August  2022, 21(8): 2857-2872. doi: 10.3934/cpaa.2022078

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

Received  November 2021 Revised  January 2022 Published  August 2022 Early access  April 2022

Fund Project: This research was supported by National Natural Science Foundation of China (Grant No. 11871278) and the National Natural Science Foundation of China (Grant No. 11571093)

In this paper, we study the following high-order Hardy-H
$ \acute{e} $
non type system:
$ \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} $
where
$ 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 $
. There are two cases to be considered. The first one is where the domain is the whole space
$ \mathbb{R}^{n} $
, and the second one is where the domain is bounded. First of all, we consider the above system in the whole space
$ \mathbb{R}^{n} $
, we show that the above system are equivalent to the integral system:
$ \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} $
Then by using the method of moving planes in integral forms, we prove that there are no positive solutions for the above integral system. In addition, while in the subcritical case
$ 1<p<\frac{n+\alpha+2a}{n-\alpha} $
,
$ 1<q<\frac{n+\alpha+2b}{n-\alpha} $
with
$ \alpha = \beta $
in the above elliptic system, we prove the nonexistence of a positive solution for the above system in
$ \mathbb{R}^{n} $
. Then, through the
$ Doubling\ Lemma $
we obtain the singularity estimates of the positive solutions on a bounded domain
$ \Omega $
.
Citation: Rong Zhang. Nonexistence of Positive Solutions for high-order Hardy-H$ \acute{e} $non Systems on $ \mathbb{R}^{n} $. Communications on Pure and Applied Analysis, 2022, 21 (8) : 2857-2872. doi: 10.3934/cpaa.2022078
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. ChenC. 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. ChenC. 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. LiP. 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. MaQ. 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. MaY. 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. PolacikP. 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. YuX. 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. ZhangX. 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. ChenC. 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. ChenC. 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. LiP. 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. MaQ. 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. MaY. 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. PolacikP. 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. YuX. 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. ZhangX. 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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