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May  2022, 21(5): 1637-1648. doi: 10.3934/cpaa.2022037

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

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

* Corresponding author

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

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

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.
Citation: Yuxia Guo, Shaolong Peng. Monotonicity and nonexistence of positive solutions for pseudo-relativistic equation with indefinite nonlinearity. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1637-1648. doi: 10.3934/cpaa.2022037
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.

[2]

H. BerestyckiI. 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. CaffarelliB. 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. 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.

[6]

W. ChenY. 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. ChenC. 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.

[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. 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]

A. Dall'AcquaT. 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. FröhlichB. 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.

[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. 

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), 18 pp. doi: 10.1063/1.4949352.

[2]

H. BerestyckiI. 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. CaffarelliB. 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. 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.

[6]

W. ChenY. 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. ChenC. 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.

[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. 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]

A. Dall'AcquaT. 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. FröhlichB. 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.

[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. 

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