# American Institute of Mathematical Sciences

March  2017, 37(3): 1631-1649. doi: 10.3934/dcds.2017067

## Nonlocal Schrödinger-Kirchhoff equations with external magnetic field

 1 College of Science, Civil Aviation University of China, Tianjin 300300, China 2 Dipartimento di Matematica e Informatica, Università di Perugia, Via Vanvitelli 1,06123 Perugia, Italy 3 Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41,25121 Brescia, Italy 4 Department of Mathematics, Heilongjiang Institute of Technology, Harbin 150050, China

Received  May 2016 Revised  October 2016 Published  December 2016

Fund Project: Mingqi Xiang was supported by the Fundamental Research Funds for the Central Universities (No. 3122015L014) and the National Natural Science Foundation of China (No. 11601515). Patrizia Pucci and Marco Squassina are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The manuscript was realized within the auspices of the INdAM – GNAMPA Project Problemi variazionali su varietà Riemanniane e gruppi di Carnot (Prot 2016 000421). Patrizia Pucci was partly supported by the Italian MIUR project Variational and perturbative aspects of nonlinear differential problems (201274FYK7). Binlin Zhang was supported by Natural Science Foundation of Heilongjiang Province of China (No. A201306).

The paper deals with the existence and multiplicity of solutions of the fractional Schrödinger-Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case
 $\begin{equation*} (a+b[u]_{s,A}^{2θ-2})(-Δ)_A^su+V(x)u=f(x,|u|)u\,\, \text{in$\mathbb{R}^N$},\end{equation*}$
where
 $s∈ (0,1)$
,
 $N>2s$
,
 $a∈ \mathbb{R}^+_0$
,
 $b∈ \mathbb{R}^+_0$
,
 $θ∈[1,N/(N-2s))$
,
 $A:\mathbb{R}^N\to\mathbb{R}^N$
is a magnetic potential,
 $V:\mathbb{R}^N\to \mathbb{R}^+$
is an electric potential,
 $(-Δ )_A^s$
is the fractional magnetic operator. In the super-and sub-linear cases, the existence of least energy solutions for the above problem is obtained by the mountain pass theorem, combined with the Nehari method, and by the direct methods respectively. In the superlinear-sublinear case, the existence of infinitely many solutions is investigated by the symmetric mountain pass theorem.
Citation: Mingqi Xiang, Patrizia Pucci, Marco Squassina, Binlin Zhang. Nonlocal Schrödinger-Kirchhoff equations with external magnetic field. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1631-1649. doi: 10.3934/dcds.2017067
##### References:
 [1] D. Applebaum, Lévy processes -From probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. [2] G. Arioli and A. Szulkin, A semilinear Schrödinger equation in the presence of a magnetic field, Arch. Rational Mech. Anal., 170 (2003), 277-295.  doi: 10.1007/s00205-003-0274-5. [3] G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714.  doi: 10.1016/j.na.2015.06.014. [4] T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on ${{\mathbb{R}}^{N}}$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149. [5] L. Caffarelli, Nonlocal equations, drifts and games, Nonlinear Partial Differential Equations, Abel Symposia, 7 (2012), 37-52.  doi: 10.1007/978-3-642-25361-4_3. [6] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [7] M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional $p$-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099-2129.  doi: 10.1007/s10231-016-0555-x. [8] G. F. Carrier, On the nonlinear vibration problem of the elastic string, Quart. Appl. Math., 3 (1945), 157-165. [9] G. F. Carrier, A note on the vibrating string, Quart. Appl. Math., 7 (1949), 97-101. [10] K. C. Chang, Critical Point Theory and Applications Shanghai Scientific and Technology Press, Shanghai, 1986. [11] S. Cingolani and S. Secchi, Semiclassical limit for nonlinear Schrodinger equations with electromagnetic fileds, J. Math. Anal. Appl., 275 (2002), 108-130.  doi: 10.1016/S0022-247X(02)00278-0. [12] F. Colasuonno and P. Pucci, Multiplicity of solutions for $p(x)$-polyharmonic Kirchhoff equations, Nonlinear Anal., 74 (2011), 5962-5974.  doi: 10.1016/j.na.2011.05.073. [13] P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262.  doi: 10.1007/BF02100605. [14] P. d'Avenia and M. Squassina, Ground states for fractional magnetic operators ESAIM Control Optim. Calc. Var. Forthcoming article. doi: 10.1051/cocv/2016071. [15] J. Di Cosmo and J. Van Schaftingen, Semiclassical stationary states for nonlinear Schrödiner equations under a strong extenal magnetic field, J. Differential Equations, 259 (2015), 596-627.  doi: 10.1016/j.jde.2015.02.016. [16] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. [17] S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $\mathbb R^n$, arXiv: 1506.01748. [18] S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Le Matematiche, 68 (2013), 201-216. [19] 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. [20] 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. [21] T. Ichinose, Essential selfadjointness of the Weyl quantized relativistic Hamiltonian, Ann. Inst. H. Poincaré Phys. Théor, 51 (1989), 265-297. [22] 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. [23] T. Ichinose and H. Tamura, Imaginary-time path integral for a relativistic spinless particle in an electromagnetic field, Comm. Math. Phys., 105 (1986), 239-257.  doi: 10.1007/BF01211101. [24] V. Iftimie, M. Măntoiu and R. Purice, Magnetic pseudodifferential operators, Publ. Res. Inst. Math. Sci., 43 (2007), 585-623.  doi: 10.2977/prims/1201012035. [25] R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal., 225 (2005), 352-370.  doi: 10.1016/j.jfa.2005.04.005. [26] G. Kirchhoff, Mechanik Teubner, Leipzig, 1883. [27] M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, MacMillan, New York, 1964. [28] K. Kurata, Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fileds, Nonlinear Anal., 41 (2000), 763-778.  doi: 10.1016/S0362-546X(98)00308-3. [29] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2. [30] N. Laskin, Fractional Schrödinger equation Phys. Rev. E 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108. [31] P. L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal., 49 (1982), 315-334.  doi: 10.1016/0022-1236(82)90072-6. [32] X. Mingqi, G. Molica Bisci, G. Tian and B. Zhang, Infinitely many solutions for the stationary Kirchhoff problems involving the fractional $p$-Laplacian, Nonlinearity, 29 (2016), 357-374.  doi: 10.1088/0951-7715/29/2/357. [33] G. Molica Bisci, V. D. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Equations Encyclopedia of Mathematics and its Applications, 162, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316282397. [34] R. Narashima, Nonlinear vibration of an elastic string, J. Sound Vib., 8 (1968), 134-146. [35] D. W. Oplinger, Frequency response of a nonlinear stretched string, J. Acoust. Soc. Am., 32 (1960), 1529-1538.  doi: 10.1121/1.1907948. [36] A. Pinamonti, M. Squassina and E. Vecchi, Magnetic BV functions and the Bourgain-Brezis-Mironescu formula, preprint. [37] A. Pinamonti, M. Squassina and E. Vecchi, The Maz'ya-Shaposhnikova limit in the magnetic setting, preprint. [38] 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. [39] 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. [40] P. Pucci, M. Xiang and B. Zhang, Existence and multiplicity of entire solutions for fractional $p$ -Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27-55.  doi: 10.1515/anona-2015-0102. [41] S. Secchi, Ground state solutions for the fractional Schrödinger in $\mathbb{R}^N$ J. Math. Phys. 54 2013,031501, 17pp. doi: 10.1063/1.4793990. [42] M. Squassina, Soliton dynamics for the nonlinear Schrödinger equation with magnetic field, Manuscripta Math., 130 (2009), 461-494.  doi: 10.1007/s00229-009-0307-y. [43] 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. [44] M. Xiang, B. Zhang and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional $p-$Laplacian, J. Math. Anal. Appl., 424 (2015), 1021-1041.  doi: 10.1016/j.jmaa.2014.11.055. [45] M. Xiang, B. Zhang and M. Ferrara, Multiplicity results for the nonhomogeneous fractional $p$-Kirchhoff equations with concave-convex nonlinearities, Proc.A, 471 (2015), 20150034-14 pp.  doi: 10.1098/rspa.2015.0034. [46] M. Xiang, B. Zhang and X. Guo, Infinitely many solutions for a fractional Kirchhoff type problem via Fountain Theorem, Nonlinear Anal., 120 (2015), 299-313.  doi: 10.1016/j.na.2015.03.015.

show all references

##### References:
 [1] D. Applebaum, Lévy processes -From probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. [2] G. Arioli and A. Szulkin, A semilinear Schrödinger equation in the presence of a magnetic field, Arch. Rational Mech. Anal., 170 (2003), 277-295.  doi: 10.1007/s00205-003-0274-5. [3] G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional operator and a critical nonlinearity, Nonlinear Anal., 125 (2015), 699-714.  doi: 10.1016/j.na.2015.06.014. [4] T. Bartsch and Z.-Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on ${{\mathbb{R}}^{N}}$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149. [5] L. Caffarelli, Nonlocal equations, drifts and games, Nonlinear Partial Differential Equations, Abel Symposia, 7 (2012), 37-52.  doi: 10.1007/978-3-642-25361-4_3. [6] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306. [7] M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional $p$-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099-2129.  doi: 10.1007/s10231-016-0555-x. [8] G. F. Carrier, On the nonlinear vibration problem of the elastic string, Quart. Appl. Math., 3 (1945), 157-165. [9] G. F. Carrier, A note on the vibrating string, Quart. Appl. Math., 7 (1949), 97-101. [10] K. C. Chang, Critical Point Theory and Applications Shanghai Scientific and Technology Press, Shanghai, 1986. [11] S. Cingolani and S. Secchi, Semiclassical limit for nonlinear Schrodinger equations with electromagnetic fileds, J. Math. Anal. Appl., 275 (2002), 108-130.  doi: 10.1016/S0022-247X(02)00278-0. [12] F. Colasuonno and P. Pucci, Multiplicity of solutions for $p(x)$-polyharmonic Kirchhoff equations, Nonlinear Anal., 74 (2011), 5962-5974.  doi: 10.1016/j.na.2011.05.073. [13] P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262.  doi: 10.1007/BF02100605. [14] P. d'Avenia and M. Squassina, Ground states for fractional magnetic operators ESAIM Control Optim. Calc. Var. Forthcoming article. doi: 10.1051/cocv/2016071. [15] J. Di Cosmo and J. Van Schaftingen, Semiclassical stationary states for nonlinear Schrödiner equations under a strong extenal magnetic field, J. Differential Equations, 259 (2015), 596-627.  doi: 10.1016/j.jde.2015.02.016. [16] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. [17] S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $\mathbb R^n$, arXiv: 1506.01748. [18] S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Le Matematiche, 68 (2013), 201-216. [19] 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. [20] 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. [21] T. Ichinose, Essential selfadjointness of the Weyl quantized relativistic Hamiltonian, Ann. Inst. H. Poincaré Phys. Théor, 51 (1989), 265-297. [22] 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. [23] T. Ichinose and H. Tamura, Imaginary-time path integral for a relativistic spinless particle in an electromagnetic field, Comm. Math. Phys., 105 (1986), 239-257.  doi: 10.1007/BF01211101. [24] V. Iftimie, M. Măntoiu and R. Purice, Magnetic pseudodifferential operators, Publ. Res. Inst. Math. Sci., 43 (2007), 585-623.  doi: 10.2977/prims/1201012035. [25] R. Kajikiya, A critical point theorem related to the symmetric mountain pass lemma and its applications to elliptic equations, J. Funct. Anal., 225 (2005), 352-370.  doi: 10.1016/j.jfa.2005.04.005. [26] G. Kirchhoff, Mechanik Teubner, Leipzig, 1883. [27] M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, MacMillan, New York, 1964. [28] K. Kurata, Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fileds, Nonlinear Anal., 41 (2000), 763-778.  doi: 10.1016/S0362-546X(98)00308-3. [29] N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2. [30] N. Laskin, Fractional Schrödinger equation Phys. Rev. E 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108. [31] P. L. Lions, Symétrie et compacité dans les espaces de Sobolev, J. Funct. Anal., 49 (1982), 315-334.  doi: 10.1016/0022-1236(82)90072-6. [32] X. Mingqi, G. Molica Bisci, G. Tian and B. Zhang, Infinitely many solutions for the stationary Kirchhoff problems involving the fractional $p$-Laplacian, Nonlinearity, 29 (2016), 357-374.  doi: 10.1088/0951-7715/29/2/357. [33] G. Molica Bisci, V. D. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Equations Encyclopedia of Mathematics and its Applications, 162, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316282397. [34] R. Narashima, Nonlinear vibration of an elastic string, J. Sound Vib., 8 (1968), 134-146. [35] D. W. Oplinger, Frequency response of a nonlinear stretched string, J. Acoust. Soc. Am., 32 (1960), 1529-1538.  doi: 10.1121/1.1907948. [36] A. Pinamonti, M. Squassina and E. Vecchi, Magnetic BV functions and the Bourgain-Brezis-Mironescu formula, preprint. [37] A. Pinamonti, M. Squassina and E. Vecchi, The Maz'ya-Shaposhnikova limit in the magnetic setting, preprint. [38] 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. [39] 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. [40] P. Pucci, M. Xiang and B. Zhang, Existence and multiplicity of entire solutions for fractional $p$ -Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27-55.  doi: 10.1515/anona-2015-0102. [41] S. Secchi, Ground state solutions for the fractional Schrödinger in $\mathbb{R}^N$ J. Math. Phys. 54 2013,031501, 17pp. doi: 10.1063/1.4793990. [42] M. Squassina, Soliton dynamics for the nonlinear Schrödinger equation with magnetic field, Manuscripta Math., 130 (2009), 461-494.  doi: 10.1007/s00229-009-0307-y. [43] 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. [44] M. Xiang, B. Zhang and M. Ferrara, Existence of solutions for Kirchhoff type problem involving the non-local fractional $p-$Laplacian, J. Math. Anal. Appl., 424 (2015), 1021-1041.  doi: 10.1016/j.jmaa.2014.11.055. [45] M. Xiang, B. Zhang and M. Ferrara, Multiplicity results for the nonhomogeneous fractional $p$-Kirchhoff equations with concave-convex nonlinearities, Proc.A, 471 (2015), 20150034-14 pp.  doi: 10.1098/rspa.2015.0034. [46] M. Xiang, B. Zhang and X. Guo, Infinitely many solutions for a fractional Kirchhoff type problem via Fountain Theorem, Nonlinear Anal., 120 (2015), 299-313.  doi: 10.1016/j.na.2015.03.015.
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