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Uniqueness of positive radial solutions of a semilinear elliptic equation in an annulus
Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian
1. | School of mathematics and statistics, Huanghuai University, Zhumadian, Henan 463000, China |
2. | Department of Mathematics, Baylor University, Waco, TX 76798, USA |
$ R^n$ |
$\left\{ {\begin{array}{*{20}{l}}{{{( - \Delta )}^{\frac{\alpha }{2}}}u(x) = {u^{{\beta _1}}}(x){v^{{\tau _1}}}(x),}&{{\rm{in}}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {R^n},}\\{{{( - \Delta )}^{\frac{\gamma }{2}}}v(x) = {u^{{\beta _2}}}(x){v^{{\tau _2}}}(x),}& \ \ \ {{\rm{ in}}{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {R^n},}\end{array}} \right.\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right)$ |
$ α$ |
$ γ$ |
$ α$ |
$ γ$ |
References:
[1] |
H. Berestycki and L. Nirenberg,
On the method of moving planes and the sliding method, Bol. Soc. Brazil. Mat. (N.S.), 22 (1991), 1-37.
doi: 10.1007/BF01244896. |
[2] |
J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996. |
[3] |
J. P. Bouchard and A. Georges,
Anomalous diffusion in disordered media, Statistical mechanics, models and physical applications, Physics reports, 195 (1990), 127-293.
doi: 10.1016/0370-1573(90)90099-N. |
[4] |
L. Caffarelli and L. Vasseur,
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[5] |
L. Caffarelli, B. Gidas and J. Spruck,
Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.
doi: 10.1002/cpa.3160420304. |
[6] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. PDE., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[7] |
L. Caffarelli and L. Silvestre,
Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.
doi: 10.1002/cpa.20274. |
[8] |
M. Cai and L. Ma,
Moving planes for nonlinear fractional Laplacian equation with negative powers, Disc. Cont. Dyn. Sys., 38 (2018), 4603-4615.
doi: 10.3934/dcds.2018201. |
[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] |
W. Chen and C. Li,
Classification of solutions of some nonlinear elliptic equations, Duke Mathematical Journal, 63 (1991), 615-622.
doi: 10.1215/S0012-7094-91-06325-8. |
[11] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS book series on Diffreential Equatons & Dynamical Systerms, Volume 4, 2010. |
[12] |
W. Chen and C. Li,
A priori estimates for prescribing scalar curvature equations, Annals of Math., 145 (1997), 547-564.
doi: 10.2307/2951844. |
[13] |
W. Chen and C. Li,
An integral system and the Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 24 (2009), 1167-1184.
doi: 10.3934/dcds.2009.24.1167. |
[14] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Advances in Mathematics, 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[15] |
W. Chen, C. Li and B. Ou,
Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.
doi: 10.3934/dcds.2005.12.347. |
[16] |
W. Chen, C. Li and B. Ou,
Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[17] |
C. Li and L. Ma,
Uniqueness of positive bound states to Shrodinger systems with critical exponents, SIAM J. Math. Analysis, 40 (2008), 1049-1057.
doi: 10.1137/080712301. |
[18] |
T. Lin and J. Wei,
Spikes in two coupled nonlinear Schrodinger equations, Ann. Inst. H. Poincare Anal. Non Linéaire, 22 (2005), 403-439.
doi: 10.1016/j.anihpc.2004.03.004. |
[19] |
B. Liu,
Direct method of moving planes for logarithmic Laplacian system in bounded domains, Disc. Cont. Dyn. Sys., 38 (2018), 5339-5349.
doi: 10.3934/dcds.2018235. |
[20] |
L. Ma and L. Zhao, Sharp thresholds of blow up and global existence for the coupled nonlinear Schrödinger system, J. Math. Phys., 49 (2008), 062103, 17 pp.
doi: 10.1063/1.2939238. |
[21] |
L. Silvestre,
Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[22] |
D. Tang and Y. Fang,
Method of sub-super solutions for fractional elliptic equations, Dis. Con. Dyn. Sys., 23 (2018), 3153-3165.
doi: 10.3934/dcdsb.2017212. |
[23] |
V. Tarasov and G. Zaslasvky,
Fractional dynamics of systems with long-range interaction, Comm. Nonl. Sci. Numer. Simul., 11 (2006), 885-889.
doi: 10.1016/j.cnsns.2006.03.005. |
[24] |
R. Zhuo and F. Li,
Liouville type theorems for Schrödinger systems in a half space, Science China Mathematics, 58 (2015), 179-196.
doi: 10.1007/s11425-014-4925-9. |
[25] |
R. Zhuo, F. Li and B. Lv,
Liouville type theorems for Schrödinger system with Navier boundary conditions in a half space, Comm. Pure Appl. Anal., 13 (2014), 977-990.
doi: 10.3934/cpaa.2014.13.977. |
[26] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan,
Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Dis. Con. Dyn. Sys., 36 (2016), 1125-1141.
doi: 10.3934/dcds.2016.36.1125. |
show all references
References:
[1] |
H. Berestycki and L. Nirenberg,
On the method of moving planes and the sliding method, Bol. Soc. Brazil. Mat. (N.S.), 22 (1991), 1-37.
doi: 10.1007/BF01244896. |
[2] |
J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, 121 Cambridge University Press, Cambridge, 1996. |
[3] |
J. P. Bouchard and A. Georges,
Anomalous diffusion in disordered media, Statistical mechanics, models and physical applications, Physics reports, 195 (1990), 127-293.
doi: 10.1016/0370-1573(90)90099-N. |
[4] |
L. Caffarelli and L. Vasseur,
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[5] |
L. Caffarelli, B. Gidas and J. Spruck,
Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989), 271-297.
doi: 10.1002/cpa.3160420304. |
[6] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. PDE., 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[7] |
L. Caffarelli and L. Silvestre,
Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.
doi: 10.1002/cpa.20274. |
[8] |
M. Cai and L. Ma,
Moving planes for nonlinear fractional Laplacian equation with negative powers, Disc. Cont. Dyn. Sys., 38 (2018), 4603-4615.
doi: 10.3934/dcds.2018201. |
[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] |
W. Chen and C. Li,
Classification of solutions of some nonlinear elliptic equations, Duke Mathematical Journal, 63 (1991), 615-622.
doi: 10.1215/S0012-7094-91-06325-8. |
[11] |
W. Chen and C. Li, Methods on Nonlinear Elliptic Equations, AIMS book series on Diffreential Equatons & Dynamical Systerms, Volume 4, 2010. |
[12] |
W. Chen and C. Li,
A priori estimates for prescribing scalar curvature equations, Annals of Math., 145 (1997), 547-564.
doi: 10.2307/2951844. |
[13] |
W. Chen and C. Li,
An integral system and the Lane-Emden conjecture, Disc. Cont. Dyn. Sys., 24 (2009), 1167-1184.
doi: 10.3934/dcds.2009.24.1167. |
[14] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Advances in Mathematics, 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[15] |
W. Chen, C. Li and B. Ou,
Qualitative properties of solutions for an integral equation, Disc. Cont. Dyn. Sys., 12 (2005), 347-354.
doi: 10.3934/dcds.2005.12.347. |
[16] |
W. Chen, C. Li and B. Ou,
Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[17] |
C. Li and L. Ma,
Uniqueness of positive bound states to Shrodinger systems with critical exponents, SIAM J. Math. Analysis, 40 (2008), 1049-1057.
doi: 10.1137/080712301. |
[18] |
T. Lin and J. Wei,
Spikes in two coupled nonlinear Schrodinger equations, Ann. Inst. H. Poincare Anal. Non Linéaire, 22 (2005), 403-439.
doi: 10.1016/j.anihpc.2004.03.004. |
[19] |
B. Liu,
Direct method of moving planes for logarithmic Laplacian system in bounded domains, Disc. Cont. Dyn. Sys., 38 (2018), 5339-5349.
doi: 10.3934/dcds.2018235. |
[20] |
L. Ma and L. Zhao, Sharp thresholds of blow up and global existence for the coupled nonlinear Schrödinger system, J. Math. Phys., 49 (2008), 062103, 17 pp.
doi: 10.1063/1.2939238. |
[21] |
L. Silvestre,
Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[22] |
D. Tang and Y. Fang,
Method of sub-super solutions for fractional elliptic equations, Dis. Con. Dyn. Sys., 23 (2018), 3153-3165.
doi: 10.3934/dcdsb.2017212. |
[23] |
V. Tarasov and G. Zaslasvky,
Fractional dynamics of systems with long-range interaction, Comm. Nonl. Sci. Numer. Simul., 11 (2006), 885-889.
doi: 10.1016/j.cnsns.2006.03.005. |
[24] |
R. Zhuo and F. Li,
Liouville type theorems for Schrödinger systems in a half space, Science China Mathematics, 58 (2015), 179-196.
doi: 10.1007/s11425-014-4925-9. |
[25] |
R. Zhuo, F. Li and B. Lv,
Liouville type theorems for Schrödinger system with Navier boundary conditions in a half space, Comm. Pure Appl. Anal., 13 (2014), 977-990.
doi: 10.3934/cpaa.2014.13.977. |
[26] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan,
Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian, Dis. Con. Dyn. Sys., 36 (2016), 1125-1141.
doi: 10.3934/dcds.2016.36.1125. |
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