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Monotonicity and nonexistence of positive solutions for pseudo-relativistic equation with indefinite nonlinearity
Semi-classical states for fractional Schrödinger equations with magnetic fields and fast decaying potentials
1. | School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, 550025, China |
2. | School of Mathematics and Statistics, Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China |
$ \varepsilon^{2s}(-\Delta)^s_{A/\varepsilon} u +V(x)u = |u|^{p-2}u, \ x\in{\mathbb R}^N, $ |
$ \varepsilon>0 $ |
$ s\in(0,1) $ |
$ N\geq3 $ |
$ 2+2s/(N-2s)<p<2_s^*: = 2N/(N-2s) $ |
$ A\in C^{0,\alpha}({\mathbb R}^N,{\mathbb R}^N) $ |
$ \alpha\in(0,1] $ |
$ V:{\mathbb R}^N\to{\mathbb R} $ |
$ V $ |
$ \varepsilon\to 0 $ |
$ V $ |
$ V $ |
$ A $ |
$ (-\Delta)^s $ |
$ A\equiv 0 $ |
References:
[1] |
C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in ${\mathbb R}^N$ via penalization method, Calc. Var. Partial Differ. Equ., 55 (2016), Art.47, 19pp.
doi: 10.1007/s00526-016-0983-x. |
[2] |
A. Ambrosetti, A. Malchiodi and W. M. Ni,
Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part I, Commun. Math. Phys., 235 (2003), 427-466.
doi: 10.1007/s00220-003-0811-y. |
[3] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[4] |
V. Ambrosio, A local mountain pass approach for a class of fractional NLS equations with magnetic fields, Nonlinear Anal., 190 (2020), 111622, 14pp.
doi: 10.1016/j.na.2019.111622. |
[5] |
V. Ambrosio,
Existence and concentration results for some fractional Schrödinger equations in ${\mathbb R}^N$ with magnitic fields, Commun. Partial Differ. Equ., 44 (2019), 637-680.
doi: 10.1080/03605302.2019.1581800. |
[6] |
V. Ambrosio and P. d'Avenia,
Nonlinear fractional magnetic Schrödinger equation: Existence and multiplicity, J. Differ. Equ., 264 (2018), 3336-3368.
doi: 10.1016/j.jde.2017.11.021. |
[7] |
X. An, L. Duan and Y. Peng,
Semi-classical analysis for fractional Schrödinger equations with fast decaying potentials., Appl. Anal., (2021), 1-18.
doi: 10.1080/00036811.2021.1880571. |
[8] |
X. An, S. Peng and C. Xie, Semi-classical solutions for fractional Schrödinger equations with potential vanishing at infinity, J. Math. Phys., 60 (2019), 021501, 18pp.
doi: 10.1063/1.5037126. |
[9] |
D. Bonheure, S. Cingolani and M. Nys, Nonlinear Schrödinger equation: concentration on circles driven by an external magnetic filed, Calc. Var. Partial Differ. Equ., 55 (2016), 33pp.
doi: 10.1007/s00526-016-1013-8. |
[10] |
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. |
[11] |
S. Cingolani and M. Lazzo,
Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal., 10 (1997), 1-13.
doi: 10.12775/TMNA.1997.019. |
[12] |
P. d'Avenia and M. Squassina,
Ground states for fractional magnetic operators, ESAIM Control Optim. Calc. Var., 24 (2018), 1-24.
doi: 10.1051/cocv/2016071. |
[13] |
J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci,
Concentration phenomena for the nonlocal Schrödinger equation with dirichlet datum, Anal. Partial Differ. Equ., 8 (2015), 1165-1235.
doi: 10.2140/apde.2015.8.1165. |
[14] |
J. Dávila, M. del Pino and J. Wei,
Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differ. Equ., 256 (2014), 858-892.
doi: 10.1016/j.jde.2013.10.006. |
[15] |
M. del Pino and P. L. Felmer,
Local Mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121-137.
doi: 10.1007/BF01189950. |
[16] |
M. del Pino and P. L. Felmer,
Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265.
doi: 10.1006/jfan.1996.3085. |
[17] |
M. del Pino, M. Kowalczyk and J. Wei,
Concentration on curves for nonlinear Schrödinger equations, Commun. Pure Appl. Math., 60 (2007), 113-146.
doi: 10.1002/cpa.20135. |
[18] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[19] |
M. M. Fall, F. Mahmoudi and E. Valdinoci,
Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961.
doi: 10.1088/0951-7715/28/6/1937. |
[20] |
P. Felmer, A. Quaas and J. Tan,
Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinb. Sect. A, 142 (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[21] |
A. Fiscella, A. Pinamonti and E. Vecchi,
Multiplicity results for magnetic fractional problems, J. Differ. Equ., 263 (2017), 4617-4633.
doi: 10.1016/j.jde.2017.05.028. |
[22] |
L. Frank, E. Lenzmann and L. Silvestre,
Uniqueness of radial solutions for the fractional Laplacians, Commun. Pure. Appl. Math., 69 (2016), 1671-1726.
doi: 10.1002/cpa.21591. |
[23] |
L. Frank and R. Seiringer,
Nonlinear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255 (2008), 3407-3430.
doi: 10.1016/j.jfa.2008.05.015. |
[24] |
T. Ichinose, Magnetic relativistic Schrödinger operators and imaginary-time path integrals, in Mathematical Physics, Spectral Theory and Stochastic Analysis, Birkhäuser/Springer Basel AG, 2013.
doi: 10.1007/978-3-0348-0591-9_5. |
[25] |
K. Kurata,
Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields, Nonlinear Anal.: Theory, Methods and Applications, 41 (2000), 763-778.
doi: 10.1016/S0362-546X(98)00308-3. |
[26] |
Y. G. Oh,
Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Commun. Partial Differ. Equ., 13 (1988), 1499-1519.
doi: 10.1080/03605308808820585. |
[27] |
A. Pinamonti, M. Squassina and E. Vecchi,
Magnetic BV functions and the Bourgain-Brezis-Mironescu formula, Adv. Calc. Var., 12 (2017), 225-252.
doi: 10.1515/acv-2017-0019. |
[28] |
A. Pinamonti, M. Squassina and E. Vecchi,
The Maz'ya-Shaposhnikova limit in the magnetic setting, J. Math. Anal. Appl., 449 (2017), 1152-1159.
doi: 10.1016/j.jmaa.2016.12.065. |
[29] |
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in ${\mathbb R}^N$, J. Math. Phys., 54 (2013), 031501, 17pp.
doi: 10.1063/1.4793990. |
[30] |
M. Squassina and B. Volzone,
Bourgain-Brezis-Mironescu formula for magnetic operators, C. R. Math. Acad. Sci. Paris, 354 (2016), 825-831.
doi: 10.1016/j.crma.2016.04.013. |
[31] |
M. Squassina, B. Zhang and X. Zhang,
Fractional NLS equations with magnetic field, critical frequency and critical growth, Manuscr. Math., 155 (2018), 115-140.
doi: 10.1007/s00229-017-0937-4. |
[32] |
M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications 24, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
show all references
References:
[1] |
C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in ${\mathbb R}^N$ via penalization method, Calc. Var. Partial Differ. Equ., 55 (2016), Art.47, 19pp.
doi: 10.1007/s00526-016-0983-x. |
[2] |
A. Ambrosetti, A. Malchiodi and W. M. Ni,
Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part I, Commun. Math. Phys., 235 (2003), 427-466.
doi: 10.1007/s00220-003-0811-y. |
[3] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[4] |
V. Ambrosio, A local mountain pass approach for a class of fractional NLS equations with magnetic fields, Nonlinear Anal., 190 (2020), 111622, 14pp.
doi: 10.1016/j.na.2019.111622. |
[5] |
V. Ambrosio,
Existence and concentration results for some fractional Schrödinger equations in ${\mathbb R}^N$ with magnitic fields, Commun. Partial Differ. Equ., 44 (2019), 637-680.
doi: 10.1080/03605302.2019.1581800. |
[6] |
V. Ambrosio and P. d'Avenia,
Nonlinear fractional magnetic Schrödinger equation: Existence and multiplicity, J. Differ. Equ., 264 (2018), 3336-3368.
doi: 10.1016/j.jde.2017.11.021. |
[7] |
X. An, L. Duan and Y. Peng,
Semi-classical analysis for fractional Schrödinger equations with fast decaying potentials., Appl. Anal., (2021), 1-18.
doi: 10.1080/00036811.2021.1880571. |
[8] |
X. An, S. Peng and C. Xie, Semi-classical solutions for fractional Schrödinger equations with potential vanishing at infinity, J. Math. Phys., 60 (2019), 021501, 18pp.
doi: 10.1063/1.5037126. |
[9] |
D. Bonheure, S. Cingolani and M. Nys, Nonlinear Schrödinger equation: concentration on circles driven by an external magnetic filed, Calc. Var. Partial Differ. Equ., 55 (2016), 33pp.
doi: 10.1007/s00526-016-1013-8. |
[10] |
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. |
[11] |
S. Cingolani and M. Lazzo,
Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal., 10 (1997), 1-13.
doi: 10.12775/TMNA.1997.019. |
[12] |
P. d'Avenia and M. Squassina,
Ground states for fractional magnetic operators, ESAIM Control Optim. Calc. Var., 24 (2018), 1-24.
doi: 10.1051/cocv/2016071. |
[13] |
J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci,
Concentration phenomena for the nonlocal Schrödinger equation with dirichlet datum, Anal. Partial Differ. Equ., 8 (2015), 1165-1235.
doi: 10.2140/apde.2015.8.1165. |
[14] |
J. Dávila, M. del Pino and J. Wei,
Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differ. Equ., 256 (2014), 858-892.
doi: 10.1016/j.jde.2013.10.006. |
[15] |
M. del Pino and P. L. Felmer,
Local Mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121-137.
doi: 10.1007/BF01189950. |
[16] |
M. del Pino and P. L. Felmer,
Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245-265.
doi: 10.1006/jfan.1996.3085. |
[17] |
M. del Pino, M. Kowalczyk and J. Wei,
Concentration on curves for nonlinear Schrödinger equations, Commun. Pure Appl. Math., 60 (2007), 113-146.
doi: 10.1002/cpa.20135. |
[18] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[19] |
M. M. Fall, F. Mahmoudi and E. Valdinoci,
Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937-1961.
doi: 10.1088/0951-7715/28/6/1937. |
[20] |
P. Felmer, A. Quaas and J. Tan,
Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinb. Sect. A, 142 (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[21] |
A. Fiscella, A. Pinamonti and E. Vecchi,
Multiplicity results for magnetic fractional problems, J. Differ. Equ., 263 (2017), 4617-4633.
doi: 10.1016/j.jde.2017.05.028. |
[22] |
L. Frank, E. Lenzmann and L. Silvestre,
Uniqueness of radial solutions for the fractional Laplacians, Commun. Pure. Appl. Math., 69 (2016), 1671-1726.
doi: 10.1002/cpa.21591. |
[23] |
L. Frank and R. Seiringer,
Nonlinear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255 (2008), 3407-3430.
doi: 10.1016/j.jfa.2008.05.015. |
[24] |
T. Ichinose, Magnetic relativistic Schrödinger operators and imaginary-time path integrals, in Mathematical Physics, Spectral Theory and Stochastic Analysis, Birkhäuser/Springer Basel AG, 2013.
doi: 10.1007/978-3-0348-0591-9_5. |
[25] |
K. Kurata,
Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields, Nonlinear Anal.: Theory, Methods and Applications, 41 (2000), 763-778.
doi: 10.1016/S0362-546X(98)00308-3. |
[26] |
Y. G. Oh,
Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Commun. Partial Differ. Equ., 13 (1988), 1499-1519.
doi: 10.1080/03605308808820585. |
[27] |
A. Pinamonti, M. Squassina and E. Vecchi,
Magnetic BV functions and the Bourgain-Brezis-Mironescu formula, Adv. Calc. Var., 12 (2017), 225-252.
doi: 10.1515/acv-2017-0019. |
[28] |
A. Pinamonti, M. Squassina and E. Vecchi,
The Maz'ya-Shaposhnikova limit in the magnetic setting, J. Math. Anal. Appl., 449 (2017), 1152-1159.
doi: 10.1016/j.jmaa.2016.12.065. |
[29] |
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in ${\mathbb R}^N$, J. Math. Phys., 54 (2013), 031501, 17pp.
doi: 10.1063/1.4793990. |
[30] |
M. Squassina and B. Volzone,
Bourgain-Brezis-Mironescu formula for magnetic operators, C. R. Math. Acad. Sci. Paris, 354 (2016), 825-831.
doi: 10.1016/j.crma.2016.04.013. |
[31] |
M. Squassina, B. Zhang and X. Zhang,
Fractional NLS equations with magnetic field, critical frequency and critical growth, Manuscr. Math., 155 (2018), 115-140.
doi: 10.1007/s00229-017-0937-4. |
[32] |
M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications 24, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
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