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 & Continuous Dynamical Systems - A, 2020, 40 (2) : 781-815. doi: 10.3934/dcds.2020062
References:
[1]

C. O. AlvesG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

V. Ambrosio and T. Isernia, Concentration phenomena for a fractional Schrödinger-Kirchhoff type problem, Math. Methods Appl. Sci., 41 (2018), 615-645.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. DávilaM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

E. Di NezzaG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

P. FelmerA. 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

[27]

A. FiscellaA. 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.  Google Scholar

[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.  Google Scholar

[29]

Y. HeG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

T. Kato, Schrödinger operators with singular potentials, Israel J. Math., 13 (1972), 135-148.  doi: 10.1007/BF02760233.  Google Scholar

[33]

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

S. LiangD. 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.  Google Scholar

[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.  Google Scholar

[39]

X. MingqiP. PucciM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

N. Nyamoradi, Existence of three solutions for Kirchhoff nonlocal operators of elliptic type, Math. Commun., 18 (2013), 489-502.   Google Scholar

[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.  Google Scholar

[44]

S. I. Pohožaev, A certain class of quasilinear hyperbolic equations, Mat. Sb., 96 (1975), 152-166.   Google Scholar

[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.  Google Scholar

[46]

P. PucciM. 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

[47]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

[48] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I, Functional analysis, Academic Press, Inc., New York, 1980.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[51]

J. WangL. TianJ. 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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

C. O. AlvesG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

V. Ambrosio and T. Isernia, Concentration phenomena for a fractional Schrödinger-Kirchhoff type problem, Math. Methods Appl. Sci., 41 (2018), 615-645.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

J. DávilaM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

E. Di NezzaG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

P. FelmerA. 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

[27]

A. FiscellaA. 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.  Google Scholar

[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.  Google Scholar

[29]

Y. HeG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

T. Kato, Schrödinger operators with singular potentials, Israel J. Math., 13 (1972), 135-148.  doi: 10.1007/BF02760233.  Google Scholar

[33]

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

S. LiangD. 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.  Google Scholar

[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.  Google Scholar

[39]

X. MingqiP. PucciM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

N. Nyamoradi, Existence of three solutions for Kirchhoff nonlocal operators of elliptic type, Math. Commun., 18 (2013), 489-502.   Google Scholar

[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.  Google Scholar

[44]

S. I. Pohožaev, A certain class of quasilinear hyperbolic equations, Mat. Sb., 96 (1975), 152-166.   Google Scholar

[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.  Google Scholar

[46]

P. PucciM. 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

[47]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

[48] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I, Functional analysis, Academic Press, Inc., New York, 1980.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[51]

J. WangL. TianJ. 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.  Google Scholar

[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.  Google Scholar

[1]

Raffaella Servadei, Enrico Valdinoci. Variational methods for non-local operators of elliptic type. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2105-2137. doi: 10.3934/dcds.2013.33.2105

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