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Structure of positive solutions to a class of Schrödinger systems
A direct method of moving planes for fully nonlinear nonlocal operators and applications
Department of Mathematics, Tsinghua University, Beijing, 100084, China |
$ F_{s,m}(u(x)) = c_{N,s} m^{ \frac{N}{2}+s} P.V. \int_{\mathbb{R}^{N}} \frac{G(u(x)-u(y))}{|x-y|^{ \frac{N}{2}+s}} K_{ \frac{N}{2}+s}(m|x-y|)dy+m^{2s}u(x), $ |
$ s\in (0,1) $ |
$ m>0 $ |
$ \mathbb{R}^{N} $ |
$ \mathbb{R}^{N}_{+} $ |
$ \Omega $ |
$ \mathbb{R}^N $ |
References:
[1] |
V. Ambrosio, Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator, J. Math. Phys., 57 (2016), 051502, 18 pp.
doi: 10.1063/1.4949352. |
[2] |
H. Berestycki, L. A. Caffarelli and L. Nirenberg,
Inequalitites for second-order elliptic equations with applications to unbounded domains I, Duke Math. J., 81 (1996), 467-494.
doi: 10.1215/S0012-7094-96-08117-X. |
[3] |
H. Berestycki, L. A. Caffarelli and L. Nirenberg,
Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math., 50 (1997), 1089-1111.
doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6. |
[4] |
C. Brandle, E. Colorado, A. de Pablo and U. Sánchez,
A concave-convex elliptic problem involving the fractional Laplacian, Proc. Royal Soc. of Edinburgh-A: Math., 143 (2013), 39-71.
doi: 10.1017/S0308210511000175. |
[5] |
J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1996.
![]() ![]() |
[6] |
H. Berestycki, F. Hamel and R. Monneau,
One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J., 103 (2000), 375-396.
doi: 10.1215/S0012-7094-00-10331-6. |
[7] |
H. Berestycki and L. Nirenberg,
Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. Phys., 5 (1988), 237-275.
doi: 10.1016/0393-0440(88)90006-X. |
[8] |
H. Berestycki and L. Nirenberg, Some Qualitative Properties of Solutions of Semilinear Elliptic Equations in Cylindrical Domains, Analysis, et Cetera, Academic Press, Boston, MA, 1990.
![]() ![]() |
[9] |
H. Berestycki and L. Nirenberg,
On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.), 22 (1991), 1-37.
doi: 10.1007/BF01244896. |
[10] |
S.-Y. A. Chang and M. del Mar Gonzàlez,
Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432.
doi: 10.1016/j.aim.2010.07.016. |
[11] |
Y. Chen and B. Liu,
Symmetry and non-existence of positive solutions for fractional p-Laplacian systems, Nonlinear Anal., 183 (2019), 303-322.
doi: 10.1016/j.na.2019.02.023. |
[12] |
W. Chen and C. Li,
Maximum principles for the fractional $p$-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.
doi: 10.1016/j.aim.2018.07.016. |
[13] |
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. |
[14] |
W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co. Pte. Ltd., Singapore, 2020.
doi: 10.1142/10550. |
[15] |
W. Chen and C. Li,
Moving planes, moving spheres, and a priori estimates, J. Differential Equations, 195 (2003), 1-13.
doi: 10.1016/j.jde.2003.06.004. |
[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] |
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. |
[18] |
P. Constantin, Euler equations, Navier-Stokes equations and turbulence, in mathematical foundation of turbulent viscous flows, Lecture Notes in Math., Springer, Berlin, 1871 (2006), 1-43.
doi: 10.1007/11545989_1. |
[19] |
W. Chen and S. Qi,
Direct methods on fractional equations, Disc. Cont. Dyn. Syst.-A, 39 (2019), 1269-1310.
doi: 10.3934/dcds.2019055. |
[20] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. PDEs, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[21] |
X. Cabré and J. Tan,
Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[22] |
L. Caffarelli and L. Vasseur,
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Annals of Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[23] |
W. Chen and L. Wu, The sliding methods for the fractional $p$-Laplacian, Adv. Math., 361 (2020), 106933, 26 pp.
doi: 10.1016/j.aim.2019.106933. |
[24] |
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. |
[25] |
R. Carmona, W. C. Masters and B. Simon,
Relativistic Schrödinger operators: Asymptotic behavior of the eigenfunctions, J. Funct. Anal., 91 (1990), 117-142.
doi: 10.1016/0022-1236(90)90049-Q. |
[26] |
W. Dai, Y. Fang, J. Huang, Y. Qin and B. Wang,
Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete and Continuous Dynamical Systems-A, 39 (2019), 1389-1403.
doi: 10.3934/dcds.2018117. |
[27] |
W. Dai and Z. Liu, Classification of nonnegative solutions to static Schrödinger-Hartree and Schrödinger-Maxwell equations with combined nonlinearities, Calc. Var. & PDEs, 58 (2019), Art. 156, 24pp.
doi: 10.1007/s00526-019-1595-z. |
[28] |
W. Dai, Z. Liu and G. Qin, Classification of nonnegative solutions to static Schrödinger-Hartree-Maxwell type equations, preprint, submitted for publication, arXiv: 1909.00492. |
[29] |
W. Dai and G. Qin, Liouville type theorems for fractional and higher order Hénon-Hardy equations via the method of scaling spheres, preprint, submitted for publication, arXiv: 1810.02752. |
[30] |
W. Dai and G. Qin,
Classification of nonnegative classical solutions to third-order equations, Adv. Math., 328 (2018), 822-857.
doi: 10.1016/j.aim.2018.02.016. |
[31] |
W. Dai and G. Qin,
Liouville type theorems for elliptic equations with Dirichlet conditions in exterior domains, J. Differential Equations, 269 (2020), 7231-7252.
doi: 10.1016/j.jde.2020.05.026. |
[32] |
W. Dai, G. Qin and D. Wu, Direct methods for pseudo-relativistic Schrödinger operators, Journal of Geometric Analysis, (2020).
doi: 10.1007/s12220-020-00492-1. |
[33] |
S. Dipierro, N. Soave and E. Valdinoci,
On fractional elliptic equations in Lipschitz sets and epigraphs: Regularity, monotonicity and rigidity results, Math. Ann., 369 (2017), 1283-1326.
doi: 10.1007/s00208-016-1487-x. |
[34] |
A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions: Vol. Ⅱ, Journal of the Franklin Institute, McGraw-Hill, New York, 257 (1954), 150.
doi: 10.1016/0016-0032(54)90080-0. |
[35] |
M. M. Fall and V. Felli,
Sharp essential self-adjointness of relativistic Schrödinger operators with a singular potential, J. Funct. Anal., 267 (2014), 1851-1877.
doi: 10.1016/j.jfa.2014.06.010. |
[36] |
M. M. Fall and V. Felli,
Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst.-A, 35 (2015), 5827-5867.
doi: 10.3934/dcds.2015.35.5827. |
[37] |
R. L. Frank, E. Lenzmann and L. Silvestre,
Uniqueness of radial solutions for the fractional laplacian, Comm. Pure Appl. Math., 69 (2013), 1671-1726.
doi: 10.1002/cpa.21591. |
[38] |
E. De Giorgi, Convergence problems for functionals and operators, in Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, Rome, (1978), Pitagora, (1979). |
[39] |
Y. Guo and J. Liu,
Liouville type theorems for positive solutions of elliptic system in $R^{N}$, Commun. Partial Differ. Equ., 33 (2008), 263-284.
doi: 10.1080/03605300701257476. |
[40] |
I. W. Herbst,
Spectral theory of the operator $(p^{2}+m^{2})^{1/2}-Ze^{2}/r$, Comm. Math. Phys., 53 (1977), 285-294.
|
[41] |
J. Liu, Y. Guo and Y. Zhang,
Liouville-type theorems for polyharmonic systems in $R^{N}$, J. Differ. Equ., 225 (2006), 685-709.
doi: 10.1016/j.jde.2005.10.016. |
[42] |
B. Liu and L. Ma,
Radial symmetry results for fractional Laplacian systems,, Nonlinear Anal., 146 (2016), 120-135.
doi: 10.1016/j.na.2016.08.022. |
[43] |
L. Ma and L. Zhao,
Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467.
doi: 10.1007/s00205-008-0208-3. |
[44] |
S. Peng,
Liouville theorems for fractional and higher order Hénon-Hardy systems on $\mathbb{R}^{n}$, Complex Variables and Elliptic Equations, 11 (2020), 1-25.
doi: 10.1080/17476933.2020.1783661. |
[45] |
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. |
[46] |
M. Qu and L. Yang, Solutions to the nonlinear Schrödinger systems involving the fractional Laplacian, J. Inequal. Appl., 297 (2018), 16pp.
doi: 10.1186/s13660-018-1874-9. |
show all references
References:
[1] |
V. Ambrosio, Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator, J. Math. Phys., 57 (2016), 051502, 18 pp.
doi: 10.1063/1.4949352. |
[2] |
H. Berestycki, L. A. Caffarelli and L. Nirenberg,
Inequalitites for second-order elliptic equations with applications to unbounded domains I, Duke Math. J., 81 (1996), 467-494.
doi: 10.1215/S0012-7094-96-08117-X. |
[3] |
H. Berestycki, L. A. Caffarelli and L. Nirenberg,
Monotonicity for elliptic equations in unbounded Lipschitz domains, Comm. Pure Appl. Math., 50 (1997), 1089-1111.
doi: 10.1002/(SICI)1097-0312(199711)50:11<1089::AID-CPA2>3.0.CO;2-6. |
[4] |
C. Brandle, E. Colorado, A. de Pablo and U. Sánchez,
A concave-convex elliptic problem involving the fractional Laplacian, Proc. Royal Soc. of Edinburgh-A: Math., 143 (2013), 39-71.
doi: 10.1017/S0308210511000175. |
[5] |
J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 1996.
![]() ![]() |
[6] |
H. Berestycki, F. Hamel and R. Monneau,
One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J., 103 (2000), 375-396.
doi: 10.1215/S0012-7094-00-10331-6. |
[7] |
H. Berestycki and L. Nirenberg,
Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. Phys., 5 (1988), 237-275.
doi: 10.1016/0393-0440(88)90006-X. |
[8] |
H. Berestycki and L. Nirenberg, Some Qualitative Properties of Solutions of Semilinear Elliptic Equations in Cylindrical Domains, Analysis, et Cetera, Academic Press, Boston, MA, 1990.
![]() ![]() |
[9] |
H. Berestycki and L. Nirenberg,
On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.), 22 (1991), 1-37.
doi: 10.1007/BF01244896. |
[10] |
S.-Y. A. Chang and M. del Mar Gonzàlez,
Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432.
doi: 10.1016/j.aim.2010.07.016. |
[11] |
Y. Chen and B. Liu,
Symmetry and non-existence of positive solutions for fractional p-Laplacian systems, Nonlinear Anal., 183 (2019), 303-322.
doi: 10.1016/j.na.2019.02.023. |
[12] |
W. Chen and C. Li,
Maximum principles for the fractional $p$-Laplacian and symmetry of solutions, Adv. Math., 335 (2018), 735-758.
doi: 10.1016/j.aim.2018.07.016. |
[13] |
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. |
[14] |
W. Chen, Y. Li and P. Ma, The Fractional Laplacian, World Scientific Publishing Co. Pte. Ltd., Singapore, 2020.
doi: 10.1142/10550. |
[15] |
W. Chen and C. Li,
Moving planes, moving spheres, and a priori estimates, J. Differential Equations, 195 (2003), 1-13.
doi: 10.1016/j.jde.2003.06.004. |
[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] |
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. |
[18] |
P. Constantin, Euler equations, Navier-Stokes equations and turbulence, in mathematical foundation of turbulent viscous flows, Lecture Notes in Math., Springer, Berlin, 1871 (2006), 1-43.
doi: 10.1007/11545989_1. |
[19] |
W. Chen and S. Qi,
Direct methods on fractional equations, Disc. Cont. Dyn. Syst.-A, 39 (2019), 1269-1310.
doi: 10.3934/dcds.2019055. |
[20] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. PDEs, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[21] |
X. Cabré and J. Tan,
Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[22] |
L. Caffarelli and L. Vasseur,
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Annals of Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[23] |
W. Chen and L. Wu, The sliding methods for the fractional $p$-Laplacian, Adv. Math., 361 (2020), 106933, 26 pp.
doi: 10.1016/j.aim.2019.106933. |
[24] |
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. |
[25] |
R. Carmona, W. C. Masters and B. Simon,
Relativistic Schrödinger operators: Asymptotic behavior of the eigenfunctions, J. Funct. Anal., 91 (1990), 117-142.
doi: 10.1016/0022-1236(90)90049-Q. |
[26] |
W. Dai, Y. Fang, J. Huang, Y. Qin and B. Wang,
Regularity and classification of solutions to static Hartree equations involving fractional Laplacians, Discrete and Continuous Dynamical Systems-A, 39 (2019), 1389-1403.
doi: 10.3934/dcds.2018117. |
[27] |
W. Dai and Z. Liu, Classification of nonnegative solutions to static Schrödinger-Hartree and Schrödinger-Maxwell equations with combined nonlinearities, Calc. Var. & PDEs, 58 (2019), Art. 156, 24pp.
doi: 10.1007/s00526-019-1595-z. |
[28] |
W. Dai, Z. Liu and G. Qin, Classification of nonnegative solutions to static Schrödinger-Hartree-Maxwell type equations, preprint, submitted for publication, arXiv: 1909.00492. |
[29] |
W. Dai and G. Qin, Liouville type theorems for fractional and higher order Hénon-Hardy equations via the method of scaling spheres, preprint, submitted for publication, arXiv: 1810.02752. |
[30] |
W. Dai and G. Qin,
Classification of nonnegative classical solutions to third-order equations, Adv. Math., 328 (2018), 822-857.
doi: 10.1016/j.aim.2018.02.016. |
[31] |
W. Dai and G. Qin,
Liouville type theorems for elliptic equations with Dirichlet conditions in exterior domains, J. Differential Equations, 269 (2020), 7231-7252.
doi: 10.1016/j.jde.2020.05.026. |
[32] |
W. Dai, G. Qin and D. Wu, Direct methods for pseudo-relativistic Schrödinger operators, Journal of Geometric Analysis, (2020).
doi: 10.1007/s12220-020-00492-1. |
[33] |
S. Dipierro, N. Soave and E. Valdinoci,
On fractional elliptic equations in Lipschitz sets and epigraphs: Regularity, monotonicity and rigidity results, Math. Ann., 369 (2017), 1283-1326.
doi: 10.1007/s00208-016-1487-x. |
[34] |
A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions: Vol. Ⅱ, Journal of the Franklin Institute, McGraw-Hill, New York, 257 (1954), 150.
doi: 10.1016/0016-0032(54)90080-0. |
[35] |
M. M. Fall and V. Felli,
Sharp essential self-adjointness of relativistic Schrödinger operators with a singular potential, J. Funct. Anal., 267 (2014), 1851-1877.
doi: 10.1016/j.jfa.2014.06.010. |
[36] |
M. M. Fall and V. Felli,
Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst.-A, 35 (2015), 5827-5867.
doi: 10.3934/dcds.2015.35.5827. |
[37] |
R. L. Frank, E. Lenzmann and L. Silvestre,
Uniqueness of radial solutions for the fractional laplacian, Comm. Pure Appl. Math., 69 (2013), 1671-1726.
doi: 10.1002/cpa.21591. |
[38] |
E. De Giorgi, Convergence problems for functionals and operators, in Proc. Int. Meeting on Recent Methods in Nonlinear Analysis, Rome, (1978), Pitagora, (1979). |
[39] |
Y. Guo and J. Liu,
Liouville type theorems for positive solutions of elliptic system in $R^{N}$, Commun. Partial Differ. Equ., 33 (2008), 263-284.
doi: 10.1080/03605300701257476. |
[40] |
I. W. Herbst,
Spectral theory of the operator $(p^{2}+m^{2})^{1/2}-Ze^{2}/r$, Comm. Math. Phys., 53 (1977), 285-294.
|
[41] |
J. Liu, Y. Guo and Y. Zhang,
Liouville-type theorems for polyharmonic systems in $R^{N}$, J. Differ. Equ., 225 (2006), 685-709.
doi: 10.1016/j.jde.2005.10.016. |
[42] |
B. Liu and L. Ma,
Radial symmetry results for fractional Laplacian systems,, Nonlinear Anal., 146 (2016), 120-135.
doi: 10.1016/j.na.2016.08.022. |
[43] |
L. Ma and L. Zhao,
Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Rational Mech. Anal., 195 (2010), 455-467.
doi: 10.1007/s00205-008-0208-3. |
[44] |
S. Peng,
Liouville theorems for fractional and higher order Hénon-Hardy systems on $\mathbb{R}^{n}$, Complex Variables and Elliptic Equations, 11 (2020), 1-25.
doi: 10.1080/17476933.2020.1783661. |
[45] |
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. |
[46] |
M. Qu and L. Yang, Solutions to the nonlinear Schrödinger systems involving the fractional Laplacian, J. Inequal. Appl., 297 (2018), 16pp.
doi: 10.1186/s13660-018-1874-9. |
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