that of the usual NLS,
and indeed there is no end-point estimates as proved by Burq, Planchon, Stalker, and
Tahvildar-Zadeh [5]. Fortunately, the following non-end-point estimates are sufficient for
our purpose.
As usual, let d denote the dimension of the space Rd
, we call a pair (q, r) admissible
if
q, r 2,
2
q
+
d
r
=
d
2
, (q, r, d) = (2, ∞, 2). (4.1)
We recall the following results of Suzuki [32, Proposition 4.8] (see also [23, Corollary
2.3]).
2
Conditions (G1)–(G5) in [25, Theorem 2.2] are verified under the assumption p < 2∗
.
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 27
Lemma 9 (Strichartz estimates). Assume (q, r) and (q, r) are admissible pairs and q, q >
2. Let I ⊂ R be a time interval containing 0. Then the following estimates hold
e−itH
ψ Lq(I;Lr(Rd)) C ψ L2(Rd), (4.2)
t
0
e−i(t−s)H
Ψ(s)
Lq(I;Lr(Rd))
C Ψ Lq (I;Lr (Rd)), (4.3)
for all ψ ∈ L2
(Rd
) and Ψ ∈ Lq
(I; Lr
(Rd
)).
Using the above Strichartz estimates we obtain the following technical result.
Lemma 10. Assume that u, v ∈ C([0, T); Q) ∩ C1
([0, T); Q∗
) be two weak solutions of
(1.1) in (0, T), possible with different initial data u(0) and v(0), such that
u(τ) = v(τ), for some τ ∈ [0, T).
Then there exists θ ∈ (0, T − τ) such that
u(t) = v(t), for all t ∈ [τ, τ + θ].
Proof. By Duhamel’s formula we can write, for 0 t T − τ,
u(t + τ) = e−i(τ+t)H
u0 + i
τ+t
0
e−i(τ+t−s)H
|u(s)|p−1
u(s)ds
= e−itH
⎛
⎝e−iτH
u0 + i
τ
0
e−i(τ−s)H
|u(s)|p−1
u(s)ds
⎞
⎠
+ i
t
0
e−i(t−s)H
|u(s + τ)|p−1
u(s + τ)ds
= e−itH
u(τ) + i
t
0
e−i(t−s)H
|u(s + τ)|p−1
u(s + τ)ds.
Similarly, we have
v(t + τ) = e−itH
v(τ) + i
t
0
e−i(t−s)H
|v(s + τ)|p−1
v(s + τ)ds.
Let θ ∈ (0, T − τ] and put u(t) = u(t + τ) and v(t) = v(t + τ). For t ∈ [0, θ],
28 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
|u(t) − v(t)| =
t
0
e−i(t−s)H
(|u(s)|p−1
u(s) − |v(s)|p−1
v(s))ds .
Therefore, by using (4.3), the elementary inequality
||a|p−1
a − |b|p−1
b| p(|a|p−1
+ |b|p−1
)|a − b| ∀a, b ∈ R
and Hölder inequality, we obtain
u(t) − v(t) Lq((0,θ);Lr(Rd))
C |u|p−1
u − |v|p−1
v Lq ((0,θ);Lr (Rd))
Cθ
q−q
qq u p−1
L∞((0,θ);Lr(Rd))
+ v p−1
L∞((0,θ);Lr(Rd))
u − v Lq((0,θ);Lr(Rd))
Cθ
q−q
qq u p−1
L∞((0,T );Q) + v p−1
L∞((0,T );Q) u − v Lq((0,θ);Lr(Rd)).
Let θ0 > 0 be such that
Cθ
q−q
qq
0 u p−1
L∞((0,T );Q) + v p−1
L∞((0,T );Q) =
1
2
.
Here we note that θ0 does not depend on τ.
Hence, for any θ min{θ0, T − τ}, it follows that
u(t) − v(t) Lq((0,θ);Lr(Rd))
1
2
u(t) − v(t) Lq((0,θ);Lr(Rd)),
which in turn implies u = v on [0, θ]. Therefore u = v in [τ, τ + θ]. This completes the
proof of the technical lemma.
Now we can conclude the uniqueness.
Lemma 11 (Uniqueness). For any given initial datum u0 ∈ Q and for any T > 0, the
equation (1.1) has at most one weak solution u ∈ C([0, T); Q) ∩ C1
([0, T); Q∗
) in (0, T).
Proof. Assume that u, v ∈ C([0, T); Q) ∩ C1
([0, T); Q∗
) are two solutions with the same
initial datum u0. Set
τ∗
:= sup{τ ∈ (0, T) : u = v in (0, τ)}.
We know that 0 < τ∗
T. We suppose by contradiction that τ∗
< T. Then there exist
τ and ε such that 0 < ε < θ0 and τ < τ∗
< min{τ + ε, T − ε} and u = v in (0, τ]. By
Lemma 10, one can choose θ = min{θ0, T − τ∗
} such that u = v in [τ, τ + θ]. Therefore
u = v in (0, τ + θ]. However,
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 29
τ + θ = τ + min{θ0, T − τ∗
} > τ∗
,
which leads to a contradiction. Thus τ∗
= T and consequently u = v in [0, T).
Unique continuation. Next, for every given initial datum u0 ∈ Q, we can define
T∗
= T∗
(u0) := sup{ T > 0 : there exists a local weak solution of (1.1) in (0, T)}.
From the above analysis we obtain the uniqueness of the weak solution in (0, T∗
). This,
combined with [25, Theorem 2.3], implies that u ∈ C([0, T∗
); Q)∩C1
([0, T∗
); Q∗
). Moreover,
the conservation laws in (2.9) hold for every t ∈ (0, T∗
). Thus in summary, for
given u0 ∈ Q, there exists a unique weak solution u of (1.1) in the maximal time interval
[0, T∗
).
Blow-up alternative. Now for any u0 ∈ Q, let u be a weak solution of (1.1) in the maximal
time interval [0, T∗
). We prove that if T∗
< ∞, then
lim
t T ∗
√
Hu L2 = ∞. (4.4)
We observe from [25, Theorem 2.2] that for any τ ∈ (0, T∗
) and ϕ ∈ Q with ϕ Q M
for some M > 0, there exists TM > 0 independent of τ such that problem
i∂tu(t, x) = Hu(t, x) − |u(t, x)|p−1
u(t, x) x ∈ Rd
, t > τ,
u(τ, x) = ϕ(x) x ∈ Rd
,
(4.5)
admits a local weak solution in (τ, τ + TM ).
If T∗
< ∞, we suppose by contradiction that there exist M > 0 and an increasing
sequence {τk} converging to T∗
such that u(τk) Q M for all k 1. We can choose
k large enough such that τk + TM > T∗
. By the above observation, problem (4.5) with
τ = τk admits a solution in (τk, τk + TM ). Consequently, problem (1.1) has a weak
solution in (0, τk + TM ), which contradicts the maximality of T∗
. Thus (4.4) holds true.
4.2. Global existence
Now we come to part (ii) of Theorem 3. By the blowup alternative, it is sufficient
to show that
√
Hu(t) L2 remains bounded uniformly in t. Our starting point is the
following estimate
E(u0) = E(u(t)) =
1
2
√
Hu(t) 2
L2 −
1
p
u(t) p
Lp
1
2
√
Hu(t) 2
L2 −
1
pCp
HGN
√
Hu(t) pθ
L2 u(t)
p(1−θ)
L2
=
1
2
√
Hu(t) 2
L2 −
1
pCp
HGN
√
Hu(t) pθ
L2 u0
p(1−θ)
L2 (4.6)
30 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
which follows from the conservation laws (2.9) and the Hardy-Gagliardo-Nirenberg inequality
(2.6). Here recall that θ = d/2 − d/p.
Case 1: 2 < p < 2 + 4/d. In this case pθ = pd/2 − d < 2, and hence for any ε > 0 small
we have
√
Hu(t)
pd/2−d
L2 ε
√
Hu(t) 2
L2 + Cε.
Inserting this in (4.6) implies that
√
Hu(t) L2 is bounded uniformly in t.
Case 2: p = 2 + 4/d. In this case, pθ = 2 and the sharp constant in (2.7) satisfies
pCp
HGN = 2 Q
4
d
L2 .
Therefore, the lower bound (4.6) boils down to
E(u0) = E(u(t))
1
2
√
Hu(t) 2
L2 −
1
2
√
Hu(t) 2
L2
u0 L2
Q L2
4
d
.
Consequently, if u0 L2 < Q L2 , then
√
Hu(t) L2 is bounded uniformly in t.
Case 3: 2 + 4/d < p < 2∗
. Multiplying (4.6) by u0
q
L2 with q determined by
p(1 − θ) + q =
qpθ
2
, namely q =
4d − 2p(d − 2)
dp − 2d − 4
we obtain
E(u0) u0
q
L2
1
2
√
Hu(t) 2
L2 u0
q
L2 −
1
pCp
HGN
√
Hu(t) 2
L2 u0
q
L2
pθ
2
= f(
√
Hu(t) 2
L2 u0
q
L2 ) (4.7)
with
f(s) :=
1
2
s −
1
pCp
HGN
s
pθ
2 , s 0.
Following the argument of Holmer-Roudenko [13] (the function f in [13] is defined slightly
different from ours), we will use the fact that f is strictly increasing in [0, s0] and strictly
decreasing in [s0, ∞) where
s0 =
Cp
HGN
θ
2
pθ−2
=
√
HQ 2
L2 Q q
L2 .
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 31
Now we prove that if
E(u0) u0
q
L2 < E(Q) Q q
L2 ,
√
Hu0
2
L2 u0
q
L2 < s0, (4.8)
then
√
Hu(t) 2
L2 u0
q
L2 < s0 (4.9)
for all t > 0. First, (4.9) holds at t = 0 by the second condition in (4.8). Moreover, from
(4.7) and the first condition in (4.8) it follows that
f(
√
Hu(t) 2
L2 u0
q
L2 ) E(u0) u0
q
L2 < E(Q) Q q
L2 = f(s0).
Therefore, since f is strictly increasing in [0, s0], by the continuity of t →
√
Hu(t) 2
L2
u0
q
L2 we conclude that
√
Hu(t) 2
L2 u0
q
L2 will never reach the maximum point s0,
namely (4.9) holds true for all t. Consequently, (4.9) implies that
√
Hu(t) L2 is bounded
uniformly in t, which ensures the global existence of u(t).
4.3. Finite time blowup
We will use the following result of Suzuki [33, Subsection 3.1].
Lemma 12 (Virial identities). Let d 3 and 2 < p < 2∗
. Let u ∈ Q be a solution of (1.1)
on [0, T). If |x|u0 ∈ L2
(Rd
), then |x|u(t, x) ∈ L2
(Rd
) for all t ∈ [0, T) and the function
Γ(t) :=
Rd
|x|2
|u(t, x)|2
dx (4.10)
satisfies the following identities for all t ∈ [0, T),
Γ (t) = 4
Rd
xu(t, x) · ∇u(t, x)dx, (4.11)
Γ (t) = 16E(u0) +
4 + 2d − dp
p
Rd
|u(t, x)|p
dx. (4.12)
Moreover, for any v ∈ Q and for any real-valued, radial function ϕ such that |x|ϕv ∈
L2
(Rd
) we have
Rd
xϕv(t, x) · ∇v(t, x)dx xϕv L2 (
√
Hv L2 + v L2 ). (4.13)
32 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
Note that (4.13) ensures that the right side of (4.11) is finite as soon as u(t) ∈ Q.
Now we prove part (iii) of Theorem 3. Assume that the solution u(t) of (1.1) exists
on [0, T). We will show that, with Γ(t) in (4.10),
Γ (t) −λ < 0 (4.14)
for all t ∈ [0, T), where λ > 0 is a constant depending only on u0. Note that by Taylor’s
expansion
0 Γ(t) = Γ(0) + tΓ (0) +
t2
2
Γ (st)
(for some st ∈ [0, t]) the bound (4.14) implies that
0 Γ(t) Γ(0) + tΓ (0) −
t2
2
λ
for all t ∈ [0, T). Since the latter bound cannot hold true for large t, we conclude that
u(t) must blow up at a finite time.
It remains to prove (4.14). If E(u0) < 0, then (4.14) follows immediately from the
Virial identity (4.12) and the fact that 4 + 2d − dp 0:
Γ (t) = 16E(u0) +
4 + 2d − dp
p
Rd
|u(t, x)|p
dx 16E(u0) < 0.
Now instead of E(u0) < 0, we assume
E(u0) u0
q
L2 < E(Q) Q q
L2 ,
√
Hu0
2
L2 u0
q
L2 >
√
HQ 2
L2 Q q
L2 .
We use again the argument of Holmer-Roudenko [13]. Note that
√
Hu0
2
L2 u0
q
L2 >
√
HQ 2
L2 Q q
L2 = s0
at time t = 0, and
f(
√
Hu(t) 2
L2 u0
q
L2 ) E(u0) u0
q
L2 < E(Q) Q q
L2 = f(s0)
for all t < T due to (4.7). Since f is strictly decreasing in [s0, ∞), by the continuity of
t →
√
Hu(t) 2
L2 u0
q
L2 we conclude that
√
Hu(t) 2
L2 u0
q
L2 > s0 (4.15)
for all t < T. Finally, multiplying the Virial identity (4.12) with u0
q
L2 , then using
(4.15) together with the facts that p > 2 and 4 − d(p − 2) 0 we obtain
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 33
Γ (t) u0
q
L2 = 4d(p − 2)E(u0) u0
q
L2 + 2(4 − d(p − 2))
√
Hu(t) 2
L2 u0
q
L2
4d(p − 2)E(u0) u0
q
L2 + 2(4 − d(p − 2))
√
HQ 2
L2 Q q
L2
= 4d(p − 2) E(u0) u0
q
L2 − E(Q) Q q
L2 < 0. (4.16)
Here in the last equality we have used
4d(p − 2)E(Q) Q q
L2 + 2(4 − d(p − 2))
√
HQ 2
L2 Q q
L2 = 0.
Thus (4.14) holds true, and hence u(t) blows up at a finite time. This ends the proof of
Theorem 3.
5. Minimal mass blowup solutions
5.1. Compactness of minimizing sequences
In this subsection we offer another, free-rearrangement proof of the existence of
minimizers of the Hardy-Gagliardo-Nirenberg inequality (2.6). This proof implies an important
consequence, that any (normalized) minimizing sequence of (2.6) is pre-compact
(without the radial assumption). This will be a crucial ingredient of our analysis of finite
time blow-up solutions in Theorem 4. For the completeness we will work on the general
case 2 < p < 2∗
(instead of focusing on the mass-critical case p = 2 + 4/d). We have
Theorem 13 (Compactness of minimizing sequences). Let d 3 and 2 < p < 2∗
. Let
{un} be a minimizing sequence for the Hardy-Gagliardo-Nirenberg inequality (2.6) such
that
lim inf
n→∞
un L2 > 0, lim sup
n→∞
un Q < ∞.
Then there exist a subsequence of {un} and constants λ > 0, z ∈ C such that
un(x) → zQ(λx) strongly in Q.
Proof. By dilations, we can assume that
un L2 =
√
Hun L2 = 1, un Lp → C−1
HGN.
Since {un} is bounded in Q, thanks to (2.3) and Sobolev’s embedding theorem we have,
up to a subsequence when n → ∞, there exists u0 ∈ Q such that
un u0 weakly in Q and Hs
(Rd
) for all 0 < s < 1,
un(x) → u0(x) for a.e. x ∈ Rd
,
1B(0,R)un → 1B(0,R)u0 strongly for all 2 p < 2∗
, for all R > 0.
34 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
We need to show that u0 L2 = 1. This will imply that un → u0 strongly in L2
(Rd
),
and hence un → u0 strongly in Lq
(Rd
) for all 2 q < 2∗
by interpolation, which allows
us to conclude by Fatou’s lemma as in the above argument.
We assume, for the sake of contradiction, that u0 L2 < 1. Then from the local
convergence, we can find a sequence Rn → ∞ such that when n → ∞,
Rn |x| 2Rn
|un(x)|2
dx → 0,
|x| Rn
|un(x)|2
dx → m < 1.
Fix smooth functions χ, η : Rd
→ [0, 1] such that
χ2
+ η2
= 1, χ(x) = 1 if |x| 1, χ(x) = 0 if |x| 2
and define
χn(x) = χ(x/Rn), ηn(x) = η(x/Rn), n ∈ N.
By the IMS formula we have the decomposition
un, Hun = χnun, H(χnun) + ηnun, −Δ(ηnun)
− c∗
Rd
η2
n
|x|2
|un(x)|2
dx −
Rd
(|∇χn(x)|2
+ |∇ηn(x)|2
)|un(x)|2
dx
χnun, H(χnun) + ηnun, −Δ(ηnun) + o(1)n→∞. (5.1)
Then by Hölder’s inequality,
un, Hun
θ
⎛
⎝
Rd
|un|2
dx
⎞
⎠
1−θ
χnun, H(χnun) + ηnun, −Δ(ηnun) + o(1)n→∞
θ
×
×
⎛
⎝
Rd
|χnun|2
dx +
Rd
|ηnun|2
dx
⎞
⎠
1−θ
χnun, H(χnun)
θ
⎛
⎝
Rd
|χnun|2
dx
⎞
⎠
1−θ
+ ηnun, −Δ(ηnun)
θ
⎛
⎝
Rd
|ηnun|2
dx
⎞
⎠
1−θ
+ o(1)n→∞. (5.2)
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 35
The first term on the right-hand side of (5.2) can be estimated using (2.6):
χnun, H(χnun)
θ
⎛
⎝
Rd
|χnun|2
dx
⎞
⎠
1−θ
C2
HGN χnun
2
Lp .
For the second term on the right-hand side of (5.2), we use
ηnun, −Δ(χnun)
θ
⎛
⎝
Rd
|ηnun|2
dx
⎞
⎠
1−θ
C2
GN ηnun
2
Lp
where
CGN = inf
ϕ=0
√
−Δϕ θ
L2 ϕ 1−θ
L2
ϕ Lp
.
Since it is well-known that CGN has a minimizer (which is indeed unique up to translations
and dilations), we must have CGN > CHGN. Denote
C2
GN
C2
HGN
= 1 + ε0 > 1.
Thus in summary, from (5.2) we deduce that
un, Hun
θ
⎛
⎝
Rd
|un|2
dx
⎞
⎠
1−θ
C2
HGN χnun
2
Lp + (1 + ε0) ηnun
2
Lp + o(1)n→∞.
(5.3)
Next, from
Rn |x| 2Rn
|un(x)|2
dx → 0
we deduce that
Rn |x| 2Rn
|un(x)|p
dx → 0.
Combining with the elementary estimate as
+bs
(a+b)s
with a, b 0 and s = 2/p < 1,
we obtain
36 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
χnun
2
Lp + ηnun
2
Lp =
⎛
⎝
Rd
|χnun|p
dx
⎞
⎠
2
p
+
⎛
⎝
Rd
|ηnun|p
dx
⎞
⎠
2
p
⎛
⎝
Rd
|χnun|p
dx +
Rd
|ηnun|p
dx
⎞
⎠
2
p
=
⎛
⎝
Rd
|un|p
dx + o(1)n→∞
⎞
⎠
2
p
= un
2
Lp + o(1)n→∞.
Therefore, (5.3) reduces to
un, Hun
θ
⎛
⎝
Rd
|un|2
dx
⎞
⎠
1−θ
C2
HGN un
2
Lp + ε0 ηnun
2
Lp + o(1)n→∞. (5.4)
Since {un} is a minimizing sequence for (2.6), (5.4) implies that
ηnun
2
Lp → 0,
and hence
χnun
2
Lp = un
2
Lp + o(1)n→∞. (5.5)
To conclude, we come back to use (5.1):
un, Hun χnun, H(χnun) + o(1)n→∞
and the fact
Rd
|χnun|2
dx → m < 1 =
Rd
|un|2
dx
together with (2.6) and (5.5). All the above estimates give
un, Hun
θ
⎛
⎝
Rd
|un|2
dx
⎞
⎠
1−θ
( χnun, H(χnun) + o(1)n→∞)
θ Rd |χnun|2
dx + o(1)n→∞
m
1−θ
mθ−1
C2
HGN χnun
2
Lp + o(1)n→∞
mθ−1
C2
HGN un
2
Lp + o(1)n→∞ + o(1)n→∞.
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 37
Thus
un, Hun
θ
Rd |un|2
dx
1−θ
un
2
Lp
mθ−1
C2
HGN + o(1)n→∞.
Since m < 1, this is a contradiction to the fact that {un} is a minimizing sequence for
(2.6).
Thus we must have un → u0 strongly in L2
(Rd
), and hence un → u0 in Lp
(Rd
) by
interpolation. This allows us to conclude that u0 is a minimizer, and that un → u0
strongly in Q. By Theorem 2, we know that u0(x) = zQ(λx) for some constants λ > 0
and z ∈ C. This ends the proof.
5.2. Blowup profile at t → T
Now we come back to Theorem 4. We will focus on the mass-critical case p = 2+4/d,
where the sharp constant in the Hardy-Gagliardo-Nirenberg inequality (2.6) is
CHGN =
d
d + 2
d
2(d+2)
Q
2
d+2
L2 . (5.6)
We will prove
Lemma 14 (Mass concentration as t → T). Assume p = 2 + 4/d. Let u be a solution
of (1.1) in [0, T) such that u0 L2 = Q L2 and limt T u(t) Q = ∞. Let {tn} be an
increasing sequence converging to T and denote un(x) = u(tnx). Then
|un|2
→ Q 2
L2 δ0 in (D(Rd
))
in the sense of distributions.
Here (D(Rd
)) denotes the space of distributions in Rd
and δ0 is the Dirac measure
concentrated at x = 0.
Proof. Denote
vn(x) = λ
d
2
n un(λnx) with λn =
√
HQ L2
√
Hun L2
.
Then limn→∞ λn = 0 and
vn L2 = un L2 = u0 L2 = Q L2 ,
√
Hvn L2 = λn
√
Hun L2 =
√
HQ L2 ,
vn
2+ 4
d
L2+ 4
d
= λ2
n un
2+ 4
d
L2+ 4
d
.
38 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
Using the above identities and the energy conservation, we obtain
E(vn) =
1
2
√
Hvn
2
L2 −
d
2(d + 2)
vn
2+ 4
d
L2+ 4
d
= λ2
n
1
2
√
Hun
2
L2 −
d
2(d + 2)
un
2+ 4
d
L2+ 4
d
= λ2
nE(un) = λ2
nE(u0) → 0.
Consequently,
lim
n→∞
vn
2+ 4
d
L2+ 4
d
= lim
n→∞
d + 2
d
√
Hvn L2 =
d + 2
d
√
HQ L2 ,
and hence
lim
n→∞
√
Hvn
d
d+2
L2 vn
2
d+2
L2
vn
L2+ 4
d
=
d
d + 2
d
2(d+2)
Q
2
d+2
L2 = CHGN.
It means that {vn} is a minimizing sequence for (2.6).
By Theorem 13, there exist a subsequence, still denoted by {vn}, and constants λ > 0,
z ∈ C such that vn(x) → zQ(λx) strongly in Q. Since
vn L2 = Q L2 ,
√
Hvn L2 =
√
HQ L2
we know that
λ = |z| = 1.
In particular, we obtain
|vn|2
→ |Q|2
in L1
(Rd
).
Next, for any φ ∈ C∞
c (Rd
), we can write
|un|2
, φ =
Rd
|un(y)|2
φ(y)dy
=
Rd
|vn(x)|2
φ(λnx)dx
=
Rd
(|vn(x)|2
− |Q(x)|2
)φ(λnx)dx +
Rd
|Q(x)|2
φ(0)dx
+
Rd
|Q(x)|2
(φ(λnx) − φ(0))dx.
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 39
It follows that
| |un|2
, φ − Q 2
L2 φ(0)| φ L∞
Rd
||vn(x)|2
− |Q(x)|2
|dx
+
Rd
|Q(x)|2
|φ(λnx) − φ(0)|dx. (5.7)
Since |vn|2
→ Q2
in L1
(Rd
), the first term on the right-hand side of (5.7) tends to zero
as n → ∞. Moreover, since λn → 0 and φ ∈ C∞
c (Rd
) it follows that φ(λnx) → φ(0) as
n → ∞ for all x ∈ Rd
. By invoking dominated convergence theorem, we derive that the
second term on the right-hand side of (5.7) tends to zero as n → ∞. As a consequence,
|un|2
→ Q 2
L2 δ0 in (D(Rd
)) .
5.3. Virial identities
To conclude the proof, we will use the Virial identities in Lemma 12. Strictly speaking,
the results in Lemma 12 hold under the condition |x|u0 ∈ L2
(Rd
), which is not available
here. However, this condition can be relaxed in the mass-critical case.
Lemma 15 (Virial identity in the mass-critical case). Let u be a solution of (1.1) in [0, T)
such that u0 L2 = Q L2 and limt T u(t) Q = ∞. Then for all t ∈ [0, T) we have
|x|u(t) ∈ L2
(Rd
) and
Rd
|x|2
|u(t, x)|2
dx = 8E(u0)(T − t)2
. (5.8)
First, by using the a-priori estimate (4.13) in Lemma 12 we can easily adapt a result
of Banica [1, Lemma 2.1] to our case.
Lemma 16. Assume p = 2 + 4/d. Let u ∈ Q such that u L2 = Q L2 . Then for any
θ ∈ C∞
0 (Rd
), there holds
Rd
∇θ · (¯u∇u)dx 2E(u)
⎛
⎝
Rd
|∇θ|2
|u|2
dx
⎞
⎠
1
2
.
Now we provide
Proof of Lemma 15. Let φ ∈ C∞
0 (Rd
) be a radial, non-negative function such that
φ(x) = |x|2
for |x| 1. Since φ is non-negative radially symmetric, we can write
φ(x) = ζ(r) with r = |x|.
40 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
By Taylor’s Theorem for all r, ρ ∈ R, there exists r ∈ [r, r + ρ] such that
0 ζ(r + ρ) = ζ(r) + ρζ (r) +
ρ2
2
ζ (r) ζ(r) + ρζ (r) + c1ρ2
, (5.9)
where c1 = 1 + maxr∈R
|ζ (r)|
2 . We note that, the right-hand side of (5.9) is a second
degree polynomial in ρ, hence |ζ (r)|2
− 4c1ζ(r) 0, which implies |ζ (r)|2
Cζ(r) with
C = 4c1. Therefore, we have
|∇φ(x)|2
Cφ(x) for x ∈ Rd
. (5.10)
For R > 0, define ψR(x) = R2
ψ( x
R ) and
ΓR(t) :=
R
ψR(x)|u(t, x)|2
dx ∀t ∈ [0, T).
An easy computation yields
ΓR(t) = 2
Rd
∇ψR (¯u∇u)dx ∀t ∈ [0, T).
Since u(t) L2 = Q L2 , it follows from Lemma 16 and (5.10) that
|ΓR(t)| 2 2E(u)
Rd
|∇ψR|2
|u|2
dx
1
2
2 2E(u0)
Rd
C2
ψR|u|2
dx
1
2
C E(u0) ΓR(t).
Integrating between fixed t ∈ [0, T) and tn we obtain,
| ΓR(t) − ΓR(tn)| C|t − tn|. (5.11)
By the mass concentration in Lemma 14 we derive
ΓR(tn) =
Rd
ψR(x)|un(x)|2
dx → Q 2
L2 ψR(0) = 0. (5.12)
By letting n → ∞ in (5.12) and employing (5.11), we deduce that ΓR(t) C(T − t)2
.
This implies
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 41
Rd
ψR(x)|u(t, x)|2
dx C(T − t)2
.
Letting R → ∞ and using monotone convergence theorem lead to |x|u(t) ∈ L2
(Rd
) for
all t ∈ [0, T) and
Γ(t) :=
Rd
|x|2
|u(t, x)|2
dx C(T − t)2
.
This allows to extend Γ(t) by continuity at t = T by setting Γ(T) = 0 and consequently
Γ (T) = 0. We obtain the (4.12), which in the mass-critical case p = 2 + 4/d boils down
to
Γ (t) = 16E(u0).
Combining with Γ(T) = Γ (T) = 0 we find that
Γ(t) = 8E(u0)(T − t)2
.
Consequently,
Γ(0) =
Rd
|x|2
|u0|2
dx = 8E(u0)T,
Γ (0) = 4
Rd
xu0(x) · ∇u0(x)dx = −16E(u0)T.
5.4. Conclusion
For any T > 0, λ > 0 and γ ∈ R, define
ST,λ,γ(t, x) := eiγ
ei λ2
T −t e−i |x|2
4(T −t)
λ
T − t
d
2
Q
λx
T − t
x ∈ Rd
, t ∈ [0, T). (5.13)
It is straightforward to check that for any T > 0, λ > 0 and γ ∈ R, the function ST,λ,γ
is a minimal-mass solution of (1.1) which blows up at finite time T > 0.
Next we conclude by using the strategy of Hmidi-Keraani [12]. We observe that for
any u ∈ Q, for any real-valued function θ ∈ C∞
0 (Rd
, R) and s ∈ R, there holds
E(ueisθ
) = E(u) + s
Rd
u∇θ · ∇udx +
s2
2
Rd
|∇θ|2
|u|2
dx. (5.14)
We can take s = 1/(2T) and choose θ(x) approaching |x|2
/2 (using appropriate cut-off
functions). This gives
42 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
E(u0e
i|x|2
4T ) = E(u0) +
1
2T
Rd
xu0 · ∇u0dx +
1
8T2
Rd
|x|2
|u0|2
dx
= E(u0) −
4E(u0)
2T
+
1
8T2
(8E(u0)T2
) = 0.
Here we have used the Virial identity (5.8) in the last equality. Thus the function
v0 = u0e
i|x|2
4T
satisfies that v0 L2 = Q L2 and E(v0) = 0. Hence v0 is a minimizer for the HardyGagliardo-Nirenberg
inequality (2). By Theorem 2, there exist λ1 > 0, γ1 ∈ R such
that
u0(x)e
i|x|2
4T = eiγ1
λ
d
2
1 Q(λ1x), x ∈ Rd
,
which is equivalent to
u0(x) = eiγ1
e− i|x|2
4T λ
d
2
1 Q(λ1x) x ∈ Rd
.
Define λ0 = λ1T > 0, γ0 = γ1 − λ2
1T, then we can write
u0(x) = eiγ0
ei
λ2
0
T e−i |x|2
4T
λ0
T
d
2
Q
λ0x
T
= ST,λ0,γ0
(0, x) x ∈ Rd
.
By the uniqueness, we conclude that u(t, x) = ST,λ0,γ0
(t, x) for all t ∈ [0, T). This
completes the proof of Theorem 4.
6. Extension to the case c|x|−2
with c < c∗
Instead of (1.1), we may also consider the NLS with subcritical inverse square potential
i∂tu(t, x) = (−Δ − c|x|−2
)u(t, x) − |u(t, x)|p−2
u(t, x), x ∈ Rd
, t > 0,
u(0, x) = u0(x), x ∈ Rd
,
(6.1)
with d 3, 2 < p < 2∗
= 2d/(d − 2) and
c < c∗ =
(d − 2)2
4
.
All the results in Theorems 1, 2, 3, 4 hold true in this subcritical case. In fact, the
proof in this case is often simpler since now the quadratic form domain of −Δ − c|x|−2
is simply H1
(Rd
).
D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092 43
Let us quickly explain how to adapt our proof to this case. We will focus on Theorem 1,
namely the existence and uniqueness of the positive radial solution to
−ΔQ − c|x|−2
Q − Qp−1
+ Q = 0. (6.2)
(our result is new for the case c < c∗ as well). The existence part is easy and we only
consider the uniqueness part. The main difference when c < c∗ is that we have the
following analogue of the asymptotic formula (3.6)
lim
|x|→0
|x|κ
Q(x) ∈ (0, ∞) (6.3)
with
κ :=
d − 2
2
−
d − 2
2
2
− c. (6.4)
Then in the proof of the uniqueness of Q, when c < c∗ we will replace (3.18) by
v(r) = rκ
Q(r),
and proceed exactly the same as in the critical case to get the desired result.
Let us explain (6.3) in more detail. If Q ∈ H1
(Rd
) is a positive radial solution of (6.2),
then using the polar coordinate r = |x| we find that Q satisfies
d2
dr2
Q +
N − 1
r
d
dr
Q +
c
r2
Q − Q + Qp−1
= 0 in (0, ∞).
Put
:=
d − 2 − 2κ
d − 2
and
W(s) := rκ
Q(r) with s = r ,
then W satisfies
d2
ds2
W +
d − 1
s
d
ds
W −
1
2
s
2(1− )
W +
1
2
s
2(1− )−κ(p−2)
Wp−1
= 0 in (0, ∞).
Equivalently, we can write
ΔW −
1
2
|x|
2(1− )
W +
1
2
|x|
2(1− )−κ(p−2)
Wp−1
= 0 in Rd
\ {0}.
44 D. Mukherjee et al. / Journal of Functional Analysis 281 (2021) 109092
Since ∈ (0, 1) and p < 2∗
= 2d
d−2 , we have
lim
|x|→0
|x|
2(1− )
= lim
|x|→0
|x|
2(1− )−κ(p−2)
= 0.
Therefore, by a similar argument as in [36, Proof of Theorem 1.2], we deduce that W(0)
is well defined as a positive number. Thus (6.3) holds true.
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