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On a coupled Cahn–Hilliard/Cahn–Hilliard model for the proliferative-to-invasive transition of hypoxic glioma cells
On weak solutions to a fractional Hardy–Hénon equation: Part I: Nonexistence
1. | Department of Mathematics, School of Fundamental Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo 169-8555, Japan |
2. | Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 223-8522, Japan |
3. | Applied Mathematics and Informatics Course, Faculty of Advanced Science and Technology, Ryukoku University, 1-5 Yokotani, Seta Oe-cho, Otsu, Shiga 520-2194, Japan |
This paper and [
References:
[1] |
B. Barrios and A. Quaas, The sharp exponent in the study of the nonlocal Hénon equation in $ \mathbb{R}^N$: a Liouville theorem and an existence result, Calc. Var. Partial Differ. Equ., 59 (2020), 22 pp.
doi: 10.1007/s00526-020-01763-z. |
[2] |
H. Berestycki and P.-L. Lions,
Nonlinear scalar field equations. I Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[3] |
M. Chipot, M. Chlebík, M. Fila and I. Shafrir,
Existence of positive solutions of a semilinear elliptic equation, J. Math. Anal. Appl., 223 (1998), 429-471.
doi: 10.1006/jmaa.1998.5958. |
[4] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[5] |
X. Cabré and Y. Sire,
Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[6] |
W. Dai and G. Qin, Liouville type theorems for fractional and higher order Hénon-Hardy type equations via the method of scaling spheres, preprint, arXiv: 1810.02752.
doi: 10.1016/j.jmaa.2011.08.081. |
[7] |
E. N. Dancer, Y. Du and Z. Guo,
Finite Morse index solutions of an elliptic equation with supercritical exponent, J. Differ. Equ., 250 (2011), 3281-3310.
doi: 10.1016/j.jde.2011.02.005. |
[8] |
J. Dávila, L. Dupaigne and M. Montenegro,
The extremal solution of a boundary reaction problem, Commun. Pure Appl. Anal., 7 (2008), 795-817.
doi: 10.3934/cpaa.2008.7.795. |
[9] |
J. Dávila, L. Dupaigne and J. Wei,
On the fractional Lane–Emden equation, Trans. Amer. Math. Soc., 369 (2017), 6087-6104.
doi: 10.1090/tran/6872. |
[10] |
F. Demengel and G. Demengel, Functional spaces for the theory of elliptic partial differential equations, Translated from the 2007 French original by Reinie Erné. Universitext. Springer, London; EDP Sciences, Les Ulis, 2012.
doi: 10.1007/978-1-4471-2807-6. |
[11] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[12] |
M. M. Fall, Semilinear elliptic equations for the fractional Laplacian with Hardy potential, Nonlinear Anal., 193 (2020), 29 pp.
doi: 10.1016/j.na.2018.07.008. |
[13] |
M. M. Fall and V. Felli,
Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Commun. Partial Differ. Equ., 39 (2014), 354-397.
doi: 10.1080/03605302.2013.825918. |
[14] |
M. M. Fall and V. Felli,
Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst., 35 (2015), 5827-5867.
doi: 10.3934/dcds.2015.35.5827. |
[15] |
M. M. Fall and T. Weth,
Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal., 263 (2012), 2205-2227.
doi: 10.1016/j.jfa.2012.06.018. |
[16] |
A. Farina,
On the classification of solutions of the Lane–Emden equation on unbounded domains of $\mathbb R^N$, J. Math. Pures Appl., 87 (2007), 537-561.
doi: 10.1016/j.matpur.2007.03.001. |
[17] |
M. Fazly and J. Wei, On stable solutions of the fractional Hénon-Lane-Emden equation, Commun. Contemp. Math., 18 (2016), 24 pp.
doi: 10.1142/S021919971650005X. |
[18] |
R. L. Frank, E. H. Lieb and R. Seiringer,
Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators, J. Amer. Math. Soc., 21 (2008), 925-950.
doi: 10.1090/S0894-0347-07-00582-6. |
[19] |
J. Harada,
Positive solutions to the Laplace equation with nonlinear boundary conditions on the half space, Calc. Var. Partial Differ. Equ., 50 (2014), 399-435.
doi: 10.1007/s00526-013-0640-6. |
[20] |
S. Hasegawa, N. Ikoma and T. Kawakami, On weak solutions to a fractional Hardy–Hénon equation: Part 2: Existence, preprint, arXiv: 2102.05873.
doi: 10.1093/integr/xyy013. |
[21] |
T. Jin, Y. Y. Li and J. Xiong,
On a fractional Nirenberg problem, part I: Blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111-1171.
doi: 10.4171/JEMS/456. |
[22] |
Y. Li and J. Bao,
Fractional Hardy–Hénon equations on exterior domains, J. Differ. Equ., 266 (2019), 1153-1175.
doi: 10.1016/j.jde.2018.07.062. |
[23] |
J. L. Lions,
Théorémes de trace et d'interpolation. I, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 389-403.
|
[24] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.
|
[25] |
C. Wang and D. Ye, Some Liouville theorems for Hénon type elliptic equations, J. Funct. Anal., 262 (2012), no. 4, 1705–1727.
doi: 10.1016/j.jfa.2012.05.025. |
[26] |
C. Wang and D. Ye, Corrigendum to "Some Liouville theorems for Hénon type elliptic equations" [J. Funct. Anal. 262 (4) (2012) 1705–1727] [MR2873856], J. Funct. Anal., 263 (2012), no. 6, 1766–1768. |
[27] |
J. Yang,
Fractional Sobolev-Hardy inequality in $ \mathbb{R}^N$, Nonlinear Anal., 119 (2015), 179-185.
doi: 10.1016/j.na.2014.09.009. |
show all references
References:
[1] |
B. Barrios and A. Quaas, The sharp exponent in the study of the nonlocal Hénon equation in $ \mathbb{R}^N$: a Liouville theorem and an existence result, Calc. Var. Partial Differ. Equ., 59 (2020), 22 pp.
doi: 10.1007/s00526-020-01763-z. |
[2] |
H. Berestycki and P.-L. Lions,
Nonlinear scalar field equations. I Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.
doi: 10.1007/BF00250555. |
[3] |
M. Chipot, M. Chlebík, M. Fila and I. Shafrir,
Existence of positive solutions of a semilinear elliptic equation, J. Math. Anal. Appl., 223 (1998), 429-471.
doi: 10.1006/jmaa.1998.5958. |
[4] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[5] |
X. Cabré and Y. Sire,
Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.
doi: 10.1016/j.anihpc.2013.02.001. |
[6] |
W. Dai and G. Qin, Liouville type theorems for fractional and higher order Hénon-Hardy type equations via the method of scaling spheres, preprint, arXiv: 1810.02752.
doi: 10.1016/j.jmaa.2011.08.081. |
[7] |
E. N. Dancer, Y. Du and Z. Guo,
Finite Morse index solutions of an elliptic equation with supercritical exponent, J. Differ. Equ., 250 (2011), 3281-3310.
doi: 10.1016/j.jde.2011.02.005. |
[8] |
J. Dávila, L. Dupaigne and M. Montenegro,
The extremal solution of a boundary reaction problem, Commun. Pure Appl. Anal., 7 (2008), 795-817.
doi: 10.3934/cpaa.2008.7.795. |
[9] |
J. Dávila, L. Dupaigne and J. Wei,
On the fractional Lane–Emden equation, Trans. Amer. Math. Soc., 369 (2017), 6087-6104.
doi: 10.1090/tran/6872. |
[10] |
F. Demengel and G. Demengel, Functional spaces for the theory of elliptic partial differential equations, Translated from the 2007 French original by Reinie Erné. Universitext. Springer, London; EDP Sciences, Les Ulis, 2012.
doi: 10.1007/978-1-4471-2807-6. |
[11] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[12] |
M. M. Fall, Semilinear elliptic equations for the fractional Laplacian with Hardy potential, Nonlinear Anal., 193 (2020), 29 pp.
doi: 10.1016/j.na.2018.07.008. |
[13] |
M. M. Fall and V. Felli,
Unique continuation property and local asymptotics of solutions to fractional elliptic equations, Commun. Partial Differ. Equ., 39 (2014), 354-397.
doi: 10.1080/03605302.2013.825918. |
[14] |
M. M. Fall and V. Felli,
Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst., 35 (2015), 5827-5867.
doi: 10.3934/dcds.2015.35.5827. |
[15] |
M. M. Fall and T. Weth,
Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal., 263 (2012), 2205-2227.
doi: 10.1016/j.jfa.2012.06.018. |
[16] |
A. Farina,
On the classification of solutions of the Lane–Emden equation on unbounded domains of $\mathbb R^N$, J. Math. Pures Appl., 87 (2007), 537-561.
doi: 10.1016/j.matpur.2007.03.001. |
[17] |
M. Fazly and J. Wei, On stable solutions of the fractional Hénon-Lane-Emden equation, Commun. Contemp. Math., 18 (2016), 24 pp.
doi: 10.1142/S021919971650005X. |
[18] |
R. L. Frank, E. H. Lieb and R. Seiringer,
Hardy–Lieb–Thirring inequalities for fractional Schrödinger operators, J. Amer. Math. Soc., 21 (2008), 925-950.
doi: 10.1090/S0894-0347-07-00582-6. |
[19] |
J. Harada,
Positive solutions to the Laplace equation with nonlinear boundary conditions on the half space, Calc. Var. Partial Differ. Equ., 50 (2014), 399-435.
doi: 10.1007/s00526-013-0640-6. |
[20] |
S. Hasegawa, N. Ikoma and T. Kawakami, On weak solutions to a fractional Hardy–Hénon equation: Part 2: Existence, preprint, arXiv: 2102.05873.
doi: 10.1093/integr/xyy013. |
[21] |
T. Jin, Y. Y. Li and J. Xiong,
On a fractional Nirenberg problem, part I: Blow up analysis and compactness of solutions, J. Eur. Math. Soc., 16 (2014), 1111-1171.
doi: 10.4171/JEMS/456. |
[22] |
Y. Li and J. Bao,
Fractional Hardy–Hénon equations on exterior domains, J. Differ. Equ., 266 (2019), 1153-1175.
doi: 10.1016/j.jde.2018.07.062. |
[23] |
J. L. Lions,
Théorémes de trace et d'interpolation. I, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 389-403.
|
[24] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.
|
[25] |
C. Wang and D. Ye, Some Liouville theorems for Hénon type elliptic equations, J. Funct. Anal., 262 (2012), no. 4, 1705–1727.
doi: 10.1016/j.jfa.2012.05.025. |
[26] |
C. Wang and D. Ye, Corrigendum to "Some Liouville theorems for Hénon type elliptic equations" [J. Funct. Anal. 262 (4) (2012) 1705–1727] [MR2873856], J. Funct. Anal., 263 (2012), no. 6, 1766–1768. |
[27] |
J. Yang,
Fractional Sobolev-Hardy inequality in $ \mathbb{R}^N$, Nonlinear Anal., 119 (2015), 179-185.
doi: 10.1016/j.na.2014.09.009. |
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