The nonlinear fractional relativistic Schrödinger equation: Existence, multiplicity, decay and concentration results

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December 2021, 41(12): 5659-5705. doi: 10.3934/dcds.2021092

The nonlinear fractional relativistic Schrödinger equation: Existence, multiplicity, decay and concentration results

Vincenzo Ambrosio

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

Received December 2020 Revised April 2021 Published December 2021 Early access June 2021

In this paper we study the following class of fractional relativistic Schrödinger equations:

$ \begin{equation*} \left\{ \begin{array}{ll} (-\Delta+m^{2})^{s}u + V(\varepsilon x) u = f(u) &\text{ in } \mathbb{R}^{N}, \\ u\in H^{s}( \mathbb{R}^{N}), \quad u>0 &\text{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} $

where

$ \varepsilon >0 $

is a small parameter,

$ s\in (0, 1) $

,

$ m>0 $

,

$ N> 2s $

,

$ (-\Delta+m^{2})^{s} $

is the fractional relativistic Schrödinger operator,

$ V: \mathbb{R}^{N} \rightarrow \mathbb{R} $

is a continuous potential satisfying a local condition, and

$ f: \mathbb{R} \rightarrow \mathbb{R} $

is a continuous subcritical nonlinearity. By using a variant of the extension method and a penalization technique, we first prove that, for

$ \varepsilon >0 $

small enough, the above problem admits a weak solution

$ u_{\varepsilon } $

which concentrates around a local minimum point of

$ V $

as

$ \varepsilon \rightarrow 0 $

. We also show that

$ u_{\varepsilon } $

has an exponential decay at infinity by constructing a suitable comparison function and by performing some refined estimates. Secondly, by combining the generalized Nehari manifold method and Ljusternik-Schnirelman theory, we relate the number of positive solutions with the topology of the set where the potential

$ V $

attains its minimum value.

Keywords: fractional relativistic Schrödinger operator, extension method, variational methods, Ljusternik-Schnirelman theory.

Mathematics Subject Classification: Primary: 35R11, 35J10, 35J20; Secondary: 35J60, 35B09, 58E05.

Citation: Vincenzo Ambrosio. The nonlinear fractional relativistic Schrödinger equation: Existence, multiplicity, decay and concentration results. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5659-5705. doi: 10.3934/dcds.2021092

References:

[1]	R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65 Academic Press, New York-London, 1975.
[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, Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator, J. Math. Phys., 57 (2016), 051502, 18 pp. doi: 10.1063/1.4949352.
[5]	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.
[6]	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.
[7]	V. Ambrosio, Concentration phenomena for a class of fractional Kirchhoff equations in $ \mathbb{R}^{N}$ with general nonlinearities, Nonlinear Anal., 195 (2020), 111761, 39 pp. doi: 10.1016/j.na.2020.111761.
[8]	N. Aronszajn and K. T. Smith, Theory of Bessel potentials. I, Ann. Inst. Fourier (Grenoble), 11 (1961), 385-475. doi: 10.5802/aif.116.
[9]	C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer, Unione Matematica Italiana, Bologna, 2016. xii+155 pp. doi: 10.1007/978-3-319-28739-3.
[10]	H. Bueno, O. H. Miyagaki and G. A. Pereira, Remarks about a generalized pseudo-relativistic Hartree equation, J. Differential Equations, 266 (2019), 876-909. doi: 10.1016/j.jde.2018.07.058.
[11]	T. Byczkowski, J. Malecki and M. Ryznar, Bessel potentials, hitting distributions and Green functions, Trans. Amer. Math. Soc., 361 (2009), 4871-4900. doi: 10.1090/S0002-9947-09-04657-1.
[12]	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.
[13]	A.-P. Calderón, Lebesgue spaces of differentiable functions and distributions, Proc. Sympos. Pure Math., American Mathematical Society, Providence, R.I., 4 (1961), 33-49.
[14]	R. Carmona, W. C. Masters and B. Simon, Relativistic Schrödinger operators: Asymptotic behavior of the eigenfunctions, J. Func. Anal., 91 (1990), 117-142. doi: 10.1016/0022-1236(90)90049-Q.
[15]	S. Cingolani and S. Secchi, Semiclassical analysis for pseudo-relativistic Hartree equations, J. Differential Equations, 258 (2015), 4156-4179. doi: 10.1016/j.jde.2015.01.029.
[16]	V. Coti Zelati and M. Nolasco, Existence of ground states for nonlinear, pseudo-relativistic Schrödinger equations, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 22 (2011), 51-72. doi: 10.4171/RLM/587.
[17]	V. Coti Zelati and M. Nolasco, Ground states for pseudo-relativistic Hartree equations of critical type, Rev. Mat. Iberoam., 29 (2013), 1421-1436. doi: 10.4171/RMI/763.
[18]	J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Anal. PDE, 8 (2015), 1165-1235. doi: 10.2140/apde.2015.8.1165.
[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]	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.
[21]	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.
[22]	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. doi: 10.1007/978-88-7642-601-8.
[23]	A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions. Vol. II, Based on notes left by Harry Bateman. Reprint of the 1953 original. Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981.
[24]	E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116. doi: 10.1080/03605308208820218.
[25]	M. M. Fall and V. Felli, Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst., 35 (2015), 5827-5867. doi: 10.3934/dcds.2015.35.5827.
[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]	P. Felmer and I. Vergara, Scalar field equation with non-local diffusion, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1411-1428. doi: 10.1007/s00030-015-0328-z.
[28]	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.
[29]	G. M. Figueiredo and G. Siciliano, A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrödinger equation in $ \mathbb{R}^{N}$, NoDEA Nonlinear Differential Equations Appl., 23 (2016), art. 12, 22 pp. doi: 10.1007/s00030-016-0355-4.
[30]	L. Grafakos, Modern Fourier analysis, Third edition. Graduate Texts in Mathematics, 250. Springer, New York, 2014. doi: 10.1007/978-1-4939-1230-8.
[31]	T. Grzywny and M. Ryznar, Two-sided optimal bounds for Green functions of half-spaces for relativistic $\alpha$-stable process, Potential Anal., 28 (2008), 201-239. doi: 10.1007/s11118-007-9071-3.
[32]	I. W. Herbst, Spectral theory of the operator $(p^{2}+m^{2})^{1/2}-Ze^{2}/r$, Comm. Math. Phys., 53 (1977), 285-294.
[33]	L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], 26. Springer-Verlag, Berlin, 1997.
[34]	T. Jin, Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: Blow up analysis and compactness of solutions, J. Eur. Math. Soc. (JEMS), 16 (2014), 1111-1171. doi: 10.4171/JEMS/456.
[35]	E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 1997.
[36]	E. H. Lieb and H. T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys., 112 (1987), 147-174. doi: 10.1007/BF01217684.
[37]	P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283. doi: 10.1016/S0294-1449(16)30422-X.
[38]	G. Molica Bisci, V. Rǎdulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, 162 Cambridge, 2016. doi: 10.1017/CBO9781316282397.
[39]	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.
[40]	D. Mugnai, Pseudorelativistic Hartree equation with general nonlinearity: existence, non-existence and variational identities, Adv. Nonlinear Stud., 13 (2013), 799-823. doi: 10.1515/ans-2013-0403.
[41]	M. Ryznar, Estimate of Green function for relativistic $\alpha$-stable processes, Potential Analysis, 17 (2002), 1-23. doi: 10.1023/A:1015231913916.
[42]	S. Secchi, On some nonlinear fractional equations involving the Bessel potential, J. Dynam. Differential Equations, 29 (2017), 1173-1193. doi: 10.1007/s10884-016-9521-y.
[43]	E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970.
[44]	P. R. Stinga, User's guide to the fractional Laplacian and the method of semigroups, Handbook of Fractional Calculus with Applications, De Gruyter, Berlin, 2 (2019), 235–265.
[45]	P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122. doi: 10.1080/03605301003735680.
[46]	A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, Int. Press, Somerville, MA, 2010,597–632.
[47]	M. H. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean $n$-space. I. Principal properties, J. Math. Mech., 13 (1964), 407-479.
[48]	R. A. Weder, Spectral properties of one-body relativistic spin-zero Hamiltonians, Ann. Inst. H. Poincaré Sect. A (N.S.), 20 (1974), 211-220.
[49]	R. A. Weder, Spectral analysis of pseudodifferential operators, J. Functional Analysis, 20 (1975), 319-337. doi: 10.1016/0022-1236(75)90038-5.
[50]	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.

show all references

References:

[1]	R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65 Academic Press, New York-London, 1975.
[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, Ground states solutions for a non-linear equation involving a pseudo-relativistic Schrödinger operator, J. Math. Phys., 57 (2016), 051502, 18 pp. doi: 10.1063/1.4949352.
[5]	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.
[6]	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.
[7]	V. Ambrosio, Concentration phenomena for a class of fractional Kirchhoff equations in $ \mathbb{R}^{N}$ with general nonlinearities, Nonlinear Anal., 195 (2020), 111761, 39 pp. doi: 10.1016/j.na.2020.111761.
[8]	N. Aronszajn and K. T. Smith, Theory of Bessel potentials. I, Ann. Inst. Fourier (Grenoble), 11 (1961), 385-475. doi: 10.5802/aif.116.
[9]	C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 20. Springer, Unione Matematica Italiana, Bologna, 2016. xii+155 pp. doi: 10.1007/978-3-319-28739-3.
[10]	H. Bueno, O. H. Miyagaki and G. A. Pereira, Remarks about a generalized pseudo-relativistic Hartree equation, J. Differential Equations, 266 (2019), 876-909. doi: 10.1016/j.jde.2018.07.058.
[11]	T. Byczkowski, J. Malecki and M. Ryznar, Bessel potentials, hitting distributions and Green functions, Trans. Amer. Math. Soc., 361 (2009), 4871-4900. doi: 10.1090/S0002-9947-09-04657-1.
[12]	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.
[13]	A.-P. Calderón, Lebesgue spaces of differentiable functions and distributions, Proc. Sympos. Pure Math., American Mathematical Society, Providence, R.I., 4 (1961), 33-49.
[14]	R. Carmona, W. C. Masters and B. Simon, Relativistic Schrödinger operators: Asymptotic behavior of the eigenfunctions, J. Func. Anal., 91 (1990), 117-142. doi: 10.1016/0022-1236(90)90049-Q.
[15]	S. Cingolani and S. Secchi, Semiclassical analysis for pseudo-relativistic Hartree equations, J. Differential Equations, 258 (2015), 4156-4179. doi: 10.1016/j.jde.2015.01.029.
[16]	V. Coti Zelati and M. Nolasco, Existence of ground states for nonlinear, pseudo-relativistic Schrödinger equations, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 22 (2011), 51-72. doi: 10.4171/RLM/587.
[17]	V. Coti Zelati and M. Nolasco, Ground states for pseudo-relativistic Hartree equations of critical type, Rev. Mat. Iberoam., 29 (2013), 1421-1436. doi: 10.4171/RMI/763.
[18]	J. Dávila, M. del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with Dirichlet datum, Anal. PDE, 8 (2015), 1165-1235. doi: 10.2140/apde.2015.8.1165.
[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]	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.
[21]	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.
[22]	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. doi: 10.1007/978-88-7642-601-8.
[23]	A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions. Vol. II, Based on notes left by Harry Bateman. Reprint of the 1953 original. Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981.
[24]	E. B. Fabes, C. E. Kenig and R. P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations, 7 (1982), 77-116. doi: 10.1080/03605308208820218.
[25]	M. M. Fall and V. Felli, Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst., 35 (2015), 5827-5867. doi: 10.3934/dcds.2015.35.5827.
[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]	P. Felmer and I. Vergara, Scalar field equation with non-local diffusion, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1411-1428. doi: 10.1007/s00030-015-0328-z.
[28]	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.
[29]	G. M. Figueiredo and G. Siciliano, A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrödinger equation in $ \mathbb{R}^{N}$, NoDEA Nonlinear Differential Equations Appl., 23 (2016), art. 12, 22 pp. doi: 10.1007/s00030-016-0355-4.
[30]	L. Grafakos, Modern Fourier analysis, Third edition. Graduate Texts in Mathematics, 250. Springer, New York, 2014. doi: 10.1007/978-1-4939-1230-8.
[31]	T. Grzywny and M. Ryznar, Two-sided optimal bounds for Green functions of half-spaces for relativistic $\alpha$-stable process, Potential Anal., 28 (2008), 201-239. doi: 10.1007/s11118-007-9071-3.
[32]	I. W. Herbst, Spectral theory of the operator $(p^{2}+m^{2})^{1/2}-Ze^{2}/r$, Comm. Math. Phys., 53 (1977), 285-294.
[33]	L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], 26. Springer-Verlag, Berlin, 1997.
[34]	T. Jin, Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: Blow up analysis and compactness of solutions, J. Eur. Math. Soc. (JEMS), 16 (2014), 1111-1171. doi: 10.4171/JEMS/456.
[35]	E. H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 1997.
[36]	E. H. Lieb and H. T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys., 112 (1987), 147-174. doi: 10.1007/BF01217684.
[37]	P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283. doi: 10.1016/S0294-1449(16)30422-X.
[38]	G. Molica Bisci, V. Rǎdulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Cambridge University Press, 162 Cambridge, 2016. doi: 10.1017/CBO9781316282397.
[39]	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.
[40]	D. Mugnai, Pseudorelativistic Hartree equation with general nonlinearity: existence, non-existence and variational identities, Adv. Nonlinear Stud., 13 (2013), 799-823. doi: 10.1515/ans-2013-0403.
[41]	M. Ryznar, Estimate of Green function for relativistic $\alpha$-stable processes, Potential Analysis, 17 (2002), 1-23. doi: 10.1023/A:1015231913916.
[42]	S. Secchi, On some nonlinear fractional equations involving the Bessel potential, J. Dynam. Differential Equations, 29 (2017), 1173-1193. doi: 10.1007/s10884-016-9521-y.
[43]	E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970.
[44]	P. R. Stinga, User's guide to the fractional Laplacian and the method of semigroups, Handbook of Fractional Calculus with Applications, De Gruyter, Berlin, 2 (2019), 235–265.
[45]	P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122. doi: 10.1080/03605301003735680.
[46]	A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, Int. Press, Somerville, MA, 2010,597–632.
[47]	M. H. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean $n$-space. I. Principal properties, J. Math. Mech., 13 (1964), 407-479.
[48]	R. A. Weder, Spectral properties of one-body relativistic spin-zero Hamiltonians, Ann. Inst. H. Poincaré Sect. A (N.S.), 20 (1974), 211-220.
[49]	R. A. Weder, Spectral analysis of pseudodifferential operators, J. Functional Analysis, 20 (1975), 319-337. doi: 10.1016/0022-1236(75)90038-5.
[50]	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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