\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Symmetry and nonexistence of positive solutions for fractional systems

  • * Corresponding author

    * Corresponding author
The authors are supported by NSFC 11571176
Abstract Full Text(HTML) Related Papers Cited by
  • We consider the following fractional Hénonsystem

    $\left\{ \begin{array}{*{35}{l}} {}&{{(-\vartriangle )}^{\alpha /2}}u = |x{{|}^{a}}{{v}^{p}},~~&x\in {{R}^{n}}, \\ {}&{{(-\vartriangle )}^{\alpha /2}}v = |x{{|}^{b}}{{u}^{q}},~~&x\in {{R}^{n}}, \\ {}&u\ge 0,v\ge 0,&{} \\\end{array} \right.$

    for $0<α<2$ and $a, b$ $≥0$, $n≥2$. Under rather weaker assumptions, by using a direct method of moving planes, we prove the nonexistence and symmetry of positive solutions in the subcritical case where $1<p<\frac{n+α+a}{n-α}$ and $1<q<\frac{n+α+b}{n-α}$.

    Mathematics Subject Classification: 35B06, 35B09, 35B50, 35B53.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  •   A. Arthur , X. Yan  and  M. Zhao , A Liouville-type theorem for higher order elliptic systems, Disc. Cont. Dyn. Syst., 34 (2014) , 3317-3339. 
      J. Busca  and  R. Man$\acute{a}$sevich , A Liouville-type theorem for Lane-Emden system, Indiana Univ. Math. J., 51 (2002) , 37-51. 
      G. Caristi , L. D'Ambrosio  and  E. Mitidieri , Representation formula for solutions to some classes of higher order systems and related Liouville theorems, Milan Journal of Mathematics, 76 (2008) , 27-67. 
      Ph. Clément , D. G. de Figueiredo  and  E. Mitidieri , Positive solutions of semilinear elliptic systems, Commun. Partial Diff. Equ., 17 (1992) , 923-940. 
      W. Chen , C. Li  and  Y. Li , A direct method of moving planes for the fractional Laplacian, Adv. in Math., 308 (2017) , 404-437. 
      W. Chen  and  C. Li , A priori estimates for prescribing scalar curvature equations, Ann. of Math., 145 (1997) , 547-564. 
      W. Chen  and  C. Li , Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal., 12 (2013) , 2497-2514. 
      W. Chen , C. Li  and  B. Ou , Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Syst., 12 (2005) , 347-354. 
      W. Chen , C. Li  and  B. Ou , Classification of solutions for a system of integral equations, Comm. Partial Diff. Eqs., 30 (2005) , 59-65. 
      W. Chen  and  C. Li , An integral system and the Lane-Emdem conjecture, Disc. Cont. Dyn. Syst., 4 (2009) , 1167-1184. 
      W. Chen , L. D'Ambrosio  and  Y. Li , Some Liouville theorems for the fractional Laplacian, Nonlinear Anal., 121 (2015) , 370-381. 
      W. Chen , Y. Fang  and  R. Yang , Liouville theorems involving the fractional Laplacian on a half space, Adv. in Math., 274 (2014) , 167-198. 
      L. D'Ambrosio  and  E. Mitidieri , Hardy-Littlewood-Sobolev systems and related Liouville theorems, Disc. Cont. Dyn. Syst., 7 (2014) , 653-671. 
      J. Dou  and  H. Zhou , Liouville theorem for fractional Hénon equation and system on Rn, Commun. Pure Appl. Anal., 14 (2015) , 493-515. 
      M. Fazly , Liouville theorems for the polyharmonic Hénon-Lane-Emden system, Methods and Appl. Anal., 21 (2014) , 265-282. 
      D. Figueiredo  and  P. Felmer , A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Super Pisa. Cl. Sci., 21 (1994) , 387-397. 
      M. Fazly  and  N. Ghoussoub , On the Hénon-Lane-Emden conjecture, Disc. Cont. Dyn. Syst., 34 (2014) , 2513-2533. 
      Y. Guo  and  J. Liu , Liouville type theorems for positive solutions of elliptic system in Rn, Comm. Partial Diff. Equ., 33 (2008) , 263-284. 
      B. Gidas , W. Ni  and  L. Nirenberg , Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979) , 209-243. 
      B. Gidas  and  B. Spruck , A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Diff. Equ., 6 (1981) , 883-901. 
      H. He , Infinitely many solutions for Hardy-Hénon type elliptic system in hyperbolic space, Ann. Acad. Sci. Fenn. Math., 40 (2015) , 969-983. 
      F. B. Hang , On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. Res. Lett., 14 (2007) , 373-383. 
      M. Hénon , Numerical experiments on the stability of spherical stellar systems, Symposium-International Astronomical Union, 62 (1974) , 259-259. 
      T. Jin , Symmetry and nonexistence of positive solutions of elliptic equations and systems with Hardy term, Ann. inst. Henri Poincaré, 28 (2011) , 965-981. 
      C. Lin , A classification of solutions of a conformally invariant fourth order equation in Rn, Comment. Math. Helv., 73 (1998) , 206-231. 
      Y. Lei , Asymptotic properties of positive solutions of the Hardy-Sobolev type equations, J. Differential Equations, 254 (2013) , 1774-1799. 
      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. 
      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. 
      F. Liu  and  J. Yang , Non-existence of Hardy-Hénon type elliptic system, Acta math. Sci. ser. B engl. Ed., 27 (2007) , 673-688. 
      E. Mitidieri , Nonexistence of positive solutions of semilinear elliptic systems in Rn, Diff. Inte. Equ., 9 (1996) , 465-479. 
      E. Mitidieri , A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993) , 125-151. 
      P. Pol$\acute{a}\check{c}$ik , P. Quittner  and  P. Souplet , Singularity and decay estimates in superlinear problems via Liouville-type theorems, Ⅰ: Elliptic systems, Duke Math. J., 139 (2007) , 555-579. 
      Q. H. Phan  and  Ph. Souplet , Liouville-type theorems and bounds of solutions of Hardy-Hénon equations, J. Differential Equations, 252 (2012) , 2544-2562. 
      Ph. Souplet , The proof of the Lane-Emden conjecture in four space dimensions, Adv. in Math., 221 (2009) , 1409-1427. 
      J. Serrin  and  H. Zou , Non-existence of positive solutions of Lane-Emden systems, Diff. Inte. Equ., 9 (1996) , 635-653. 
      J. Serrin  and  H. Zou , Existence of positive solutions of the Lane-Emden system, Atti Semin. Mat. Fis. Univ. Modena, 46 (1998) , 369-380. 
      D. Tang  and  Y. Fang , Regularity and nonexistence of solutions for a system involving the fractional Laplacian, Commun. Pure Appl. Anal., 14 (2015) , 2431-2451. 
      R. Zhuo , W. Chen , X. Cui  and  Z. Yuan , Symmetry and nonexistence of solutions for a nonlinear system involving the fractional Laplacian, Disc. Cont. Dyn. Syst., 36 (2016) , 1125-1141. 
  • 加载中
SHARE

Article Metrics

HTML views(403) PDF downloads(303) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return