# 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 & Continuous Dynamical Systems - A, 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [8] G. F. Carrier, On the nonlinear vibration problem of the elastic string, Quart. Appl. Math., 3 (1945), 157-165. Google Scholar [9] G. F. Carrier, A note on the vibrating string, Quart. Appl. Math., 7 (1949), 97-101. Google Scholar [10] K. C. Chang, Critical Point Theory and Applications Shanghai Scientific and Technology Press, Shanghai, 1986.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [17] S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $\mathbb R^n$, arXiv: 1506.01748.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [21] T. Ichinose, Essential selfadjointness of the Weyl quantized relativistic Hamiltonian, Ann. Inst. H. Poincaré Phys. Théor, 51 (1989), 265-297. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [26] G. Kirchhoff, Mechanik Teubner, Leipzig, 1883.Google Scholar [27] M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, MacMillan, New York, 1964. Google Scholar [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. Google Scholar [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. Google Scholar [30] N. Laskin, Fractional Schrödinger equation Phys. Rev. E 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [34] R. Narashima, Nonlinear vibration of an elastic string, J. Sound Vib., 8 (1968), 134-146. Google Scholar [35] D. W. Oplinger, Frequency response of a nonlinear stretched string, J. Acoust. Soc. Am., 32 (1960), 1529-1538. doi: 10.1121/1.1907948. Google Scholar [36] A. Pinamonti, M. Squassina and E. Vecchi, Magnetic BV functions and the Bourgain-Brezis-Mironescu formula, preprint.Google Scholar [37] A. Pinamonti, M. Squassina and E. Vecchi, The Maz'ya-Shaposhnikova limit in the magnetic setting, preprint.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

show all references

##### References:
 [1] D. Applebaum, Lévy processes -From probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [8] G. F. Carrier, On the nonlinear vibration problem of the elastic string, Quart. Appl. Math., 3 (1945), 157-165. Google Scholar [9] G. F. Carrier, A note on the vibrating string, Quart. Appl. Math., 7 (1949), 97-101. Google Scholar [10] K. C. Chang, Critical Point Theory and Applications Shanghai Scientific and Technology Press, Shanghai, 1986.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [17] S. Dipierro, M. Medina and E. Valdinoci, Fractional elliptic problems with critical growth in the whole of $\mathbb R^n$, arXiv: 1506.01748.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [21] T. Ichinose, Essential selfadjointness of the Weyl quantized relativistic Hamiltonian, Ann. Inst. H. Poincaré Phys. Théor, 51 (1989), 265-297. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [26] G. Kirchhoff, Mechanik Teubner, Leipzig, 1883.Google Scholar [27] M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations, MacMillan, New York, 1964. Google Scholar [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. Google Scholar [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. Google Scholar [30] N. Laskin, Fractional Schrödinger equation Phys. Rev. E 66 (2002), 056108, 7pp. doi: 10.1103/PhysRevE.66.056108. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [34] R. Narashima, Nonlinear vibration of an elastic string, J. Sound Vib., 8 (1968), 134-146. Google Scholar [35] D. W. Oplinger, Frequency response of a nonlinear stretched string, J. Acoust. Soc. Am., 32 (1960), 1529-1538. doi: 10.1121/1.1907948. Google Scholar [36] A. Pinamonti, M. Squassina and E. Vecchi, Magnetic BV functions and the Bourgain-Brezis-Mironescu formula, preprint.Google Scholar [37] A. Pinamonti, M. Squassina and E. Vecchi, The Maz'ya-Shaposhnikova limit in the magnetic setting, preprint.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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