Uniqueness of ground state and minimal-mass blow-up solutions for focusing NLS with Hardy potential

D. Mukherjee; P.T. Nam; P.T. Nguyen

arxiv: 1907.06964 · v1 · pith:JEILEN2Ynew · submitted 2019-07-16 · 🧮 math.AP · math-ph· math.MP

Uniqueness of ground state and minimal-mass blow-up solutions for focusing NLS with Hardy potential

D. Mukherjee , P.T. Nam , P.T. Nguyen This is my paper

Pith reviewed 2026-05-24 20:56 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MP

keywords focusing NLSHardy potentialground state uniquenessblow-up solutionsGagliardo-Nirenberg inequalityminimal massinverse square potential

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The pith

The ground state solution of the focusing NLS with critical Hardy potential is unique.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that there is only one ground state solution, up to symmetries, for the focusing nonlinear Schrödinger equation that includes the critical inverse-square potential. This uniqueness directly produces the sharp constant in the Hardy-Gagliardo-Nirenberg inequality. The result also gives a precise description of all solutions that blow up in finite time with the smallest possible mass. Readers interested in dispersive PDEs care because the ground state sets the threshold separating solutions that exist for all time from those that blow up.

Core claim

We prove the uniqueness of the ground state solution for the focusing NLS equation with the critical inverse square potential. As a consequence we obtain the sharp Hardy-Gagliardo-Nirenberg interpolation inequality. We also give a complete characterization of the minimal mass blow-up solutions for the time-dependent problem.

What carries the argument

The ground state as the unique minimizer of the energy functional under a mass constraint.

If this is right

The Hardy-Gagliardo-Nirenberg inequality holds with a sharp constant determined by the ground state.
Minimal-mass blow-up solutions are completely classified for the time-dependent equation.
Existence and uniqueness together determine the exact blow-up threshold.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If uniqueness holds, similar arguments might apply to other singular potentials.
The characterization could be tested numerically by simulating low-mass initial data.
Extensions to higher dimensions or different nonlinearities remain open.

Load-bearing premise

A ground state minimizer is known to exist from earlier results.

What would settle it

Construction or numerical discovery of two distinct positive solutions with the same minimal energy and mass would disprove the uniqueness claim.

read the original abstract

We consider the focusing nonlinear Schr\"odinger 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives the first uniqueness proof for the ground state of focusing NLS with critical Hardy potential, but it rests on existence imported from prior work.

read the letter

The paper claims the first uniqueness result for the ground state solution of the focusing NLS with the critical inverse square potential. From that they derive a sharp Hardy-Gagliardo-Nirenberg inequality and give a complete characterization of the minimal mass blow-up solutions for the evolution equation. What the paper does well is to close the uniqueness gap left by earlier existence-only results. The abstract is direct about this being new, and the consequence for the inequality and the blow-up analysis is a natural next step that organizes the theory in this setting. The soft spots are limited but worth noting. The existence of the minimizer is taken from the prior literature, so the uniqueness proof applies conditionally to that object. One should check whether the cited existence holds precisely at the critical value of the Hardy coefficient and for the exact range of the nonlinearity. If there is any mismatch in the assumptions, the uniqueness statement would need adjustment. The abstract does not indicate any other technical problems, and there is no sign of circular reasoning or self-referential definitions. The math appears to be in the standard variational framework for these problems, with no free parameters or invented entities. This work is aimed at experts in nonlinear Schrödinger equations and dispersive PDEs. Readers who follow ground state uniqueness and its implications for stability and blow-up will get the most out of it. It is the kind of targeted advance that merits a serious referee report rather than a desk rejection. I would bring this to a reading group focused on dispersive equations. I would not cite it in my own work unless I am working directly in this area. It should go to peer review.

Referee Report

2 major / 1 minor

Summary. The manuscript proves uniqueness of the ground state solution to the focusing NLS equation with critical inverse-square Hardy potential. From this it derives a sharp Hardy-Gagliardo-Nirenberg inequality and gives a complete characterization of the minimal-mass blow-up solutions for the associated time-dependent problem.

Significance. If the uniqueness result holds under the precise critical parameters, the work supplies the first such uniqueness statement, yields a sharp interpolation inequality, and furnishes a blow-up characterization that is useful for the study of singular-potential NLS equations.

major comments (2)

[§1] §1 (Introduction) and the statement of the main uniqueness theorem: existence of the ground-state minimizer is imported from prior literature without re-derivation or explicit verification that the cited result applies exactly at the critical Hardy coefficient and the precise range of the nonlinearity; the uniqueness argument therefore stands conditionally on that transfer.
[Main theorem proof] The variational characterization used for uniqueness (presumably in the proof of the main theorem) must be checked to confirm it does not inadvertently rely on strict subcriticality of the potential or on regularity that fails at the critical value; otherwise the claimed uniqueness does not automatically apply to the equation studied here.

minor comments (1)

Notation for the Hardy potential and the energy functional should be introduced once and used consistently; cross-references to the cited existence result should include the precise parameter range stated in that reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We respond to each major comment below.

read point-by-point responses

Referee: [§1] §1 (Introduction) and the statement of the main uniqueness theorem: existence of the ground-state minimizer is imported from prior literature without re-derivation or explicit verification that the cited result applies exactly at the critical Hardy coefficient and the precise range of the nonlinearity; the uniqueness argument therefore stands conditionally on that transfer.

Authors: We agree that an explicit verification strengthens the presentation. The existence result cited in the introduction applies directly at the critical Hardy coefficient μ = (N-2)^2/4 and for the precise range of the nonlinearity exponent p considered here, as these parameters fall within the hypotheses of the referenced theorem. In the revised manuscript we will insert a short paragraph in §1 that confirms the parameter match. revision: yes
Referee: [Main theorem proof] The variational characterization used for uniqueness (presumably in the proof of the main theorem) must be checked to confirm it does not inadvertently rely on strict subcriticality of the potential or on regularity that fails at the critical value; otherwise the claimed uniqueness does not automatically apply to the equation studied here.

Authors: The variational characterization employed in the uniqueness proof is formulated directly with the critical Hardy potential and does not invoke strict subcriticality. The argument relies on the quadratic form associated with the critical inverse-square term together with the concentration-compactness lemma adapted to this setting; the required regularity holds for the admissible range of nonlinearities at criticality. We will add a clarifying remark in the proof section to make this applicability explicit. revision: partial

Circularity Check

0 steps flagged

No significant circularity; uniqueness proof presented as independent of fitted inputs or self-referential definitions

full rationale

The paper's central claim is a first proof of uniqueness for the ground state solution of the focusing NLS with critical Hardy potential, from which a sharp interpolation inequality and characterization of minimal-mass blow-up solutions follow. No equations or steps in the provided abstract reduce a prediction or result to its own inputs by construction, nor does any load-bearing premise collapse to a self-citation chain that itself lacks external verification. Existence of the minimizer is referenced from prior literature (as noted in the reader's summary), but this is standard and does not render the uniqueness argument circular; the derivation remains self-contained against functional-analytic benchmarks external to the present work. No self-definitional, fitted-input, or ansatz-smuggling patterns are exhibited.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only view shows reliance on standard Sobolev and Hardy inequality theory; no free parameters, new entities, or ad-hoc axioms are visible.

axioms (1)

domain assumption Standard existence and properties of ground states for NLS equations with Hardy potential from prior literature.
The uniqueness claim presupposes the ground state exists as a minimizer.

pith-pipeline@v0.9.0 · 5580 in / 1123 out tokens · 20300 ms · 2026-05-24T20:56:35.940598+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages

[1]

Banica , Remarks on the blow-up for the Schr¨ odinger equation with cr itical mass on a plane domain , Ann

V. Banica , Remarks on the blow-up for the Schr¨ odinger equation with cr itical mass on a plane domain , Ann. Sc. Norm. Super. Pisa Cl. Sci. 3 (2004), 139–170

work page 2004
[2]

Bensouilah, V.D

A. Bensouilah, V.D. Dinh, S. Zhu , On stability and instability of standing waves for the nonli near Schr¨ odinger equation with an inverse-square potential, J. Math. Phys., 59 (2018), 101505

work page 2018
[3]

Brezis and J.-L

H. Brezis and J.-L. V ´azquez, Blow-up solutions of some nonlinear elliptic problems , Rev. Mat. Univ. Madrid 10 (1997), 443–469

work page 1997
[4]

Burchard , A Short Course on Rearrangement Inequalities , Lecture Notes 2009

A. Burchard , A Short Course on Rearrangement Inequalities , Lecture Notes 2009. http://www.math.toronto.edu/almut/rearrange.pdf

work page 2009
[5]

N. Burq, F. Planchon, J.G. Stalker and A.S. Tahvildar-Zadeh , Strichartz estimates for the wave and Schr¨ odinger equations with the inverse-square potential, J. Funct. Anal. 203 (2003), 519–549

work page 2003
[6]

Cazenave , Semilinear Schrodinger Equations , Courant Lecture Notes Volume 10 (2003), American Mathematical Society

T. Cazenave , Semilinear Schrodinger Equations , Courant Lecture Notes Volume 10 (2003), American Mathematical Society

work page 2003
[7]

Coffman ,Uniqueness of the ground state solution for ∆u − u + u3 = 0 and a variational charac- terization of other solutions , Arch

C.V. Coffman ,Uniqueness of the ground state solution for ∆u − u + u3 = 0 and a variational charac- terization of other solutions , Arch. Ration. Mech. Anal. 46 (1972), 81–95

work page 1972
[8]

Csobo and F

E. Csobo and F. Genoud , Minimal mass blow-up solutions for the L2 critical NLS with inverse-square potential, Nonlinear Anal. 168 (2018), 110–129

work page 2018
[9]

R. L. Frank , A simple proof of Hardy-Lieb-Thirring inequalities , Comm. Math. Phys. 290 (2009), 789–800

work page 2009
[10]

Gidas, W.M

B. Gidas, W.M. Ni, L. Nirenberg , Symmetry of positive solutions of nonlinear elliptic equat ions in Rn, Mathematical analysis and applications. Part A, Adv. in Ma th. Suppl. Stud. Vol. 7 (1981), Academic Press, New York, 369–402

work page 1981
[11]

Hmidi and S

T. Hmidi and S. Keraani , Blowup theory for the critical nonlinear Schr¨ odinger equa tions revisited , Int. Math. Res. Not. 46 (2005), 2815–2828

work page 2005
[12]

Holmer and S

J. Holmer and S. Roudenko , On blow-up solutions to the 3d cubic nonlinear Schr¨ odinger equation, AMRX Appl. Math. Res. Express (2007), Art. ID abm004. 34 D. MUKHERJEE, P.T. NAM, AND P.T. NGUYEN

work page 2007
[13]

H. Kalf, U. W. Schmincke, J. W alter, R. W ¨ust, On the spectral theory of Schr¨ odinger and Dirac operators with strongly singular potentials . In Spectral Theory and Diﬀerential Equations, pp. 182-226 . Lecture Notes in Mathematics 448 (1975) Springer, Berlin

work page 1975
[14]

C. E. Kenig, F. Merle , Global well-posedness, scattering and blow-up for the ener gy-critical focusing non-linear wave equation , Acta Math. 201 (2008), 147–212

work page 2008
[15]

Killip, C

R. Killip, C. Miao, M. Visan, J. Zhang, J. Zheng , The energy-critical NLS with inverse-square potential, Discrete Contin. Dyn. Syst., 37 (2017), 3831–3866

work page 2017
[16]

Killip, J

R. Killip, J. Murphy, M. Visan, J. Zheng , The focusing cubic NLS with inverse-square potential in three space dimensions , Diﬀerential Integral Equations, 30 (2017), 161–206

work page 2017
[17]

M. K. Kwong , Uniqueness of positive solutions of ∆u − u + up = 0 in Rn, Arch. Ration. Mech. Anal. 105 (1989), 243–266

work page 1989
[18]

E. H. Lieb and M. Loss , Analysis. Second edition. Graduate Studies in Mathematics , 14. American Mathematical Society, Providence, RI, 2001. xxii+346 pp

work page 2001
[19]

J. Lu, C. Miao, J. Murphy , Scattering in H 1 for the intercritical NLS with an inverse-square potential , J. Diﬀerential Equations, 264, (2018), 3174–3211

work page 2018
[20]

McLeod, J

K. McLeod, J. Serrin , Uniqueness of positive radial solutions of ∆u + f (u) = 0 in Rn, Arch. Ration. Mech. Anal. 99 (1987), 115–145

work page 1987
[21]

Merle , Determination of blow-up solutions with minimal mass for no nlinear Schr¨ odinger equations with critical power , Duke Math

F. Merle , Determination of blow-up solutions with minimal mass for no nlinear Schr¨ odinger equations with critical power , Duke Math. J. 69 (1993) 427–454

work page 1993
[22]

Mizutani , Remarks on endpoint Strichartz estimates for Schr¨ odinger equations with the critical inverse-square potential, J

H. Mizutani , Remarks on endpoint Strichartz estimates for Schr¨ odinger equations with the critical inverse-square potential, J. Diﬀerential Equations 263 (2017), 3832–3853

work page 2017
[23]

P´ olya, G

G. P´ olya, G. Szeg¨ o,Isoperimetric Inequalities in Mathematical Physics , Annals of Mathematics Studies, Princeton University Press (1951)

work page 1951
[24]

Okazaw a, T

N. Okazaw a, T. Suzuki and T. Yokota , Cauchy problem for nonlinear Schr¨ odinger equations with inverse-square potentials, Appl. Anal. 91 (2012), 1605–1629

work page 2012
[25]

Okazaw a, T

N. Okazaw a, T. Suzuki and T. Yokota , Energy methods for abstract nonlinear Schr¨ odinger equa- tions, Evolution equations and control theory 1 (2012), 337–354

work page 2012
[26]

Ogaw a, Y

T. Ogaw a, Y. Tsutsumi, Blow-up of H 1 solution for the nonlinear Schr¨ odinger equation, J. Diﬀ. Equa. 92 (1991), 317–330

work page 1991
[27]

Rodnianski, W

I. Rodnianski, W. Schlag , Time decay for solutions of Schrdinger equations with rough and time- dependent potentials, Invent. Math. 155 (2004), 451–513

work page 2004
[28]

J. P. Solovej, T. Ø. Sørensen and W. L. Spitzer , Relativistic Scott correction for atoms and molecules, Comm. Pure Appl. Math. 63 (2010), 39–118

work page 2010
[29]

W. A. Strauss , Existence of solitary waves in higher dimensions , Commun. Math. Phys. 55 (1977), no. 2, 149–162

work page 1977
[30]

Su, Z.-Q

J. Su, Z.-Q. W ang, M. Willem , Nonlinear Schr¨ odinger equations with unbounded and decaying radial potentials, Commun. Contemp. Math. 9 (2007), 571–583

work page 2007
[31]

Suzuki , Solvability of nonlinear Schr¨ odinger equations with some critical singular potential via gen- eralized Hardy-Rellich inequalities , Funkcialaj Ekvacioj 59 (2016), 1–34

T. Suzuki , Solvability of nonlinear Schr¨ odinger equations with some critical singular potential via gen- eralized Hardy-Rellich inequalities , Funkcialaj Ekvacioj 59 (2016), 1–34

work page 2016
[32]

Suzuki , Virial identities for nonlinear Schr¨ odinger equations with a critical coeﬃcient inverse-square potential, Diﬀer

T. Suzuki , Virial identities for nonlinear Schr¨ odinger equations with a critical coeﬃcient inverse-square potential, Diﬀer. Equ. Appl. 9 (2017), 327–352

work page 2017
[33]

Shioji, K

N. Shioji, K. W atanabe , A generalized Pohozaev identity and uniqueness of positive radial solutions of ∆u + g(r)u + h(r)up = 0, J. Diﬀ. Equa. 255 (2013), 4448–4475

work page 2013
[34]

Tao , Nonlinear dispersive equations, Local and global analysis , CBMS Regional Conference Series in Mathematics, 106 (2006), xvi+373

T. Tao , Nonlinear dispersive equations, Local and global analysis , CBMS Regional Conference Series in Mathematics, 106 (2006), xvi+373

work page 2006
[35]

Tsutsumi , L2-solutions for nonlinear Schr¨ odinger equations and nonlinear groups, Funkcial

Y. Tsutsumi , L2-solutions for nonlinear Schr¨ odinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987), 115–125

work page 1987
[36]

G. P. Trachanas and N. B. Zographopoulos , Orbital stability for the Schr¨ odinger operator involv- ing inverse square potential , J. Diﬀerential Equations 259 (2015), 4989–5016

work page 2015
[37]

J. L. V azquez, E. Zuazua , The Hardy inequality and the asymptotic behaviour of the hea t equation with an inverse-square potential , J. Funct. Anal. 173 (2000), 103-153

work page 2000
[38]

M. I. Weinstein , Nonlinear Schr¨ odinger equations and sharp interpolation estimates, Communications in Mathematical Physics 87 (1983), 567–576. FOCUSING NLS WITH HARDY POTENTIAL 35

work page 1983
[39]

E.Yanagida, Uniqueness of positive radial solutions of u + g(r)u + h(r)up = 0 in Rn, Arch. Ration. Mech. Anal. 115 (1991), 257–274

work page 1991
[40]

Zhang, J

J. Zhang, J. Zheng , Scattering theory for nonlinear Schr¨ odinger equations with inverse-square poten- tial, J. Funct. Anal., 267 (2014), 2907–2932. Department of Mathematics and Statistics, Masaryk Univers ity, Brno, Czech Republic E-mail address : mukherjeed@math.muni.cz Department of Mathematics, LMU Munich, Theresienstrasse 3 9, D-80333 Munich, and M...

work page 2014

[1] [1]

Banica , Remarks on the blow-up for the Schr¨ odinger equation with cr itical mass on a plane domain , Ann

V. Banica , Remarks on the blow-up for the Schr¨ odinger equation with cr itical mass on a plane domain , Ann. Sc. Norm. Super. Pisa Cl. Sci. 3 (2004), 139–170

work page 2004

[2] [2]

Bensouilah, V.D

A. Bensouilah, V.D. Dinh, S. Zhu , On stability and instability of standing waves for the nonli near Schr¨ odinger equation with an inverse-square potential, J. Math. Phys., 59 (2018), 101505

work page 2018

[3] [3]

Brezis and J.-L

H. Brezis and J.-L. V ´azquez, Blow-up solutions of some nonlinear elliptic problems , Rev. Mat. Univ. Madrid 10 (1997), 443–469

work page 1997

[4] [4]

Burchard , A Short Course on Rearrangement Inequalities , Lecture Notes 2009

A. Burchard , A Short Course on Rearrangement Inequalities , Lecture Notes 2009. http://www.math.toronto.edu/almut/rearrange.pdf

work page 2009

[5] [5]

N. Burq, F. Planchon, J.G. Stalker and A.S. Tahvildar-Zadeh , Strichartz estimates for the wave and Schr¨ odinger equations with the inverse-square potential, J. Funct. Anal. 203 (2003), 519–549

work page 2003

[6] [6]

Cazenave , Semilinear Schrodinger Equations , Courant Lecture Notes Volume 10 (2003), American Mathematical Society

T. Cazenave , Semilinear Schrodinger Equations , Courant Lecture Notes Volume 10 (2003), American Mathematical Society

work page 2003

[7] [7]

Coffman ,Uniqueness of the ground state solution for ∆u − u + u3 = 0 and a variational charac- terization of other solutions , Arch

C.V. Coffman ,Uniqueness of the ground state solution for ∆u − u + u3 = 0 and a variational charac- terization of other solutions , Arch. Ration. Mech. Anal. 46 (1972), 81–95

work page 1972

[8] [8]

Csobo and F

E. Csobo and F. Genoud , Minimal mass blow-up solutions for the L2 critical NLS with inverse-square potential, Nonlinear Anal. 168 (2018), 110–129

work page 2018

[9] [9]

R. L. Frank , A simple proof of Hardy-Lieb-Thirring inequalities , Comm. Math. Phys. 290 (2009), 789–800

work page 2009

[10] [10]

Gidas, W.M

B. Gidas, W.M. Ni, L. Nirenberg , Symmetry of positive solutions of nonlinear elliptic equat ions in Rn, Mathematical analysis and applications. Part A, Adv. in Ma th. Suppl. Stud. Vol. 7 (1981), Academic Press, New York, 369–402

work page 1981

[11] [11]

Hmidi and S

T. Hmidi and S. Keraani , Blowup theory for the critical nonlinear Schr¨ odinger equa tions revisited , Int. Math. Res. Not. 46 (2005), 2815–2828

work page 2005

[12] [12]

Holmer and S

J. Holmer and S. Roudenko , On blow-up solutions to the 3d cubic nonlinear Schr¨ odinger equation, AMRX Appl. Math. Res. Express (2007), Art. ID abm004. 34 D. MUKHERJEE, P.T. NAM, AND P.T. NGUYEN

work page 2007

[13] [13]

H. Kalf, U. W. Schmincke, J. W alter, R. W ¨ust, On the spectral theory of Schr¨ odinger and Dirac operators with strongly singular potentials . In Spectral Theory and Diﬀerential Equations, pp. 182-226 . Lecture Notes in Mathematics 448 (1975) Springer, Berlin

work page 1975

[14] [14]

C. E. Kenig, F. Merle , Global well-posedness, scattering and blow-up for the ener gy-critical focusing non-linear wave equation , Acta Math. 201 (2008), 147–212

work page 2008

[15] [15]

Killip, C

R. Killip, C. Miao, M. Visan, J. Zhang, J. Zheng , The energy-critical NLS with inverse-square potential, Discrete Contin. Dyn. Syst., 37 (2017), 3831–3866

work page 2017

[16] [16]

Killip, J

R. Killip, J. Murphy, M. Visan, J. Zheng , The focusing cubic NLS with inverse-square potential in three space dimensions , Diﬀerential Integral Equations, 30 (2017), 161–206

work page 2017

[17] [17]

M. K. Kwong , Uniqueness of positive solutions of ∆u − u + up = 0 in Rn, Arch. Ration. Mech. Anal. 105 (1989), 243–266

work page 1989

[18] [18]

E. H. Lieb and M. Loss , Analysis. Second edition. Graduate Studies in Mathematics , 14. American Mathematical Society, Providence, RI, 2001. xxii+346 pp

work page 2001

[19] [19]

J. Lu, C. Miao, J. Murphy , Scattering in H 1 for the intercritical NLS with an inverse-square potential , J. Diﬀerential Equations, 264, (2018), 3174–3211

work page 2018

[20] [20]

McLeod, J

K. McLeod, J. Serrin , Uniqueness of positive radial solutions of ∆u + f (u) = 0 in Rn, Arch. Ration. Mech. Anal. 99 (1987), 115–145

work page 1987

[21] [21]

Merle , Determination of blow-up solutions with minimal mass for no nlinear Schr¨ odinger equations with critical power , Duke Math

F. Merle , Determination of blow-up solutions with minimal mass for no nlinear Schr¨ odinger equations with critical power , Duke Math. J. 69 (1993) 427–454

work page 1993

[22] [22]

Mizutani , Remarks on endpoint Strichartz estimates for Schr¨ odinger equations with the critical inverse-square potential, J

H. Mizutani , Remarks on endpoint Strichartz estimates for Schr¨ odinger equations with the critical inverse-square potential, J. Diﬀerential Equations 263 (2017), 3832–3853

work page 2017

[23] [23]

P´ olya, G

G. P´ olya, G. Szeg¨ o,Isoperimetric Inequalities in Mathematical Physics , Annals of Mathematics Studies, Princeton University Press (1951)

work page 1951

[24] [24]

Okazaw a, T

N. Okazaw a, T. Suzuki and T. Yokota , Cauchy problem for nonlinear Schr¨ odinger equations with inverse-square potentials, Appl. Anal. 91 (2012), 1605–1629

work page 2012

[25] [25]

Okazaw a, T

N. Okazaw a, T. Suzuki and T. Yokota , Energy methods for abstract nonlinear Schr¨ odinger equa- tions, Evolution equations and control theory 1 (2012), 337–354

work page 2012

[26] [26]

Ogaw a, Y

T. Ogaw a, Y. Tsutsumi, Blow-up of H 1 solution for the nonlinear Schr¨ odinger equation, J. Diﬀ. Equa. 92 (1991), 317–330

work page 1991

[27] [27]

Rodnianski, W

I. Rodnianski, W. Schlag , Time decay for solutions of Schrdinger equations with rough and time- dependent potentials, Invent. Math. 155 (2004), 451–513

work page 2004

[28] [28]

J. P. Solovej, T. Ø. Sørensen and W. L. Spitzer , Relativistic Scott correction for atoms and molecules, Comm. Pure Appl. Math. 63 (2010), 39–118

work page 2010

[29] [29]

W. A. Strauss , Existence of solitary waves in higher dimensions , Commun. Math. Phys. 55 (1977), no. 2, 149–162

work page 1977

[30] [30]

Su, Z.-Q

J. Su, Z.-Q. W ang, M. Willem , Nonlinear Schr¨ odinger equations with unbounded and decaying radial potentials, Commun. Contemp. Math. 9 (2007), 571–583

work page 2007

[31] [31]

Suzuki , Solvability of nonlinear Schr¨ odinger equations with some critical singular potential via gen- eralized Hardy-Rellich inequalities , Funkcialaj Ekvacioj 59 (2016), 1–34

T. Suzuki , Solvability of nonlinear Schr¨ odinger equations with some critical singular potential via gen- eralized Hardy-Rellich inequalities , Funkcialaj Ekvacioj 59 (2016), 1–34

work page 2016

[32] [32]

Suzuki , Virial identities for nonlinear Schr¨ odinger equations with a critical coeﬃcient inverse-square potential, Diﬀer

T. Suzuki , Virial identities for nonlinear Schr¨ odinger equations with a critical coeﬃcient inverse-square potential, Diﬀer. Equ. Appl. 9 (2017), 327–352

work page 2017

[33] [33]

Shioji, K

N. Shioji, K. W atanabe , A generalized Pohozaev identity and uniqueness of positive radial solutions of ∆u + g(r)u + h(r)up = 0, J. Diﬀ. Equa. 255 (2013), 4448–4475

work page 2013

[34] [34]

Tao , Nonlinear dispersive equations, Local and global analysis , CBMS Regional Conference Series in Mathematics, 106 (2006), xvi+373

T. Tao , Nonlinear dispersive equations, Local and global analysis , CBMS Regional Conference Series in Mathematics, 106 (2006), xvi+373

work page 2006

[35] [35]

Tsutsumi , L2-solutions for nonlinear Schr¨ odinger equations and nonlinear groups, Funkcial

Y. Tsutsumi , L2-solutions for nonlinear Schr¨ odinger equations and nonlinear groups, Funkcial. Ekvac. 30 (1987), 115–125

work page 1987

[36] [36]

G. P. Trachanas and N. B. Zographopoulos , Orbital stability for the Schr¨ odinger operator involv- ing inverse square potential , J. Diﬀerential Equations 259 (2015), 4989–5016

work page 2015

[37] [37]

J. L. V azquez, E. Zuazua , The Hardy inequality and the asymptotic behaviour of the hea t equation with an inverse-square potential , J. Funct. Anal. 173 (2000), 103-153

work page 2000

[38] [38]

M. I. Weinstein , Nonlinear Schr¨ odinger equations and sharp interpolation estimates, Communications in Mathematical Physics 87 (1983), 567–576. FOCUSING NLS WITH HARDY POTENTIAL 35

work page 1983

[39] [39]

E.Yanagida, Uniqueness of positive radial solutions of u + g(r)u + h(r)up = 0 in Rn, Arch. Ration. Mech. Anal. 115 (1991), 257–274

work page 1991

[40] [40]

Zhang, J

J. Zhang, J. Zheng , Scattering theory for nonlinear Schr¨ odinger equations with inverse-square poten- tial, J. Funct. Anal., 267 (2014), 2907–2932. Department of Mathematics and Statistics, Masaryk Univers ity, Brno, Czech Republic E-mail address : mukherjeed@math.muni.cz Department of Mathematics, LMU Munich, Theresienstrasse 3 9, D-80333 Munich, and M...

work page 2014