Monotonicity and nonexistence of positive solutions for pseudo-relativistic equation with indefinite nonlinearity

Yuxia Guo; Shaolong Peng

doi:10.3934/cpaa.2022037

Article Contents

2022, Volume 21, Issue 5: 1637-1648. Doi: 10.3934/cpaa.2022037

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Monotonicity and nonexistence of positive solutions for pseudo-relativistic equation with indefinite nonlinearity

Yuxia Guo and
Shaolong Peng^,

Department of Mathematics, Tsinghua University, Beijing, 100084, China

^* Corresponding author
^* Corresponding author

Received: July 31, 2021

Revised: December 31, 2021

Early access: February 2022

Published: May 2022

The first author is supported by NSFC (No.11771235, 12031015). The second author is supported by NSFC (No.11971049)

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Abstract

In this paper, we consider the following general pseudo-relativistic Schrödinger equation with indefinite nonlinearities:

$ (-\Delta+m^{2})^{s}u = a(x_1)f\left(u,\nabla u\right),\quad {\rm{in}} \,\,\mathbb R^{N}, $

where $ s\in(0,1) $, mass $ m>0 $ and $ a $ is a non-decreasing functions. We prove the nonexistence and the monotonicity of the positive bounded solution for the above equation via the direct method of moving planes.

Keywords:

Monotonicity,
nonexistence,
pseudo-relativistic Schrödinger equation,
indefinite nonlinearity,
moving planes.

Mathematics Subject Classification: Primary: 35R11; Secondary: 35B06, 35B53.

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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), 18 pp. doi: 10.1063/1.4949352. doi: 10.1063/1.4949352.
[2]	H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal., 4 (1994), 59-78. doi: 10.12775/TMNA.1994.023.
[3]	L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Commun. Pure Appl. Math, 42 (1989), 271-297. doi: 10.1002/cpa.3160420304.
[4]	W. Chen and C. Li, On Nirenberg and related problems - a necessary and sufficient condition, Commun. Pure Appl. Math, 48 (1995), 657-667. doi: 10.1002/cpa.3160480606.
[5]	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.
[6]	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.
[7]	W. Chen, C. Li and J. Zhu, Fractional equations with indefinite nonlinearities, Discrete Contin. Dyn. Syst., 39 (2019), 1257-1268. doi: 10.3934/dcds.2019054.
[8]	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.
[9]	W. Choi and J. Seok, Nonrelativistic limit of standing waves for pseudo-relativistic nonlinear Schrödinger equations, J. Math. Phys., 57 (2016), 15 pp. doi: 10.1063/1.4941037. doi: 10.1063/1.4941037.
[10]	S. Y. A. Chang and P. C. Yang, On uniqueness of solutions of $n$-th order differential equations in conformal geometry, Math. Res. Lett., 4 (1997), 91-102. doi: 10.4310/MRL.1997.v4.n1.a9.
[11]	W. Chen and J. Zhu, Indefinite fractional elliptic problem and Liouville theorems, J. Differ. Equ., 260 (2016), 475-4785. doi: 10.1016/j.jde.2015.11.029.
[12]	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.
[13]	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.
[14]	A. Dall'Acqua, T. Sorensen and E. Stockmeyer, Hartree-Fock theory for pseudo-relativistic atoms, Ann. Henri Poincare, 9 (2008), 711-742. doi: 10.1007/s00023-008-0370-z.
[15]	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.
[16]	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.
[17]	J. Fröhlich, B. L. G. Jonsson and E. Lenzmann, Boson stars as solitary waves, Commun. Math. Phys, 274 (2007), 1-30. doi: 10.1007/s00220-007-0272-9.
[18]	J. Fröhlich and E. Lenzmann, Blowup for nonlinear wave equations describing boson stars, Commun. Pure Appl. Math, 60 (2007), 1691-1705. doi: 10.1002/cpa.20186.
[19]	Y. Guo and S. Peng, Symmetry and monotonicity of nonnegative solutions to pseudo-relativistic Choquard equations, Z. Angew. Math. Phys., 72 (2021), 20 pp. doi: 10.1007/s00033-021-01551-5. doi: 10.1007/s00033-021-01551-5.
[20]	E. H. Lieb and W. E. Thirring, Gravitational collapse in quantum mechanics with relativistic kinetic energy, Ann. Phys., 155 (1984), 494-512. doi: 10.1016/0003-4916(84)90010-1.
[21]	E. H. Lieb and H. T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., 112 (1987), 147-174.