Symmetry and monotonicity of positive solutions to systems involving general pseudo-relativistic operators

Xueying Chen; Guanfeng Li

doi:10.3934/cpaa.2023045

Article Contents

2023, Volume 22, Issue 6: 1745-1762. Doi: 10.3934/cpaa.2023045

This issue Previous Article Next Article Periodic solutions to a nonlinear Dirac-Klein-Gordon system with concave and convex nonlinearities

Symmetry and monotonicity of positive solutions to systems involving general pseudo-relativistic operators

Xueying Chen^1, , and
Guanfeng Li^2,

1.
College of Science, Xi'an University of Science and Technology, Xi'an 710054, China
2.
School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

^*Corresponding author: Xueying Chen
^*Corresponding author: Xueying Chen

Received: November 2022

Revised: February 2023

Early access: March 11, 2023

Published online: March 11, 2023

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Abstract

The aim of this paper is to derive some properties of solutions to general pseudo-relativistic systems. Some maximum principles are the key ingredients in the proof of the main results. For this purpose, a decay at infinity and a narrow region principle are established for the system. Then we utilize the direct method of moving planes to obtain the symmetry and monotonicity of positive solutions. The regions involved here are the unit ball and the whole space.

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Mathematics Subject Classification: Primary: 35R11; Secondary: 35B06, 35B09.

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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]	R. Carmona, W. 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.
[3]	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.
[4]	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.
[5]	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.
[6]	X. Chen, G. Li and S. Bao, Symmetry and monotonicity of positive solutions for a class of general pseudo-relativistic systems, Commun. Pure Appl. Anal., 21 (2022), 1755-1772. doi: 10.3934/cpaa.2022045.
[7]	Y. Chen and B. Liu, Symmetry and non-existence of positive solutions for fractional $p$-Laplacian systems, Nonlinear Anal.-Theor., 183 (2019), 303-322. doi: 10.1016/j.na.2019.02.023.
[8]	W. Choi, Y. Hong and J. Seok, Semilinear elliptic equations with the pseudo-relativistic operator on a bounded domain, Nonlinear Anal.-Theor., 173 (2018), 123-145. doi: 10.1016/j.na.2018.03.020.
[9]	W. Dai, G. 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.
[10]	A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, Higher Transcendental Functions, Vol. Ⅱ, McGraw-Hill, New York, 1953.,
[11]	M. Fall and V. Felli, Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discret. Contin. Dyn. Syst., 35 (2015), 5827-5867. doi: 10.3934/dcds.2015.35.5827.
[12]	Y. Guo and S. Peng, Liouville-type results for positive solutions of pseudo-relativistic Schrödinger system, P. Roy. Soc. Edinb. A, 153 (2021), 196-228. doi: 10.1017/prm.2021.81.
[13]	Y. Guo and S. Peng, Monotonicity and nonexistence of positive solutions for pseudo-relativistic equation with indefinite nonlinearity, Commun. Pure Appl. Anal., 21 (2022), 1637-1648. doi: 10.3934/cpaa.2022037.
[14]	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.
[15]	B. Liu and L. Ma, Radial symmetry results for fractional Laplacian systems, Nonlinear Anal.-Theor., 146 (2016), 120-135. doi: 10.1016/j.na.2016.08.022.
[16]	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.
[17]	C. Li and Z. Wu, Radial symmetry for systems of fractional Laplacian, Acta Math. Sci., 38 (2018), 1567-1582. doi: 10.1016/S0252-9602(18)30832-4.
[18]	E. H. Lieb and H. T. Yao, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., 112 (1987), 147-174.
[19]	E. H. Lieb and H. T. Yao, The stability and instability of relativistic matter, Commun. Math. Phys., 118 (1988), 177-213.
[20]	L. Luo and Z. Zhang, Symmetry and nonexistence of positive solutions for fully nonlinear nonlocal systems, Appl. Math. Lett., 124 (2022), 107674, 7 pp. doi: 10.1016/j.aml.2021.107674.
[21]	M. Ryznar, Estimate of Green function for relativistic α-stable processes, Potent. Anal., 17 (2002), 1-23. doi: 10.1023/A:1015231913916.
[22]	P. Wang and P. Niu, A direct method of moving planes for a fully nonlinear nonlocal system, Commun. Pure. Appl. Anal., 16 (2017), 1707-1718. doi: 10.3934/cpaa.2017082.
[23]	P. Wang and M. Yu, Solutions of fully nonlinear nonlocal systems, J. Math. Anal. Appl., 450 (2017), 982-995. doi: 10.1016/j.jmaa.2017.01.070.
[24]	L. Wu and P. Niu, Symmetry and nonexistence of positive solutions to fractional $p$-Laplacian equations, Disc. Cont. Dyn. Sys., 39 (2018), 1573-1583. doi: 10.3934/dcds.2019069.
[25]	L. Zhang, B. Ahmad, G. Wang and X. Ren, Radial symmetry of solution for fractional $p$-Laplacian system, Nonlinear Anal.-Theor., 196 (2020), 111801, 16 pp. doi: 10.1016/j.na.2020.111801.