# American Institute of Mathematical Sciences

May  2022, 21(5): 1649-1672. doi: 10.3934/cpaa.2022038

## 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

* Corresponding author

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

Fund Project: This work was partially supported by NSFC grants (No.12101150; No.11931012) and the Science and Technology Foundation of Guizhou Province ([2021]ZK008)

This paper deals with the following fractional magnetic Schrödinger equations
 $\varepsilon^{2s}(-\Delta)^s_{A/\varepsilon} u +V(x)u = |u|^{p-2}u, \ x\in{\mathbb R}^N,$
where
 $\varepsilon>0$
is a parameter,
 $s\in(0,1)$
,
 $N\geq3$
,
 $2+2s/(N-2s) , $ A\in C^{0,\alpha}({\mathbb R}^N,{\mathbb R}^N) $with $ \alpha\in(0,1] $is a magnetic field, $ V:{\mathbb R}^N\to{\mathbb R} $is a nonnegative continuous potential. By variational methods and penalized idea, we show that the problem has a family of solutions concentrating at a local minimum of $ V $as $ \varepsilon\to 0 $. There is no restriction on the decay rates of $ V $. Especially, $ V $can be compactly supported. The appearance of $ A $and the nonlocal of $ (-\Delta)^s $makes the proof more difficult than that in [7], which considered the case $ A\equiv 0 $. Citation: Xiaoming An, Xian Yang. Semi-classical states for fractional Schrödinger equations with magnetic fields and fast decaying potentials. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1649-1672. doi: 10.3934/cpaa.2022038 ##### References:  [1] C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in${\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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