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Abstract

In this note we prove the validity of the Maz’ya–Shaposhnikova formula in magnetic fractional Orlicz–Sobolev spaces. This complements a previous asymptotic study of the limit as $s ↑ 1$ performed by the second author in (Studia Mathematica (2020)).

Keywords

Asymptotic behaviours Orlicz spaces fractional Sobolev norms

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References

Alberico ,

Cianchi,

Pick and

Slavíková, On the limit as

s \to 0^{+}

of fractional Orlicz–Sobolev spaces, J. Fourier Anal. Appl. 26(6) (2020), Paper No. 80, 19. doi:10.1007/s00041-020-09785-z.

Alberico,

Cianchi,

Pick and

Slavíková, Fractional Orlicz–Sobolev embeddings, preprint available at: https://arxiv.org/pdf/2001.05565.pdf.

Ambrosio, Existence and concentration results for some fractional Schrödinger equations in

R^{N}

with magnetic fields, Comm. Partial Differential Equations 44(8) (2019), 637–680. doi:10.1080/03605302.2019.1581800.

Ambrosio, Multiple concentrating solutions for a fractional Kirchhoff equation with magnetic fields, Discrete Contin. Dyn. Syst. 40(2) (2020), 781–815. doi:10.3934/dcds.2020062.

Ambrosio, Multiplicity and concentration results for a fractional Schrödinger–Poisson type equation with magnetic field, Proc. Roy. Soc. Edinburgh Sect. A 150(2) (2020), 655–694. doi:10.1017/prm.2018.153.

Ambrosio and

d’Avenia, Nonlinear fractional magnetic Schrödinger equation: Existence and multiplicity, J. Differential Equations 264(5) (2018), 3336–3368. doi:10.1016/j.jde.2017.11.021.

Bahrouni,

Bahrouni and

Xiang, On a class of nonvariational problems in fractional Orlicz–Sobolev spaces, Nonlinear Anal. 190 (2020), 111595. doi:10.1016/j.na.2019.111595.

Bahrouni,

Ounaies and

L.S.

Tavares, Basic results of fractional Orlicz–Sobolev space and applications to non-local problems, Topol. Methods Nonlinear Anal. 55(2) (2020), 681–695. doi:10.12775/tmna.2019.111.

Binlin,

Squassina and

Xia, Fractional NLS equations with magnetic field, critical frequency and critical growth, Manuscripta Math. 155(1–2) (2018), 115–140. doi:10.1007/s00229-017-0937-4.

10.

Bonanno,

Molica Bisci and

Rădulescu, Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz–Sobolev spaces, Nonlinear Anal. 75 (2012), 4441–4456. doi:10.1016/j.na.2011.12.016.

11.

Bourgain,

Brezis and

Mironescu, Another look at Sobolev spaces, in: Optimal Control and Partial Differential Equations. A Volume in Honor of Professor Alain Bensoussan’s 60th Birthday,

J.L.

Menaldi,

Rofman and

Sulem, eds, IOS Press, Amsterdam, 2001, pp. 439–455.

12.

Buseghin,

Garofalo and

Tralli, On the limiting behaviour of some nonlocal seminorms: A new phenomenon, preprint available at: https://arxiv.org/pdf/2004.01303.pdf.

13.

Capolli,

Maione,

A.M.

Salort and

Vecchi, Asymptotic behaviours in fractional Orlicz–Sobolev spaces on Carnot groups, J. Geom. Anal. (2020). doi:10.1007/s12220-020-00391-5.

14.

Cianchi, Optimal Orlicz–Sobolev embeddings, Rev. Mat. Iberoam. 20 (2004), 427–474. doi:10.4171/RMI/396.

15.

d’Avenia and

Squassina, Ground states for fractional magnetic operators, ESAIM Control Optim. Calc. Var. 24(1) (2018), 1–24. doi:10.1051/cocv/2016071.

16.

De Nápoli,

Fernández Bonder and

A.M.

Salort, A Pólya–Szegö principle for general fractional Orlicz–Sobolev spaces, Complex Var. Elliptic Equ. (2020). doi:10.1080/17476933.2020.1729139.

17.

Diening,

Harjulehto,

Hästö and

Ružička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Vol. 2017, Springer, Heidelberg, 2011.

18.

M.J.

Esteban and

P.L.

Lions, Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, in: Partial Differential Equations and the Calculus of Variations, Vol. I, Progr. Nonlinear Differential Equations Appl., 1989, pp. 401–449.

19.

Fernández Bonder,

M.P.L.

Lanos and

A.M.

Salort, A Hölder infinity Laplacian obtained as limit of Orlicz fractional Laplacians, preprint available at: https://arxiv.org/abs/1807.01669.

20.

Fernández Bonder and

A.M.

Salort, Fractional order Orlicz–Sobolev spaces, J. Funct. Anal. 277(2) (2019), 333–367. doi:10.1016/j.jfa.2019.04.003.

21.

Fernández Bonder and

A.M.

Salort, Magnetic fractional order Orlicz–Sobolev spaces, Studia Mathematica (2020). doi:10.4064/sm190713-5-7.

22.

Fiscella,

Pinamonti and

Vecchi, Multiplicity results for magnetic fractional problems, J. Differential Equations 263(8) (2017), 4617–4633. doi:10.1016/j.jde.2017.05.028.

23.

Fiscella and

Vecchi, Bifurcation and multiplicity results for critical magnetic fractional problems, Electronic J. Differential Equations 2018 (2018), No. 153, 1–18.

24.

Ichinose, Magnetic Relativistic Schrödinger Operators and Imaginary-Time Path Integrals, Mathematical Physics, Spectral Theory and Stochastic Analysis, Vol. 232, Birkhäuser/Springer, Basel, 2013, pp. 247–297. doi:10.1007/978-3-0348-0591-9_5.

25.

Ji and

Rădulescu, Multi-bump solutions for the nonlinear magnetic Schrödinger equation with exponential critical growth in

R^{2}

, Manuscripta Math. (2020). doi:10.1007/s00229-020-01195-1.

26.

Ji and

Rădulescu, Multiplicity and concentration of solutions to the nonlinear magnetic Schrödinger system, Calc. Var. Partial Differential Equations 59 (2020), 115. doi:10.1007/s00526-020-01772-y.

27.

Liang,

Repovs and

Zhang, On the fractional Schrödinger–Kirchhoff equations with electromagnetic fields and critical nonlinearity, Comput. Math. Appl. 75(5) (2018), 1778–1794. doi:10.1016/j.camwa.2017.11.033.

28.

E.H.

Lieb and

Loss, Analysis, Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 2001, p. xxii+346.

29.

E.H.

Lieb and

Seiringer, The Stability of Matter in Quantum Mechanics, Cambridge University Press, Cambridge, 2010, p. xvi+293.

30.

Liu and

Ji, Concentration results for a magnetic Schrödinger–Poisson system with critical growth, Adv. Nonlinear Anal. 10(1) (2021), 775–798. doi:10.1515/anona-2020-0159.

31.

Liu,

Li and

Ji, Multiplicity of concentrating solutions for a class of magnetic Schrödinger–Poisson type equation, Adv. Nonlinear Anal. 10(1) (2021), 131–151. doi:10.1515/anona-2020-0110.

32.

Maz’ya and

Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal. 195(2) (2002), 230–238. doi:10.1006/jfan.2002.3955.

33.

Mihailescu and

Rădulescu, Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: An Orlicz–Sobolev space setting, J. Math. Anal. Appl. 330 (2007), 416–432. doi:10.1016/j.jmaa.2006.07.082.

34.

Mihailescu,

Rădulescu and

Repovs, On a non-homogeneous eigenvalue problem involving a potential: An Orlicz–Sobolev space setting, J. Math. Pures Appl. 93 (2010), 132–148. doi:10.1016/j.matpur.2009.06.004.

35.

Milman, Notes on limits of Sobolev spaces and the continuity of interpolation scales, Trans. Amer. Math. Soc. 357(9) (2005), 3425–3442. doi:10.1090/S0002-9947-05-03937-1.

36.

Mingqi,

Pucci,

Squassina and

Zhang, Nonlocal Schrödinger–Kirchhoff equations with external magnetic field, Discrete Contin. Dyn. Syst. 37(3) (2017), 1631–1649. doi:10.3934/dcds.2017067.

37.

Mingqi,

Rădulescu and

Zhang, A critical fractional Choquard–Kirchhoff problem with magnetic field, Commun. Contemp. Math. 21(4) (2020), 1850004. doi:10.1142/S0219199718500049.

38.

H.-M.

Nguyen,

Pinamonti,

Squassina and

Vecchi, New characterizations of magnetic Sobolev spaces, Adv. Nonlinear Anal. 7(2) (2018), 227–245. doi:10.1515/anona-2017-0239.

39.

H.-M.

Nguyen,

Pinamonti,

Squassina and

Vecchi, Some characterizations of magnetic Sobolev spaces, Complex Var. Elliptic Equ. 65(7) (2020), 1104–1114. doi:10.1080/17476933.2018.1520850.

40.

Pick,

Kufner,

John and

Fučík, Function spaces. Vol. 1, De Gruyter Series in Nonlinear Analysis and Applications 129(14) (2013), xvi+479 pp.

41.

Pinamonti,

Squassina and

Vecchi, The Maz’ya–Shaposhnikova limit in the magnetic setting, J. Math. Anal. Appl. 449(2) (2017), 1152–1159. doi:10.1016/j.jmaa.2016.12.065.

42.

Pinamonti,

Squassina and

Vecchi, Magnetic BV-functions and the Bourgain–Brezis–Mironescu formula, Adv. Calc. Var. 12(3) (2019), 225–252. doi:10.1515/acv-2017-0019.

43.

A.M.

Salort, Eigenvalues and minimizers for a non-standard growth non-local operator, J. Differential Equations 268(9) (2020), 5413–5439. doi:10.1016/j.jde.2019.11.027.

44.

Squassina and

Volzone, Bourgain–Brezis–Mironescu formula for magnetic operators, C. R. Math. Acad. Sci. Paris 354(8) (2016), 825–831. doi:10.1016/j.crma.2016.04.013.

45.

Triebel, Limits of Besov norms, Arch. Math. (Basel) 96(2) (2011), 169–175. doi:10.1007/s00013-010-0214-1.

46.

Zhang,

Tang and

V.D.

Rădulescu, Small perturbations for nonlinear Schrödinger equations with magnetic potential, Milan J. Math. 88(2) (2020), 479–506. doi:10.1007/s00032-020-00322-7.