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April  2021, 20(4): 1559-1600. doi: 10.3934/cpaa.2021033

On weak solutions to a fractional Hardy–Hénon equation: Part I: Nonexistence

Shoichi Hasegawa 1, , Norihisa Ikoma 2, and  Tatsuki Kawakami 3,,

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

* Corresponding author

Received  April 2020 Revised  January 2021 Published  March 2021

Fund Project: The first author (S.H.) was supported by JSPS KAKENHI Grant Numbers JP 20J01191.The second author (N.I.) was supported by JSPS KAKENHI Grant Numbers JP 17H02851, 19H01797 and 19K03590.The third author (T.K.) was supported by JSPS KAKENHI Grant Numbers JP 19H05599 and 16K17629

This paper and [20] treat the existence and nonexistence of stable (resp. outside stable) weak solutions to a fractional Hardy–Hénon equation $ (-\Delta)^s u = |x|^\ell |u|^{p-1} u $ in $ \mathbb{R}^N $, where $ 0 < s < 1 $, $ \ell > -2s $, $ p>1 $, $ N \geq 1 $ and $ N > 2s $. In this paper, the nonexistence part is proved for the Joseph–Lundgren subcritical case.

Keywords: fractional Hardy–Hénon equation, stable (stable outside a compact set) solutions, Liouville type theorem.
Mathematics Subject Classification: Primary: 35R11, 35J61, 35B33, 35B53; Secondary: 35D30.
Citation: Shoichi Hasegawa, Norihisa Ikoma, Tatsuki Kawakami. On weak solutions to a fractional Hardy–Hénon equation: Part I: Nonexistence. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1559-1600. doi: 10.3934/cpaa.2021033
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. L. Lions, Théorémes de trace et d'interpolation. I, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 389-403.   Google Scholar

[24]

W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. L. Lions, Théorémes de trace et d'interpolation. I, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (1959), 389-403.   Google Scholar

[24]

W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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