Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres

Xiaoxue Ji; Pengcheng Niu; Pengyan Wang

doi:10.3934/cpaa.2020051

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February 2020, 19(2): 1111-1128. doi: 10.3934/cpaa.2020051

Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres

Xiaoxue Ji , Pengcheng Niu ^, and Pengyan Wang

Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaan xi, 710129, China

^* Corresponding author: Pengcheng Niu

Received April 2019 Revised June 2019 Published October 2019

Fund Project: The first author is supported by NSFC grant No. 11771354 and Natural Science Basic Research Plan in Shaanxi Province of China grant No.2017JM5140.

In this article, we consider the cooperative semi-linear fractional system

$ (-\Delta)^{\frac{\alpha}{2}}\vec {u}(x) = \vec {h}(x,\vec {u}(x)), $

where

$ 0<\alpha <2 $

$ \vec u $

and

$ \vec h $

stand for

$ k $

-dimentional vector-valued functions, and

$ \vec {h}(x,\vec {u}(x)) $

is locally Lipschitz in

$ \vec {u} $

. We first establish two narrow region principles for different cases. Based on these principles, we use the direct method of moving spheres to prove the non-existence of positive solutions of the above system in bounded star-shaped domains and the whole space.

Keywords: Semi-linear fractional system, non-existence, narrow region principle, the direct method of moving spheres, bounded star-shaped domain.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation: Xiaoxue Ji, Pengcheng Niu, Pengyan Wang. Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1111-1128. doi: 10.3934/cpaa.2020051

References:

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[2]	J. Bear, Dynamics of Fluids in Porous Media, American Elsevier, New York, 1972.
[3]	M. Belloni, V. Ferone and B. Kawohl, Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators, Special issue dedicated to Lawrence E. Payne. Z. Angew. Math. Phys., 54 (2003), 771-783. doi: 10.1007/s00033-003-3209-y.
[4]	J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics 121, Cambridge University Press, Cambridge, 1996.
[5]	P. J. Bouchard and A. Georges, Anomalous diffusion in disordered media: Statistical mechanics, models and physical applications, Phys. Rep., 195 (1990), 127-293.
[6]	C. Brandle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect., 143 (2013), 39-71. doi: 10.1017/S0308210511000175.
[7]	X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Advances in Math., 224 (2010), 2052-2053. doi: 10.1016/j.aim.2010.01.025.
[8]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Patial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[9]	L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.
[10]	L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems, Discrete Contin. Dyn. Syst., 33 (2013), 3937-3955. doi: 10.3934/dcds.2013.33.3937.
[11]	W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differential Equations, 56 (2017). doi: 10.1007/s00526-017-1110-3.
[12]	W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Advances in Math., 308 (2017), 404-437. doi: 10.1016/j.aim.2016.11.038.
[13]	W. Chen, Y. Li and R. Zhang, A direct method of moving spheres on fractional order equations, J. Funct. Anal., 272 (2017), 4131-4157. doi: 10.1016/j.jfa.2017.02.022.
[14]	W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems, J. Math. Anal. Appl., 377 (2001), 744-753. doi: 10.1016/j.jmaa.2010.11.035.
[15]	P. Constantin, Euler equations, Navier-Stokes equations and turbulence, Mathematical Foundation of Turbulent Viscous Flows, Lecture notes in Math. Springer, Berlin, 1871 (2006), 1–43. doi: 10.1007/11545989_1.
[16]	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.
[17]	S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Ann. Mat. Pura Appl., 195 (2016), 273-291. doi: 10.1007/s10231-014-0462-y.
[18]	D. Li, P. Niu and R. Zhuo, Nonexistence of positive solutions for an integral equation related to the Hardy-Sobolev inequality, Acta Appl. Math., 134 (2014), 185-200. doi: 10.1007/s10440-014-9878-z.
[19]	D. Li, P. Niu and R. Zhuo, Symmetry and non-existence of positive solutions for PDE system with Navier boundary conditions on a half space, Compex Var. Elliptic Equ., 59 (2014). doi: 10.1080/17476933.2013.854346.
[20]	D. Li, P. Niu and R. Zhuo, Symmetry and nonexistence of positive solutions of integral systems with hardy term, J. Math. Anal. Appl., 424 (2015), 915-931. doi: 10.1016/j.jmaa.2014.11.029.
[21]	G. Lu and J. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Anal., 75 (2012), 3036-3048. doi: 10.1016/j.na.2011.11.036.
[22]	W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differential Equations, 161 (2000), 219-243. doi: 10.1006/jdeq.1999.3700.
[23]	V. Tarasov and G. Zaslasvky, Fractional dynamics of systems with long-range interaction, Comm. Nonl Sci. Numer. Simul., 11 (2006), 885-898. doi: 10.1016/j.cnsns.2006.03.005.
[24]	P. Wang and P. Niu, A direct method of moving planes for a fully nonlinear nonlocal system, Commun. Pur. Appl. Anal., 16 (2017), 1707-1718.
[25]	P. Wang and M. Yu, Solutions of fully nonlinear nonlocal systems, J. Math. Anal. Appl., 450 (2017), 982-995.
[26]	R. Zhuo, W. Chen, X. Cui and Z. Yuan, Symmetry and non-existence of solutions for nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141. doi: 10.3934/dcds.2016.36.1125.
[27]	R. Zhuo and D. Li, A system of integral equations on half space, J. Math. Anal. Appl., 381 (2011), 392-401.

show all references

References:

[1]	D. Applebeaum, Lévy Processes and Stochastic Calculus, second edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009.
[2]	J. Bear, Dynamics of Fluids in Porous Media, American Elsevier, New York, 1972.
[3]	M. Belloni, V. Ferone and B. Kawohl, Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators, Special issue dedicated to Lawrence E. Payne. Z. Angew. Math. Phys., 54 (2003), 771-783. doi: 10.1007/s00033-003-3209-y.
[4]	J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics 121, Cambridge University Press, Cambridge, 1996.
[5]	P. J. Bouchard and A. Georges, Anomalous diffusion in disordered media: Statistical mechanics, models and physical applications, Phys. Rep., 195 (1990), 127-293.
[6]	C. Brandle, E. Colorado, A. de Pablo and U. Sanchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect., 143 (2013), 39-71. doi: 10.1017/S0308210511000175.
[7]	X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Advances in Math., 224 (2010), 2052-2053. doi: 10.1016/j.aim.2010.01.025.
[8]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Patial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.
[9]	L. Caffarelli and L. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.
[10]	L. Cao and W. Chen, Liouville type theorems for poly-harmonic Navier problems, Discrete Contin. Dyn. Syst., 33 (2013), 3937-3955. doi: 10.3934/dcds.2013.33.3937.
[11]	W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differential Equations, 56 (2017). doi: 10.1007/s00526-017-1110-3.
[12]	W. Chen, C. Li and Y. Li, A direct method of moving planes for the fractional Laplacian, Advances in Math., 308 (2017), 404-437. doi: 10.1016/j.aim.2016.11.038.
[13]	W. Chen, Y. Li and R. Zhang, A direct method of moving spheres on fractional order equations, J. Funct. Anal., 272 (2017), 4131-4157. doi: 10.1016/j.jfa.2017.02.022.
[14]	W. Chen and J. Zhu, Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems, J. Math. Anal. Appl., 377 (2001), 744-753. doi: 10.1016/j.jmaa.2010.11.035.
[15]	P. Constantin, Euler equations, Navier-Stokes equations and turbulence, Mathematical Foundation of Turbulent Viscous Flows, Lecture notes in Math. Springer, Berlin, 1871 (2006), 1–43. doi: 10.1007/11545989_1.
[16]	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.
[17]	S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Ann. Mat. Pura Appl., 195 (2016), 273-291. doi: 10.1007/s10231-014-0462-y.
[18]	D. Li, P. Niu and R. Zhuo, Nonexistence of positive solutions for an integral equation related to the Hardy-Sobolev inequality, Acta Appl. Math., 134 (2014), 185-200. doi: 10.1007/s10440-014-9878-z.
[19]	D. Li, P. Niu and R. Zhuo, Symmetry and non-existence of positive solutions for PDE system with Navier boundary conditions on a half space, Compex Var. Elliptic Equ., 59 (2014). doi: 10.1080/17476933.2013.854346.
[20]	D. Li, P. Niu and R. Zhuo, Symmetry and nonexistence of positive solutions of integral systems with hardy term, J. Math. Anal. Appl., 424 (2015), 915-931. doi: 10.1016/j.jmaa.2014.11.029.
[21]	G. Lu and J. Zhu, An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Anal., 75 (2012), 3036-3048. doi: 10.1016/j.na.2011.11.036.
[22]	W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differential Equations, 161 (2000), 219-243. doi: 10.1006/jdeq.1999.3700.
[23]	V. Tarasov and G. Zaslasvky, Fractional dynamics of systems with long-range interaction, Comm. Nonl Sci. Numer. Simul., 11 (2006), 885-898. doi: 10.1016/j.cnsns.2006.03.005.
[24]	P. Wang and P. Niu, A direct method of moving planes for a fully nonlinear nonlocal system, Commun. Pur. Appl. Anal., 16 (2017), 1707-1718.
[25]	P. Wang and M. Yu, Solutions of fully nonlinear nonlocal systems, J. Math. Anal. Appl., 450 (2017), 982-995.
[26]	R. Zhuo, W. Chen, X. Cui and Z. Yuan, Symmetry and non-existence of solutions for nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141. doi: 10.3934/dcds.2016.36.1125.
[27]	R. Zhuo and D. Li, A system of integral equations on half space, J. Math. Anal. Appl., 381 (2011), 392-401.

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Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres

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