# American Institute of Mathematical Sciences

February  2020, 40(2): 781-815. doi: 10.3934/dcds.2020062

## Multiple concentrating solutions for a fractional Kirchhoff equation with magnetic fields

 Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 12, 60131 Ancona, Italy

Received  January 2019 Published  November 2019

This paper is concerned with the multiplicity and concentration behavior of nontrivial solutions for the following fractional Kirchhoff equation in presence of a magnetic field:
 $\begin{equation*} \left(a\varepsilon^{2s}+b\varepsilon^{4s-3} [u]_{A/\varepsilon}^{2}\right)(-\Delta)_{A/\varepsilon}^{s}u+V(x)u = f(|u|^{2})u \quad \mbox{ in } \mathbb{R}^{3}, \end{equation*}$
where
 $\varepsilon>0$
is a small parameter,
 $a, b>0$
are constants,
 $s\in (\frac{3}{4}, 1)$
,
 $(-\Delta)^{s}_{A}$
is the fractional magnetic Laplacian,
 $A:\mathbb{R}^{3}\rightarrow \mathbb{R}^{3}$
is a smooth magnetic potential,
 $V:\mathbb{R}^{3}\rightarrow \mathbb{R}$
is a positive continuous electric potential satisfying local conditions and
 $f:\mathbb{R}\rightarrow \mathbb{R}$
is a
 $C^{1}$
subcritical nonlinearity. Applying penalization techniques, fractional Kato's type inequality and Ljusternik-Schnirelmann theory, we relate the number of nontrivial solutions with the topology of the set where the potential
 $V$
attains its minimum.
Citation: Vincenzo Ambrosio. Multiple concentrating solutions for a fractional Kirchhoff equation with magnetic fields. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 781-815. doi: 10.3934/dcds.2020062
##### References:
 [1] C. O. Alves, G. M. Figueiredo and M. F. Furtado, Multiple solutions for a nonlinear Schrödinger equation with magnetic fields, Comm. Partial Differential Equations, 36 (2011), 1565-1586.  doi: 10.1080/03605302.2011.593013. [2] 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 Differential Equations, 55 (2016), art. 47, 19 pp. doi: 10.1007/s00526-016-0983-x. [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, Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method, Ann. Mat. Pura Appl. (4), 196 (2017), 2043–2062. doi: 10.1007/s10231-017-0652-5. [5] V. Ambrosio, Concentrating solutions for a class of nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$, Rev. Mat. Iberoam., 35 (2019), 1367-1414.  doi: 10.4171/rmi/1086. [6] V. Ambrosio, Boundedness and decay of solutions for some fractional magnetic Schrödinger equations in $\mathbb{R}^{N}$, Milan J. Math., 86 (2018), 125-136.  doi: 10.1007/s00032-018-0283-3. [7] V. Ambrosio, Existence and concentration results for some fractional Schrödinger equations in $\mathbb{R}^{N}$ with magnetic fields, Comm. Partial Differential Equations, 44 (2019), 637-680.  doi: 10.1080/03605302.2019.1581800. [8] V. Ambrosio, Multiplicity and concentration of solutions for a fractional kirchhoff equation with magnetic field and critical growth, Ann. Henri Poincaré, 20 (2019), 2717-2766.  doi: 10.1007/s00023-019-00803-5. [9] V. Ambrosio, On a fractional magnetic Schrödinger equation in $\mathbb{R}$ with exponential critical growth, Nonlinear Anal., 183 (2019), 117-148.  doi: 10.1016/j.na.2019.01.016. [10] V. Ambrosio and P. d'Avenia, Nonlinear fractional magnetic Schrödinger equation: Existence and multiplicity, J. Differential Equations, 264 (2018), 3336-3368.  doi: 10.1016/j.jde.2017.11.021. [11] V. Ambrosio and T. Isernia, A multiplicity result for a fractional Kirchhoff equation in $\mathbb{R}^{N}$ with a general nonlinearity, Commun. Contemp. Math., 20 (2018), 1750054, 17 pp. doi: 10.1142/S0219199717500547. [12] V. Ambrosio and T. Isernia, Concentration phenomena for a fractional Schrödinger-Kirchhoff type problem, Math. Methods Appl. Sci., 41 (2018), 615-645. [13] V. Ambrosio and T. Isernia, Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., 38 (2018), 5835-5881.  doi: 10.3934/dcds.2018254. [14] G. Arioli and A. Szulkin, A semilinear Schrödinger equation in the presence of a magnetic field, Arch. Ration. Mech. Anal., 170 (2003), 277-295.  doi: 10.1007/s00205-003-0274-5. [15] V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2 (1994), 29-48.  doi: 10.1007/BF01234314. [16] S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées partielles, Bull. Acad. Sci. URSS. Sér. Math., [Izvestia Akad. Nauk SSSR] 4 (1940), 17–26. [17] 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. [18] S. Cingolani and S. Secchi, Semiclassical states for NLS equations with magnetic potentials having polynomial growths, J. Math. Phys., 46 (2005), 053503, 19 pp. doi: 10.1063/1.1874333. [19] J. Dávila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.  doi: 10.1016/j.jde.2013.10.006. [20] 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. [21] M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.  doi: 10.1007/BF01189950. [22] 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. [23] S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb{R}^{n}$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017. viii+152 pp. doi: 10.1007/978-88-7642-601-8. [24] M. Esteban and P. L. Lions, Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, Partial Differential Equations and the Calculus of Variations, Vol. I, 401–449, Progr. Nonlinear Differential Equations Appl., 1, Birkhäuser Boston, Boston, MA, 1989. [25] G. M. Figueiredo and J. R. Santos, Multiplicity and concentration behavior of positive solutions for a Schrödinger-Kirchhoff type problem via penalization method, ESAIM Control Optim. Calc. Var., 20 (2014), 389-415.  doi: 10.1051/cocv/2013068. [26] P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746. [27] A. Fiscella, A. Pinamonti and E. Vecchi, Multiplicity results for magnetic fractional problems, J. Differential Equations, 263 (2017), 4617-4633.  doi: 10.1016/j.jde.2017.05.028. [28] A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.  doi: 10.1016/j.na.2013.08.011. [29] Y. He, G. Li and S. Peng, Concentrating bound states for Kirchhoff type problems in $\mathbb{R}^{3}$ involving critical Sobolev exponents, Adv. Nonlinear Stud., 14 (2014), 483-510.  doi: 10.1515/ans-2014-0214. [30] X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb{R}^{3}$, J. Differ. Equ., 252 (2012), 1813-1834.  doi: 10.1016/j.jde.2011.08.035. [31] T. Ichinose, Magnetic relativistic Schrödinger operators and imaginary-time path integrals, Mathematical Physics, Spectral Theory and Stochastic Analysis, 247–297, Oper. Theory Adv. Appl. 232, Birkhäuser/Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0591-9_5. [32] T. Kato, Schrödinger operators with singular potentials, Israel J. Math., 13 (1972), 135-148.  doi: 10.1007/BF02760233. [33] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [34] K. Kurata, Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields, Nonlinear Anal., 41 (2000), 763-778.  doi: 10.1016/S0362-546X(98)00308-3. [35] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Pergamon Press Ltd., London-Paris; for U.S.A. and Canada: Addison-Wesley Publishing Co., Inc., Reading, Mass; 1958. [36] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2. [37] S. Liang, D. Repovš and B. Zhang, On the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity, Comput. Math. Appl., 75 (2018), 1778-1794.  doi: 10.1016/j.camwa.2017.11.033. [38] J. L. Lions, On some questions in boundary value problems of mathematical physics, Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), 284–346, North-Holland Math. Stud., 30, North-Holland, Amsterdam-New York, 1978. [39] X. Mingqi, P. Pucci, M. Squassina and B. Zhang, Nonlocal Schrödinger-Kirchhoff equations with external magnetic field, Discrete Contin. Dyn. Syst., 37 (2017), 1631-1649.  doi: 10.3934/dcds.2017067. [40] G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, 162 Cambridge, 2016. doi: 10.1017/CBO9781316282397. [41] J. Moser, A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math., 13 (1960), 457-468.  doi: 10.1002/cpa.3160130308. [42] N. Nyamoradi, Existence of three solutions for Kirchhoff nonlocal operators of elliptic type, Math. Commun., 18 (2013), 489-502. [43] G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. Partial Differential Equations, 50 (2014), 799-829.  doi: 10.1007/s00526-013-0656-y. [44] S. I. Pohožaev, A certain class of quasilinear hyperbolic equations, Mat. Sb., 96 (1975), 152-166. [45] P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in $\mathbb{R}^{N}$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22.  doi: 10.4171/RMI/879. [46] P. Pucci, M. Xiang and B. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional $p$-Laplacian in $\mathbb{R}^{N}$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806.  doi: 10.1007/s00526-015-0883-5. [47] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631. [48] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I, Functional analysis, Academic Press, Inc., New York, 1980. [49] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153. [50] M. Squassina and B. Volzone, Bourgain-Brezis-Mironescu formula for magnetic operators, C. R. Math., 354 (2016), 825-831.  doi: 10.1016/j.crma.2016.04.013. [51] J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations, 253 (2012), 2314-2351.  doi: 10.1016/j.jde.2012.05.023. [52] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.

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##### References:
 [1] C. O. Alves, G. M. Figueiredo and M. F. Furtado, Multiple solutions for a nonlinear Schrödinger equation with magnetic fields, Comm. Partial Differential Equations, 36 (2011), 1565-1586.  doi: 10.1080/03605302.2011.593013. [2] 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 Differential Equations, 55 (2016), art. 47, 19 pp. doi: 10.1007/s00526-016-0983-x. [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, Multiplicity of positive solutions for a class of fractional Schrödinger equations via penalization method, Ann. Mat. Pura Appl. (4), 196 (2017), 2043–2062. doi: 10.1007/s10231-017-0652-5. [5] V. Ambrosio, Concentrating solutions for a class of nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$, Rev. Mat. Iberoam., 35 (2019), 1367-1414.  doi: 10.4171/rmi/1086. [6] V. Ambrosio, Boundedness and decay of solutions for some fractional magnetic Schrödinger equations in $\mathbb{R}^{N}$, Milan J. Math., 86 (2018), 125-136.  doi: 10.1007/s00032-018-0283-3. [7] V. Ambrosio, Existence and concentration results for some fractional Schrödinger equations in $\mathbb{R}^{N}$ with magnetic fields, Comm. Partial Differential Equations, 44 (2019), 637-680.  doi: 10.1080/03605302.2019.1581800. [8] V. Ambrosio, Multiplicity and concentration of solutions for a fractional kirchhoff equation with magnetic field and critical growth, Ann. Henri Poincaré, 20 (2019), 2717-2766.  doi: 10.1007/s00023-019-00803-5. [9] V. Ambrosio, On a fractional magnetic Schrödinger equation in $\mathbb{R}$ with exponential critical growth, Nonlinear Anal., 183 (2019), 117-148.  doi: 10.1016/j.na.2019.01.016. [10] V. Ambrosio and P. d'Avenia, Nonlinear fractional magnetic Schrödinger equation: Existence and multiplicity, J. Differential Equations, 264 (2018), 3336-3368.  doi: 10.1016/j.jde.2017.11.021. [11] V. Ambrosio and T. Isernia, A multiplicity result for a fractional Kirchhoff equation in $\mathbb{R}^{N}$ with a general nonlinearity, Commun. Contemp. Math., 20 (2018), 1750054, 17 pp. doi: 10.1142/S0219199717500547. [12] V. Ambrosio and T. Isernia, Concentration phenomena for a fractional Schrödinger-Kirchhoff type problem, Math. Methods Appl. Sci., 41 (2018), 615-645. [13] V. Ambrosio and T. Isernia, Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., 38 (2018), 5835-5881.  doi: 10.3934/dcds.2018254. [14] G. Arioli and A. Szulkin, A semilinear Schrödinger equation in the presence of a magnetic field, Arch. Ration. Mech. Anal., 170 (2003), 277-295.  doi: 10.1007/s00205-003-0274-5. [15] V. Benci and G. Cerami, Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. Var. Partial Differential Equations, 2 (1994), 29-48.  doi: 10.1007/BF01234314. [16] S. Bernstein, Sur une classe d'équations fonctionnelles aux dérivées partielles, Bull. Acad. Sci. URSS. Sér. Math., [Izvestia Akad. Nauk SSSR] 4 (1940), 17–26. [17] 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. [18] S. Cingolani and S. Secchi, Semiclassical states for NLS equations with magnetic potentials having polynomial growths, J. Math. Phys., 46 (2005), 053503, 19 pp. doi: 10.1063/1.1874333. [19] J. Dávila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.  doi: 10.1016/j.jde.2013.10.006. [20] 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. [21] M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.  doi: 10.1007/BF01189950. [22] 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. [23] S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb{R}^{n}$, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], 15. Edizioni della Normale, Pisa, 2017. viii+152 pp. doi: 10.1007/978-88-7642-601-8. [24] M. Esteban and P. L. Lions, Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, Partial Differential Equations and the Calculus of Variations, Vol. I, 401–449, Progr. Nonlinear Differential Equations Appl., 1, Birkhäuser Boston, Boston, MA, 1989. [25] G. M. Figueiredo and J. R. Santos, Multiplicity and concentration behavior of positive solutions for a Schrödinger-Kirchhoff type problem via penalization method, ESAIM Control Optim. Calc. Var., 20 (2014), 389-415.  doi: 10.1051/cocv/2013068. [26] P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746. [27] A. Fiscella, A. Pinamonti and E. Vecchi, Multiplicity results for magnetic fractional problems, J. Differential Equations, 263 (2017), 4617-4633.  doi: 10.1016/j.jde.2017.05.028. [28] A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.  doi: 10.1016/j.na.2013.08.011. [29] Y. He, G. Li and S. Peng, Concentrating bound states for Kirchhoff type problems in $\mathbb{R}^{3}$ involving critical Sobolev exponents, Adv. Nonlinear Stud., 14 (2014), 483-510.  doi: 10.1515/ans-2014-0214. [30] X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb{R}^{3}$, J. Differ. Equ., 252 (2012), 1813-1834.  doi: 10.1016/j.jde.2011.08.035. [31] T. Ichinose, Magnetic relativistic Schrödinger operators and imaginary-time path integrals, Mathematical Physics, Spectral Theory and Stochastic Analysis, 247–297, Oper. Theory Adv. Appl. 232, Birkhäuser/Springer Basel AG, Basel, 2013. doi: 10.1007/978-3-0348-0591-9_5. [32] T. Kato, Schrödinger operators with singular potentials, Israel J. Math., 13 (1972), 135-148.  doi: 10.1007/BF02760233. [33] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. [34] K. Kurata, Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields, Nonlinear Anal., 41 (2000), 763-778.  doi: 10.1016/S0362-546X(98)00308-3. [35] L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Pergamon Press Ltd., London-Paris; for U.S.A. and Canada: Addison-Wesley Publishing Co., Inc., Reading, Mass; 1958. [36] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2. [37] S. Liang, D. Repovš and B. Zhang, On the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity, Comput. Math. Appl., 75 (2018), 1778-1794.  doi: 10.1016/j.camwa.2017.11.033. [38] J. L. Lions, On some questions in boundary value problems of mathematical physics, Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), 284–346, North-Holland Math. Stud., 30, North-Holland, Amsterdam-New York, 1978. [39] X. Mingqi, P. Pucci, M. Squassina and B. Zhang, Nonlocal Schrödinger-Kirchhoff equations with external magnetic field, Discrete Contin. Dyn. Syst., 37 (2017), 1631-1649.  doi: 10.3934/dcds.2017067. [40] G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, 162 Cambridge, 2016. doi: 10.1017/CBO9781316282397. [41] J. Moser, A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math., 13 (1960), 457-468.  doi: 10.1002/cpa.3160130308. [42] N. 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Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631. [48] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I, Functional analysis, Academic Press, Inc., New York, 1980. [49] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.  doi: 10.1002/cpa.20153. [50] M. Squassina and B. Volzone, Bourgain-Brezis-Mironescu formula for magnetic operators, C. R. Math., 354 (2016), 825-831.  doi: 10.1016/j.crma.2016.04.013. [51] J. Wang, L. Tian, J. Xu and F. Zhang, Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth, J. Differential Equations, 253 (2012), 2314-2351.  doi: 10.1016/j.jde.2012.05.023. [52] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.
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