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Minimal mass non-scattering solutions of the focusing L2-critical Hartree equations with radial data
A unified approach to weighted Hardy type inequalities on Carnot groups
1. | Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA |
2. | Department of Mathematics, Faculty of Art and Science, Istanbul Commerce University, Beyoglu 34445, Istanbul, Turkey |
$V(x)$ |
$W(x) $ |
$\mathbb{G},$ |
$L^{p}$ |
$\begin{equation*}\int_{\mathbb{G}}V\left( x\right) \left\vert \nabla _{\mathbb{G}}\phi \left(x\right) \right\vert ^{p}dx\geq \int_{\mathbb{G}}W\left( x\right) \left\vert\phi \left( x\right) \right\vert ^{p}dx\end{equation*}$ |
$φ ∈ C_{0}^{∞ }(\mathbb{G})$ |
$p>1.$ |
$\mathbb{G}.$ |
$L^{p}$ |
$Ω$ |
$\mathbb{G}$ |
References:
[1] |
Adimurthi, M. Ramaswamy and N. Chaudhuri,
An improved Hardy Sobolev inequality and its applications, Proc. Amer. Math. Soc., 130 (2002), 489-505.
doi: 10.1090/S0002-9939-01-06132-9. |
[2] |
Adimurthi and K. Sandeep,
Existence and non-existence of the first eigenvalue of the perturbed Hardy-Sobolev operator, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1021-1043.
doi: 10.1017/S0308210500001992. |
[3] |
Z. Balogh and J. Tyson,
Polar coordinates in Carnot groups, Math. Z., 241 (2002), 697-730.
doi: 10.1007/s00209-002-0441-7. |
[4] |
Z. Balogh, I. Holopainen and J. Tyson,
Singular solutions, homogeneous norms and quasiconformal mappings in Carnot groups, Math. Ann., 324 (2002), 159-186.
doi: 10.1007/s00208-002-0334-4. |
[5] |
P. Baras and J. A. Goldstein,
The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139.
doi: 10.1090/S0002-9947-1984-0742415-3. |
[6] |
G. Barbatis, S. Filippas and A. Tertikas,
Series expansion for Lp Hardy inequalities, Indiana Univ. Math. J., 52 (2003), 171-190.
doi: 10.1512/iumj.2003.52.2207. |
[7] |
A. Bonfiglioli, E. Lanconelli and F. Uguzzoni,
Stratified Lie Groups and Potential Theory for their Sub-Laplacians Springer-Verlag, Berlin-Heidelberg, 2007. |
[8] |
H. Brezis and J. L. Vázquez,
Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complutense Madrid, 10 (1997), 443-469.
|
[9] |
C. Cowan,
Optimal Hardy inequalities for general elliptic operators with improvements, Commun. Pure Appl. Anal., 9 (2010), 109-140.
doi: 10.3934/cpaa.2010.9.109. |
[10] |
D. Danielli, N. Garofalo and N. C. Phuc,
Hardy-Sobolev type inequalities with sharp constants in Carnot-Carath éodory spaces, Potential Anal., 34 (2011), 223-242.
doi: 10.1007/s11118-010-9190-0. |
[11] |
L. D'Ambrosio,
Hardy type inequalities related to degenerate elliptic differential operators, Ann. Sc. Norm. Super. Pisa Cl. Sci., 5 (2005), 451-486.
|
[12] |
D. E. Edmunds and H. Triebel,
Sharp Sobolev embeddings and related Hardy inequalities: the critical case, Math. Nachr., 207 (1999), 79-92.
doi: 10.1002/mana.1999.3212070105. |
[13] |
S. Filippas and A. Tertikas,
Optimizing Improved Hardy Inequalities, J. Funct. Anal., 192 (2002), 186-233.
doi: 10.1006/jfan.2001.3900. |
[14] |
G. B. Folland,
Subelliptic estimates and function spaces on nilpotent Lie groups, Arkiv für Math., 13 (1975), 161-207.
doi: 10.1007/BF02386204. |
[15] |
G. B. Folland and E. Stein,
Hardy Spaces on Homogeneous Groups Princeton University Press, Princeton, NJ, 1982. |
[16] |
G. B. Folland and A. Sitaram,
The uncertainty principle: A mathematical survey, J. Fourier Anal. Appl., 3 (1997), 207-238.
doi: 10.1007/BF02649110. |
[17] |
B. Franchi, R. Serapioni and F. Serra Cassano,
Regular hypersurfaces, intrinsic perimeter and implicit function theorem in carnot groups, Comm. Anal. Geom., 11 (2003), 909-944.
doi: 10.4310/CAG.2003.v11.n5.a4. |
[18] |
J. Garcia Azorero and I. Peral Alonso,
Hardy inequalities and some critical elliptic and parabolic problems, J. Diff. Equations, 144 (1998), 441-476.
doi: 10.1006/jdeq.1997.3375. |
[19] |
N. Ghoussoub and A. Moradifam,
Bessel potentials and optimal Hardy and Hardy-Rellich inequalities, Math. Ann., 349 (2011), 1-57.
doi: 10.1007/s00208-010-0510-x. |
[20] |
G. R. Goldstein, J. A. Goldstein and A. Rhandi,
Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential, Appl. Anal., 91 (2011), 2057-2071.
doi: 10.1080/00036811.2011.587809. |
[21] |
J. A. Goldstein, D. Hauer and A. Rhandi,
Existence and nonexistence of positive solutions of p-Kolmogorov equations perturbed by a Hardy potential, Nonlinear Anal., 131 (2016), 121-154.
doi: 10.1016/j.na.2015.07.016. |
[22] |
J. A. Goldstein and I. Kombe,
The Hardy inequality and nonlinear parabolic equations on Carnot groups, Nonlinear Anal., 69 (2008), 4643-4653.
doi: 10.1016/j.na.2007.11.020. |
[23] |
D. Hauer and A. Rhandi,
A weighted Hardy inequality and nonexistence of positive solutions, Arch. Math.(Basel), 100 (2013), 273-287.
doi: 10.1007/s00013-013-0484-5. |
[24] |
Y. Han and P. Niu, Some Hardy type inequalities in the Heisenberg group, J. Inequal. Pure Appl. Math. , 4 (2003), Article 103, 5 pp. |
[25] |
W. Heisenberg,
Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Original Scientific Papers Wissenschaftliche Originalarbeiten, A/1 (1985), 478-504.
doi: 10.1007/978-3-642-61659-4_30. |
[26] |
Y. Jin and S. Shen,
Weighted Hardy and Rellich inequality on Carnot groups, Arch. Math.(Basel), 96 (2011), 263-271.
doi: 10.1007/s00013-011-0220-y. |
[27] |
I. Kombe,
Sharp Weighted Rellich and uncertainty principle inequalities on Carnot groups, Comm. App. Anal., 14 (2010), 251-271.
|
[28] |
I. Kombe and A. Yener,
Weighted Hardy and Rellich type inequalities on Riemannian manifolds, Math. Nachr., 289 (2016), 994-1004.
doi: 10.1002/mana.201500237. |
[29] |
B. Lian,
Some sharp Rellich type inequalities on nilpotent groups and application, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 59-74.
doi: 10.1016/S0252-9602(12)60194-5. |
[30] |
P. Lindqvist,
On the equation $\text{div}(|\nabla u|^{p-2}\nabla u)+λ |u|^{p-2}u=0$, Proc. Amer. Math. Soc., 109 (1990), 157-164.
doi: 10.1090/S0002-9939-1990-1007505-7. |
[31] |
I. Skrzypczak,
Hardy type inequalities derived from $p$-harmonic problems, Nonlinear Anal., 93 (2013), 30-50.
doi: 10.1016/j.na.2013.07.006. |
[32] |
E. Stein,
Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals Princeton University Press, Princeton, NJ, 1993. |
[33] |
J. L. Vazquez and E. Zuazua,
The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse square potential, J. Funct. Anal., 173 (2000), 103-153.
doi: 10.1006/jfan.1999.3556. |
[34] |
J. Wang and P. Niu,
Sharp weighted Hardy type inequalities and Hardy-Sobolev type inequalities on polarizable Carnot groups, C. R. Math. Acad. Sci. Paris Ser. I, 346 (2008), 1231-1234.
doi: 10.1016/j.crma.2008.10.009. |
[35] |
H. Weyl,
The Theory of Groups and Quantum Mechanics Reprint of the 1931 English translation. Dover Publications, Inc. , New York, 1950. |
[36] |
A. Yener,
Weighted Hardy type inequalities on the Heisenberg group $\mathbb{H}^{n}$, Math. Inequal. Appl., 19 (2016), 671-683.
doi: 10.7153/mia-19-48. |
show all references
References:
[1] |
Adimurthi, M. Ramaswamy and N. Chaudhuri,
An improved Hardy Sobolev inequality and its applications, Proc. Amer. Math. Soc., 130 (2002), 489-505.
doi: 10.1090/S0002-9939-01-06132-9. |
[2] |
Adimurthi and K. Sandeep,
Existence and non-existence of the first eigenvalue of the perturbed Hardy-Sobolev operator, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 1021-1043.
doi: 10.1017/S0308210500001992. |
[3] |
Z. Balogh and J. Tyson,
Polar coordinates in Carnot groups, Math. Z., 241 (2002), 697-730.
doi: 10.1007/s00209-002-0441-7. |
[4] |
Z. Balogh, I. Holopainen and J. Tyson,
Singular solutions, homogeneous norms and quasiconformal mappings in Carnot groups, Math. Ann., 324 (2002), 159-186.
doi: 10.1007/s00208-002-0334-4. |
[5] |
P. Baras and J. A. Goldstein,
The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139.
doi: 10.1090/S0002-9947-1984-0742415-3. |
[6] |
G. Barbatis, S. Filippas and A. Tertikas,
Series expansion for Lp Hardy inequalities, Indiana Univ. Math. J., 52 (2003), 171-190.
doi: 10.1512/iumj.2003.52.2207. |
[7] |
A. Bonfiglioli, E. Lanconelli and F. Uguzzoni,
Stratified Lie Groups and Potential Theory for their Sub-Laplacians Springer-Verlag, Berlin-Heidelberg, 2007. |
[8] |
H. Brezis and J. L. Vázquez,
Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complutense Madrid, 10 (1997), 443-469.
|
[9] |
C. Cowan,
Optimal Hardy inequalities for general elliptic operators with improvements, Commun. Pure Appl. Anal., 9 (2010), 109-140.
doi: 10.3934/cpaa.2010.9.109. |
[10] |
D. Danielli, N. Garofalo and N. C. Phuc,
Hardy-Sobolev type inequalities with sharp constants in Carnot-Carath éodory spaces, Potential Anal., 34 (2011), 223-242.
doi: 10.1007/s11118-010-9190-0. |
[11] |
L. D'Ambrosio,
Hardy type inequalities related to degenerate elliptic differential operators, Ann. Sc. Norm. Super. Pisa Cl. Sci., 5 (2005), 451-486.
|
[12] |
D. E. Edmunds and H. Triebel,
Sharp Sobolev embeddings and related Hardy inequalities: the critical case, Math. Nachr., 207 (1999), 79-92.
doi: 10.1002/mana.1999.3212070105. |
[13] |
S. Filippas and A. Tertikas,
Optimizing Improved Hardy Inequalities, J. Funct. Anal., 192 (2002), 186-233.
doi: 10.1006/jfan.2001.3900. |
[14] |
G. B. Folland,
Subelliptic estimates and function spaces on nilpotent Lie groups, Arkiv für Math., 13 (1975), 161-207.
doi: 10.1007/BF02386204. |
[15] |
G. B. Folland and E. Stein,
Hardy Spaces on Homogeneous Groups Princeton University Press, Princeton, NJ, 1982. |
[16] |
G. B. Folland and A. Sitaram,
The uncertainty principle: A mathematical survey, J. Fourier Anal. Appl., 3 (1997), 207-238.
doi: 10.1007/BF02649110. |
[17] |
B. Franchi, R. Serapioni and F. Serra Cassano,
Regular hypersurfaces, intrinsic perimeter and implicit function theorem in carnot groups, Comm. Anal. Geom., 11 (2003), 909-944.
doi: 10.4310/CAG.2003.v11.n5.a4. |
[18] |
J. Garcia Azorero and I. Peral Alonso,
Hardy inequalities and some critical elliptic and parabolic problems, J. Diff. Equations, 144 (1998), 441-476.
doi: 10.1006/jdeq.1997.3375. |
[19] |
N. Ghoussoub and A. Moradifam,
Bessel potentials and optimal Hardy and Hardy-Rellich inequalities, Math. Ann., 349 (2011), 1-57.
doi: 10.1007/s00208-010-0510-x. |
[20] |
G. R. Goldstein, J. A. Goldstein and A. Rhandi,
Weighted Hardy's inequality and the Kolmogorov equation perturbed by an inverse-square potential, Appl. Anal., 91 (2011), 2057-2071.
doi: 10.1080/00036811.2011.587809. |
[21] |
J. A. Goldstein, D. Hauer and A. Rhandi,
Existence and nonexistence of positive solutions of p-Kolmogorov equations perturbed by a Hardy potential, Nonlinear Anal., 131 (2016), 121-154.
doi: 10.1016/j.na.2015.07.016. |
[22] |
J. A. Goldstein and I. Kombe,
The Hardy inequality and nonlinear parabolic equations on Carnot groups, Nonlinear Anal., 69 (2008), 4643-4653.
doi: 10.1016/j.na.2007.11.020. |
[23] |
D. Hauer and A. Rhandi,
A weighted Hardy inequality and nonexistence of positive solutions, Arch. Math.(Basel), 100 (2013), 273-287.
doi: 10.1007/s00013-013-0484-5. |
[24] |
Y. Han and P. Niu, Some Hardy type inequalities in the Heisenberg group, J. Inequal. Pure Appl. Math. , 4 (2003), Article 103, 5 pp. |
[25] |
W. Heisenberg,
Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Original Scientific Papers Wissenschaftliche Originalarbeiten, A/1 (1985), 478-504.
doi: 10.1007/978-3-642-61659-4_30. |
[26] |
Y. Jin and S. Shen,
Weighted Hardy and Rellich inequality on Carnot groups, Arch. Math.(Basel), 96 (2011), 263-271.
doi: 10.1007/s00013-011-0220-y. |
[27] |
I. Kombe,
Sharp Weighted Rellich and uncertainty principle inequalities on Carnot groups, Comm. App. Anal., 14 (2010), 251-271.
|
[28] |
I. Kombe and A. Yener,
Weighted Hardy and Rellich type inequalities on Riemannian manifolds, Math. Nachr., 289 (2016), 994-1004.
doi: 10.1002/mana.201500237. |
[29] |
B. Lian,
Some sharp Rellich type inequalities on nilpotent groups and application, Acta Math. Sci. Ser. B Engl. Ed., 33 (2013), 59-74.
doi: 10.1016/S0252-9602(12)60194-5. |
[30] |
P. Lindqvist,
On the equation $\text{div}(|\nabla u|^{p-2}\nabla u)+λ |u|^{p-2}u=0$, Proc. Amer. Math. Soc., 109 (1990), 157-164.
doi: 10.1090/S0002-9939-1990-1007505-7. |
[31] |
I. Skrzypczak,
Hardy type inequalities derived from $p$-harmonic problems, Nonlinear Anal., 93 (2013), 30-50.
doi: 10.1016/j.na.2013.07.006. |
[32] |
E. Stein,
Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals Princeton University Press, Princeton, NJ, 1993. |
[33] |
J. L. Vazquez and E. Zuazua,
The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse square potential, J. Funct. Anal., 173 (2000), 103-153.
doi: 10.1006/jfan.1999.3556. |
[34] |
J. Wang and P. Niu,
Sharp weighted Hardy type inequalities and Hardy-Sobolev type inequalities on polarizable Carnot groups, C. R. Math. Acad. Sci. Paris Ser. I, 346 (2008), 1231-1234.
doi: 10.1016/j.crma.2008.10.009. |
[35] |
H. Weyl,
The Theory of Groups and Quantum Mechanics Reprint of the 1931 English translation. Dover Publications, Inc. , New York, 1950. |
[36] |
A. Yener,
Weighted Hardy type inequalities on the Heisenberg group $\mathbb{H}^{n}$, Math. Inequal. Appl., 19 (2016), 671-683.
doi: 10.7153/mia-19-48. |
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