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Multi-peak semiclassical bound states for Fractional Schrödinger Equations with fast decaying potentials

  • Received: 21 September 2021 Revised: 25 November 2021 Accepted: 25 November 2021 Published: 15 February 2022
  • We study the following fractional Schrödinger equation

    $ \begin{equation*} \label{eq0.1} \varepsilon^{2s}(-\Delta)^s u + V(x)u = f(u), \,\,x\in\mathbb{R}^N, \end{equation*} $

    where $ s\in(0,1) $. Under some conditions on $ f(u) $, we show that the problem has a family of solutions concentrating at any finite given local minima of $ V $ provided that $ V\in C( \mathbb{R}^N,[0,+\infty)) $. All decay rates of $ V $ are admissible. Especially, $ V $ can be compactly supported. Different from the local case $ s = 1 $ or the case of single-peak solutions, the nonlocal effect of the operator $ (-\Delta)^s $ makes the peaks of the candidate solutions affect mutually, which causes more difficulties in finding solutions with multiple bumps. The methods in this paper are penalized technique and variational method.

    Citation: Xiaoming An, Shuangjie Peng. Multi-peak semiclassical bound states for Fractional Schrödinger Equations with fast decaying potentials[J]. Electronic Research Archive, 2022, 30(2): 585-614. doi: 10.3934/era.2022031

    Related Papers:

  • We study the following fractional Schrödinger equation

    $ \begin{equation*} \label{eq0.1} \varepsilon^{2s}(-\Delta)^s u + V(x)u = f(u), \,\,x\in\mathbb{R}^N, \end{equation*} $

    where $ s\in(0,1) $. Under some conditions on $ f(u) $, we show that the problem has a family of solutions concentrating at any finite given local minima of $ V $ provided that $ V\in C( \mathbb{R}^N,[0,+\infty)) $. All decay rates of $ V $ are admissible. Especially, $ V $ can be compactly supported. Different from the local case $ s = 1 $ or the case of single-peak solutions, the nonlocal effect of the operator $ (-\Delta)^s $ makes the peaks of the candidate solutions affect mutually, which causes more difficulties in finding solutions with multiple bumps. The methods in this paper are penalized technique and variational method.



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    [1] P. Felmer, A. Quaas, J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237–1262. https://doi.org/10.1017/S0308210511000746 doi: 10.1017/S0308210511000746
    [2] N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108. https://doi.org/10.1103/PhysRevE.66.056108 doi: 10.1103/PhysRevE.66.056108
    [3] N. Laskin, Fractional quantum mechanics and Levy path integrals, Phys. Lett. A, 268 (2000) , 298–305. https://doi.org/10.1016/S0375-9601(00)00201-2 doi: 10.1016/S0375-9601(00)00201-2
    [4] E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573. https://doi.org/10.1016/j.bulsci.2011.12.004 doi: 10.1016/j.bulsci.2011.12.004
    [5] A. Ambrosetti, M. Badiale, S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 140 (1997), 285–300. https://doi.org/10.1007/s002050050067 doi: 10.1007/s002050050067
    [6] A. Ambrosetti, A. Malchiodi, Perturbation methods and semilinear elliptic problems on $\mathbb{R}^N$, Progress in Mathmatics, vol. 240. Birfh$\ddot{a}$user Verlag, Basel, 2006. https://doi.org/10.1007/3-7643-7396-2
    [7] S. Cingolani, M. Lazzo, Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. Methods Nonlinear Anal., 10 (1997), 1–13. https://doi.org/10.12775/TMNA.1997.019 doi: 10.12775/TMNA.1997.019
    [8] S. Cingolani, M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differ. Equ., 160 (2000) , 118–138. https://doi.org/10.1006/jdeq.1999.3662 doi: 10.1006/jdeq.1999.3662
    [9] M. del Pino, P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4 (1996), 121–137. https://doi.org/10.1007/BF01189950 doi: 10.1007/BF01189950
    [10] M. del Pino, P. L. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal., 149 (1997), 245–265. https://doi.org/10.1006/jfan.1996.3085 doi: 10.1006/jfan.1996.3085
    [11] A. Floer, A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397–408. https://doi.org/10.1016/0022-1236(86)90096-0 doi: 10.1016/0022-1236(86)90096-0
    [12] Y. G. Oh, Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$, Comm. Partial Differ. Equ., 13 (1988), 1499–1519. https://doi.org/10.1080/03605308808820585 doi: 10.1080/03605308808820585
    [13] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270–291. https://doi.org/10.1007/BF00946631 doi: 10.1007/BF00946631
    [14] A. Ambrosetti, A. Malchiodi, W. M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, I. Comm. Math. Phys., 235 (2003), 427–466. https://doi.org/10.1007/s00220-003-0811-y doi: 10.1007/s00220-003-0811-y
    [15] M. del Pino, M. Kowalczyk, J. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60 (2006), 113–146. https://doi.org/10.1002/cpa.20135 doi: 10.1002/cpa.20135
    [16] D. Bonheure, S. Cingolani, M. Nys, Nonlinear Schrödinger equation: concentration on circles driven by an external maganetic field, Calc. Var. Partial Differ. Equ., 55 (2016), 1–33. https://doi.org/10.1007/s00526-016-1013-8 doi: 10.1007/s00526-016-1013-8
    [17] T. Bartsch, E. N. Dancer, S. Peng, On multi-bump semiclassical bound states of nonlinear Schrödinger euqations with electromagnetic fields, Adv. Differ. Equ., 7 (2006), 781–812.
    [18] D. Cao, E. S. Noussair, Multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations, J. Differ. Equ., 203 (2004), 292–312. https://doi.org/10.1016/j.jde.2004.05.003 doi: 10.1016/j.jde.2004.05.003
    [19] J. Wei, S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $\mathbb{R}^N$, Calc. Var. Partial Differ. Equ., 37 (2010), 423–439. https://doi.org/10.1007/s00526-009-0270-1 doi: 10.1007/s00526-009-0270-1
    [20] R. L. Frank, E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta. Math., 210 (2013), 261–318. https://doi.org/10.1007/s11511-013-0095-9 doi: 10.1007/s11511-013-0095-9
    [21] R. L. Frank, E. Lenzmann, L. Silvestre, Uniqueness of radial solutions for the fractional Laplacians, Comm. Pure. Appl. Math., 69 (2016), 1671–1726. https://doi.org/10.1002/cpa.21591 doi: 10.1002/cpa.21591
    [22] G. Chen, Y. Zheng, Concentration phenomena for fractional noninear Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 2359–2376. https://doi.org/10.3934/cpaa.2014.13.2359 doi: 10.3934/cpaa.2014.13.2359
    [23] M. M. Fall, F. Mahmoudi, E. Valdinoci, Ground states and concentration phenomena for the fractional Schrödinger equation, Nonlinearity, 28 (2015), 1937–1961. https://doi.org/10.1088/0951-7715/28/6/1937 doi: 10.1088/0951-7715/28/6/1937
    [24] X. Shang, J. Zhang, Concentrating solutions of nonlinear fractional Schrödinger equation with potentials, J. Differ. Equ., 258 (2015), 1106–1128. https://doi.org/10.1016/j.jde.2014.10.012 doi: 10.1016/j.jde.2014.10.012
    [25] C. Alves, O. 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), 1–19. https://doi.org/10.1007/s00526-016-0983-x doi: 10.1007/s00526-016-0983-x
    [26] L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differ. Equ., 32 (2007), 1245–1260. https://doi.org/10.1080/03605300600987306 doi: 10.1080/03605300600987306
    [27] X. An, S. Peng, C. Xie, Semi-classical solutions for fractional Schrödinger equations with potential vanishing at infinity, J. Math. Phys., 60 (2019), 021501. https://doi.org/10.1063/1.5037126 doi: 10.1063/1.5037126
    [28] X. An, L. Duan, Y. Peng, Semi-classical analysis for fractional Schrödinger equations with fast decaying potentials, Appl. Anal., (2021), 1–18. https://doi.org/10.1080/00036811.2021.1880571 doi: 10.1080/00036811.2021.1880571
    [29] M. del Pino, P. L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincarè, Analyse non linèaire, 15 (1998), 127–149. https://doi.org/10.1016/s0294-1449(97)89296-7 doi: 10.1016/s0294-1449(97)89296-7
    [30] J. Byeon, L. Jeanjean, Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity, Discrete Contin. Dyn. Syst., 19 (2007), 255–269. https://doi.org/10.3934/dcds.2007.19.255 doi: 10.3934/dcds.2007.19.255
    [31] T. Hu, W. Shuai, Multi-peak solutions to Kirchhoff equations in $ \mathbb{R}^3$ with general nonlinearlity, J. Differ. Equ., 265 (2018), 3587–3617. https://doi.org/10.1016/j.jde.2018.05.012 doi: 10.1016/j.jde.2018.05.012
    [32] M. Willem, Minimax theorems, Birkhäuser, 1996.
    [33] A. Ambrosetti, A. Malchiodi, Concentration phenomena for NLS: Recent results and new perspectives, perspectives in nonlinear partial differential equations, Contemp. Math., 446 (2007), 19–30. https://doi.org/10.1090/conm/446/08624 doi: 10.1090/conm/446/08624
    [34] V. Coti-Zelati, P. H. Rabinowitz, Homoclinic orbits for a second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693–727. https://doi.org/10.1090/S0894-0347-1991-1119200-3 doi: 10.1090/S0894-0347-1991-1119200-3
    [35] R. L. Frank, R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255 (2008), 3407–3430. https://doi.org/10.1016/j.jfa.2008.05.015 doi: 10.1016/j.jfa.2008.05.015
    [36] S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^N$, J. Math. Phys., 54 (2013), 031501. https://doi.org/10.1063/1.4793990 doi: 10.1063/1.4793990
    [37] M. I. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation, Comm. Partial Differ. Equ., 12 (1987), 1133–1173. https://doi.org/10.1080/03605308708820522 doi: 10.1080/03605308708820522
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