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Emergent dynamics of the fractional Cucker-Smale model under general network topologies
Nonexistence of Positive Solutions for high-order Hardy-H$ \acute{e} $non Systems on $ \mathbb{R}^{n} $
School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China |
$ \acute{e} $ |
$ \begin{cases} \ (-\Delta)^{\frac{\alpha}{2}}u(x) = |x|^{a}v^{p}(x) ,\\ \ (-\Delta)^{\frac{\beta}{2}}v(x) = |x|^{b}u^{q}(x) ,\\ \end{cases} $ |
$ 0<\alpha = s_{1}+2<n $ |
$ 0<\beta = s_{2}+2<n $ |
$ 0<s_{1},s_{2}<2 $ |
$ a>-s_{1} $ |
$ b>-s_{2} $ |
$ p,q>0 $ |
$ \mathbb{R}^{n} $ |
$ \mathbb{R}^{n} $ |
$ \begin{cases} \ u(x) = \int_{\mathbb{R}^{n}}\frac{|y|^{a}v^{p}(y)}{|x-y|^{n-\alpha}}dy,\\[1.5mm] \ v(x) = \int_{\mathbb{R}^{n}}\frac{|y|^{b}u^{q}(y)}{|x-y|^{n-\beta}}dy.\\ \end{cases} $ |
$ 1<p<\frac{n+\alpha+2a}{n-\alpha} $ |
$ 1<q<\frac{n+\alpha+2b}{n-\alpha} $ |
$ \alpha = \beta $ |
$ \mathbb{R}^{n} $ |
$ Doubling\ Lemma $ |
$ \Omega $ |
References:
[1] |
J. Bertoin, L$\acute{e}$vy Processes, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1996. |
[2] |
J. P. Bouchard and A. Georges,
Anomalous diffusion in disordered media: Statistical mechanics, models and physical applications, Phys. Report., 195 (1990), 127-293.
doi: 10.1016/0370-1573(90)90099-N. |
[3] |
L. Caffarelli and L. Vasseur,
Drift diffusion equations with fractional diffusion and the quasigeostrophic equation, Ann. Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[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] |
W. Chen and C. Li,
Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 949-960.
doi: 10.1016/S0252-9602(09)60079-5. |
[6] |
W. Chen, C. Li and B. Ou,
Qualitative problems of solutions for an integral equation, Discret. Contin. Dynam. Syst., 12 (2005), 347-354.
doi: 10.3934/dcds.2005.12.347. |
[7] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[8] |
W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co, Singapore, 2019.
doi: 10.1142/10550. |
[9] |
P. Constantin, Euler equations, navier-stokes equations and turbulence, Mathematical Foundation of Turbulent Viscous Flows Lecture Notes in Math., 1871 (2006), 1–43.
doi: 10.1007/11545989_1. |
[10] |
X. Cui and M. Yu,
Non-existence of positive solutions for a higher order fractional equation, Discret. Contin. Dynam. Syst., 39 (2019), 1379-1387.
doi: 10.3934/dcds.2019059. |
[11] |
S. Dipierro, O. Savin and E. Valdinoci, Definition of fractional Laplacian for functions with polynomial growth, Rev. Mat. Iberoam., 35 (2019), 1079-1122. |
[12] |
M. Fazly,
Liouville theorems for the polyharmonic Henon-Lane-Emden system, Methods Appl. Anal., 21 (2014), 265-282.
doi: 10.4310/MAA.2014.v21.n2.a5. |
[13] |
M. H$\acute{e}$non,
Numerical experiments on the stability of spheriocal stellar systems, Astron. Astrophys., 24 (1973), 229-238.
|
[14] |
N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag Berlin Heidelberg, New York, 1972. |
[15] |
D. Li, P. Niu and R. Zhuo,
Symmetry and nonexistence of positive solutions of integral systems with Hardy terms, J. Math. Anal. Appl., 424 (2015), 915-931.
doi: 10.1016/j.jmaa.2014.11.029. |
[16] |
Y. Li and B. Liu,
Singularity estimates for elliptic systems of m-laplacians, J. Korean Math. Soc., 55 (2018), 1423-1433.
doi: 10.4134/JKMS.j170724. |
[17] |
P. Ma, Q. F. Li and Y. Li,
A Pohozaev Identity for the Fractional H$\acute{e}$non System, Acta Math. Sin., Engl. Ser., 33 (2017), 1382-1396.
doi: 10.1007/s10114-017-6556-x. |
[18] |
P. Ma, Y. Li and J. Zhang,
Symmetry and nonexistence of positive solutions for fractional systems, Commun. Pure Appl. Anal., 17 (2018), 1053-1070.
doi: 10.3934/cpaa.2018051. |
[19] |
P. Polacik, P. Quittner and P. Souplet,
Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.
doi: 10.1215/S0012-7094-07-13935-8. |
[20] |
X. Ros-Oton and J. Serra,
The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[21] |
V. Tarasov and G. Zaslasvky,
Fractional dynamics of systems with long-range interaction, Commun. Nonlinear Sci. Numer. Simul., 11 (2006), 885-889.
doi: 10.1016/j.cnsns.2006.03.005. |
[22] |
P. Tolksdorf,
Regularity for a more general class of quasilinear elliptic equations, J. Differ. Equ., 51 (1984), 126-150.
doi: 10.1016/0022-0396(84)90105-0. |
[23] |
M. Yu, X. Zhang and B. Zhang,
Property of solutions for elliptic equation involving the higher-order fractional Laplacian in $ \mathbb{R}^{n}_{+}$, Commun. Pure Appl. Anal., 19 (2020), 3597-3612.
doi: 10.3934/cpaa.2020157. |
[24] |
R. Zhang, X. Wang and Z. D. Yang,
Symmetry and Nonexistence of Positive Solutions for an Elliptic System Involving the Fractional Laplacian, Quaest. Math., 45 (2022), 247-265.
doi: 10.2989/16073606.2020.1854363. |
show all references
References:
[1] |
J. Bertoin, L$\acute{e}$vy Processes, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1996. |
[2] |
J. P. Bouchard and A. Georges,
Anomalous diffusion in disordered media: Statistical mechanics, models and physical applications, Phys. Report., 195 (1990), 127-293.
doi: 10.1016/0370-1573(90)90099-N. |
[3] |
L. Caffarelli and L. Vasseur,
Drift diffusion equations with fractional diffusion and the quasigeostrophic equation, Ann. Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[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] |
W. Chen and C. Li,
Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 949-960.
doi: 10.1016/S0252-9602(09)60079-5. |
[6] |
W. Chen, C. Li and B. Ou,
Qualitative problems of solutions for an integral equation, Discret. Contin. Dynam. Syst., 12 (2005), 347-354.
doi: 10.3934/dcds.2005.12.347. |
[7] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Adv. Math., 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[8] |
W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co, Singapore, 2019.
doi: 10.1142/10550. |
[9] |
P. Constantin, Euler equations, navier-stokes equations and turbulence, Mathematical Foundation of Turbulent Viscous Flows Lecture Notes in Math., 1871 (2006), 1–43.
doi: 10.1007/11545989_1. |
[10] |
X. Cui and M. Yu,
Non-existence of positive solutions for a higher order fractional equation, Discret. Contin. Dynam. Syst., 39 (2019), 1379-1387.
doi: 10.3934/dcds.2019059. |
[11] |
S. Dipierro, O. Savin and E. Valdinoci, Definition of fractional Laplacian for functions with polynomial growth, Rev. Mat. Iberoam., 35 (2019), 1079-1122. |
[12] |
M. Fazly,
Liouville theorems for the polyharmonic Henon-Lane-Emden system, Methods Appl. Anal., 21 (2014), 265-282.
doi: 10.4310/MAA.2014.v21.n2.a5. |
[13] |
M. H$\acute{e}$non,
Numerical experiments on the stability of spheriocal stellar systems, Astron. Astrophys., 24 (1973), 229-238.
|
[14] |
N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag Berlin Heidelberg, New York, 1972. |
[15] |
D. Li, P. Niu and R. Zhuo,
Symmetry and nonexistence of positive solutions of integral systems with Hardy terms, J. Math. Anal. Appl., 424 (2015), 915-931.
doi: 10.1016/j.jmaa.2014.11.029. |
[16] |
Y. Li and B. Liu,
Singularity estimates for elliptic systems of m-laplacians, J. Korean Math. Soc., 55 (2018), 1423-1433.
doi: 10.4134/JKMS.j170724. |
[17] |
P. Ma, Q. F. Li and Y. Li,
A Pohozaev Identity for the Fractional H$\acute{e}$non System, Acta Math. Sin., Engl. Ser., 33 (2017), 1382-1396.
doi: 10.1007/s10114-017-6556-x. |
[18] |
P. Ma, Y. Li and J. Zhang,
Symmetry and nonexistence of positive solutions for fractional systems, Commun. Pure Appl. Anal., 17 (2018), 1053-1070.
doi: 10.3934/cpaa.2018051. |
[19] |
P. Polacik, P. Quittner and P. Souplet,
Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.
doi: 10.1215/S0012-7094-07-13935-8. |
[20] |
X. Ros-Oton and J. Serra,
The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[21] |
V. Tarasov and G. Zaslasvky,
Fractional dynamics of systems with long-range interaction, Commun. Nonlinear Sci. Numer. Simul., 11 (2006), 885-889.
doi: 10.1016/j.cnsns.2006.03.005. |
[22] |
P. Tolksdorf,
Regularity for a more general class of quasilinear elliptic equations, J. Differ. Equ., 51 (1984), 126-150.
doi: 10.1016/0022-0396(84)90105-0. |
[23] |
M. Yu, X. Zhang and B. Zhang,
Property of solutions for elliptic equation involving the higher-order fractional Laplacian in $ \mathbb{R}^{n}_{+}$, Commun. Pure Appl. Anal., 19 (2020), 3597-3612.
doi: 10.3934/cpaa.2020157. |
[24] |
R. Zhang, X. Wang and Z. D. Yang,
Symmetry and Nonexistence of Positive Solutions for an Elliptic System Involving the Fractional Laplacian, Quaest. Math., 45 (2022), 247-265.
doi: 10.2989/16073606.2020.1854363. |
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