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May  2022, 21(5): 1755-1772. doi: 10.3934/cpaa.2022045

Symmetry and monotonicity of positive solutions for a class of general pseudo-relativistic systems

School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author

Received  October 2021 Revised  January 2022 Published  May 2022 Early access  February 2022

Fund Project: This work was supported by the National Natural Science Foundation of China (Grant No. 62173111)

In this paper, we focus on a class of general pseudo-relativistic systems
$ \begin{equation*} \begin{cases} \begin{aligned} &(-\Delta+m^2)^su(x) = f(u(x), v(x)), \\ &(-\Delta+m^2)^tv(x) = g(u(x), v(x)), \end{aligned} \end{cases} \end{equation*} $
where
$ m \in (0, +\infty) $
and
$ s, t \in (0,1) $
. Before giving the main results, we first introduce a decay at infinity and a narrow region principle. Then we implement the direct method of moving planes to show the radial symmetry and monotonicity of positive solutions for the above system in both the unit ball and the whole space.
Citation: Xueying Chen, Guanfeng Li, Sijia Bao. Symmetry and monotonicity of positive solutions for a class of general pseudo-relativistic systems. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1755-1772. doi: 10.3934/cpaa.2022045
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 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.

[3]

J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differ. Equ., 163 (2000), 41-56.  doi: 10.1006/jdeq.1999.3701.

[4]

H. Bueno, AHS. Medeiros and GA. Pereira, Pohozaev-type identities for a pseudo-relativistic Schrödinger operator and applications, arXiv: 1810.07597.

[5]

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.

[6]

R. CarmonaW. 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.

[7]

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.

[8]

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.

[9]

W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differ. Equ., 56 (2017), 18 pp. doi: 10.1007/s00526-017-1110-3.

[10]

W. ChenC. Li and B. Ou, Classification of solutions for a system of integral equations, Commun. Partial Differ. Equ., 30 (2005), 59-65. 

[11]

W. ChenC. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.  doi: 10.3934/dcds.2005.12.347.

[12]

W. ChoiY. Hong and J. Seok, Semilinear elliptic equations with the pseudo-relativistic operator on a bounded domain, Nonlinear Anal., 173 (2018), 123-145.  doi: 10.1016/j.na.2018.03.020.

[13]

W. DaiG. Qin and D. Wu, Direct methods for pseudo-relativistic Schrödinger operators, J. Geom. Anal., 31 (2021), 5555-5618.  doi: 10.1007/s12220-020-00492-1.

[14]

S. DipierroN. 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.

[15]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, Higher Transcendental Functions, McGraw-Hill, New York, 1953.

[16]

M. Fall and V. Felli, Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst., 35 (2015), 5827-5867.  doi: 10.3934/dcds.2015.35.5827.

[17]

B. GidasW. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243. 

[18]

Y. Guo and S. Peng, Symmetry and monotonicity of nonnegative solutions to pseudo-relativistic Choquard equations, Z. Angew. Math. Phys., 72 (2021), 1-20.  doi: 10.1007/s00033-021-01551-5.

[19]

Y. Guo and S. Peng, Liouville-type results for positive solutions of pseudo-relativistic Schrödinger system, P. Roy. Soc. Edinb. A, (2021), 1-33. 

[20]

Y. Li and P. Ma, Symmetry of solutions for a fractional system, Sci. China Math., 60 (2017), 1805-1824.  doi: 10.1007/s11425-016-0231-x.

[21]

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.

[22]

M. Ryznar, Estimate of Green function for relativistic $\alpha$-stable processes, Potential Anal., 17 (2002), 1-23.  doi: 10.1023/A:1015231913916.

[23]

P. Wang and Y. Wang, Positive solutions for a weighted fractional system, Acta Math. Sci., 38 (2018), 935-949.  doi: 10.1016/S0252-9602(18)30794-X.

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 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.

[3]

J. Busca and B. Sirakov, Symmetry results for semilinear elliptic systems in the whole space, J. Differ. Equ., 163 (2000), 41-56.  doi: 10.1006/jdeq.1999.3701.

[4]

H. Bueno, AHS. Medeiros and GA. Pereira, Pohozaev-type identities for a pseudo-relativistic Schrödinger operator and applications, arXiv: 1810.07597.

[5]

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.

[6]

R. CarmonaW. 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.

[7]

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.

[8]

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.

[9]

W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differ. Equ., 56 (2017), 18 pp. doi: 10.1007/s00526-017-1110-3.

[10]

W. ChenC. Li and B. Ou, Classification of solutions for a system of integral equations, Commun. Partial Differ. Equ., 30 (2005), 59-65. 

[11]

W. ChenC. Li and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst., 12 (2005), 347-354.  doi: 10.3934/dcds.2005.12.347.

[12]

W. ChoiY. Hong and J. Seok, Semilinear elliptic equations with the pseudo-relativistic operator on a bounded domain, Nonlinear Anal., 173 (2018), 123-145.  doi: 10.1016/j.na.2018.03.020.

[13]

W. DaiG. Qin and D. Wu, Direct methods for pseudo-relativistic Schrödinger operators, J. Geom. Anal., 31 (2021), 5555-5618.  doi: 10.1007/s12220-020-00492-1.

[14]

S. DipierroN. 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.

[15]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, Higher Transcendental Functions, McGraw-Hill, New York, 1953.

[16]

M. Fall and V. Felli, Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst., 35 (2015), 5827-5867.  doi: 10.3934/dcds.2015.35.5827.

[17]

B. GidasW. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 68 (1979), 209-243. 

[18]

Y. Guo and S. Peng, Symmetry and monotonicity of nonnegative solutions to pseudo-relativistic Choquard equations, Z. Angew. Math. Phys., 72 (2021), 1-20.  doi: 10.1007/s00033-021-01551-5.

[19]

Y. Guo and S. Peng, Liouville-type results for positive solutions of pseudo-relativistic Schrödinger system, P. Roy. Soc. Edinb. A, (2021), 1-33. 

[20]

Y. Li and P. Ma, Symmetry of solutions for a fractional system, Sci. China Math., 60 (2017), 1805-1824.  doi: 10.1007/s11425-016-0231-x.

[21]

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.

[22]

M. Ryznar, Estimate of Green function for relativistic $\alpha$-stable processes, Potential Anal., 17 (2002), 1-23.  doi: 10.1023/A:1015231913916.

[23]

P. Wang and Y. Wang, Positive solutions for a weighted fractional system, Acta Math. Sci., 38 (2018), 935-949.  doi: 10.1016/S0252-9602(18)30794-X.

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