# American Institute of Mathematical Sciences

July  2020, 19(7): 3501-3530. doi: 10.3934/cpaa.2020153

## Multiplicity of radial and nonradial solutions to equations with fractional operators

 Department of Mathematics, Faculty of Science and Technology, Keio University, , Yagami Campus: 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, Kanagawa 2238522, JAPAN

Received  February 2019 Revised  January 2020 Published  April 2020

Fund Project: The author is supported by JSPS KAKENHI Grant Number JP16K17623 and JP17H02851

In this paper, we study the existence of radial and nonradial solutions to the scalar field equations with fractional operators. For radial solutions, we prove the existence of infinitely many solutions under $N \geq 2$. We also show the existence of least energy solution (with the Pohozaev identity) and its mountain pass characterization. For nonradial solutions, we prove the existence of at least one nonradial solution under $N \geq 4$ and infinitely many nonradial solutions under either $N = 4$ or $N \geq 6$. We treat both of the zero mass and the positive mass cases.

Citation: Norihisa Ikoma. Multiplicity of radial and nonradial solutions to equations with fractional operators. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3501-3530. doi: 10.3934/cpaa.2020153
##### References:
 [1] F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc., 2 (1989), 683-773.  doi: 10.2307/1990893.  Google Scholar [2] V. Ambrosio, Mountain pass solutions for the fractional Berestycki–Lions problem, Adv. Differ. Equ., 23 (2018), 455-488.   Google Scholar [3] V. Ambrosio, Zero mass case for a fractional Berestycki–Lions-type problem, Adv. Nonlinear Anal., 7 (2018), 365-374.  doi: 10.1515/anona-2016-0153.  Google Scholar [4] T. Bartsch and S. de Valeriola, Normalized solutions of nonlinear Schrödinger equations, Arch. Math. (Basel), 100 (2013), 75-83.  doi: 10.1007/s00013-012-0468-x.  Google Scholar [5] T. Bartsch and N. Soave, A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems, J. Funct. Anal., 272 (2017), 4998-5037.  doi: 10.1016/j.jfa.2017.01.025.  Google Scholar [6] T. Bartsch and N. Soave, Correction to: A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems [J. Funct. Anal. 272 (2017), 4998–5037], J. Funct. Anal., 275 (2018), 516-521.  doi: 10.1016/j.jfa.2018.02.007.  Google Scholar [7] T. Bartsch and N. Soave, Multiple normalized solutions for a competing system of Schrödinger equations, Calc. Var. Partial Differ. Equ., 58 (2019), Art. 58, 22 pp. doi: 10.1007/s00526-018-1476-x.  Google Scholar [8] T. Bartsch and M. Willem, Infinitely many nonradial solutions of a Euclidean scalar field equation, J. Funct. Anal., 117 (1993), 447-460.  doi: 10.1006/jfan.1993.1133.  Google Scholar [9] H. Berestycki, T. Gallouët and O. Kavian, Équations de champs scalaires euclidiens non linéaires dans le plan, C. R. Acad. Sci. Paris. Ser. I Math., 297 (1983), 307-310.   Google Scholar [10] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar [11] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal., 82 (1983), 347-375.  doi: 10.1007/BF00250556.  Google Scholar [12] H. Berestycki and P. L. Lions, Existence d'états multiples dans des équations de champs scalaires non linéaires dans le cas de masse nulle, C. R. Acad. Sci. Paris. Ser. I Math., 297 (1983), 267-270.   Google Scholar [13] X. Chang and Z. Q. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494.  doi: 10.1088/0951-7715/26/2/479.  Google Scholar [14] 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.  Google Scholar [15] 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.  Google Scholar [16] P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. R. Soc. Edinb. Sect. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.  Google Scholar [17] P. Felmer and I. Vergara, Scalar field equation with non-local diffusion, NoDea-Nonlinear Differ. Equ. Appl., 22 (2015), 1411-1428.  doi: 10.1007/s00030-015-0328-z.  Google Scholar [18] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory(With appendices by David Robinson), Cambridge Tracts in Mathematics, Vol. 107, Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511551703.  Google Scholar [19] J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in $\mathbb{R}^N$: mountain pass and symmetric mountain pass approaches, Topol. Meth. Nonlinear Anal., 35 (2010), 253-276.   Google Scholar [20] J. Hirata and K. Tanaka, Nonlinear scalar field equations with $L^2$ constraint: Mountain pass and symmetric mountain pass approaches, Adv. Nonlinear Stud., 19 (2019), 263-290.  doi: 10.1515/ans-2018-2039.  Google Scholar [21] N. Ikoma, Existence of solutions of scalar field equations with fractional operator, J. Fixed Point Theory Appl., 19 (2017), 649-690.  doi: 10.1007/s11784-016-0369-x.  Google Scholar [22] N. Ikoma, Erratum to: Existence of solutions of scalar field equations with fractional operator, J. Fixed Point Theory Appl., 19 (2017), 1649-1652.  doi: 10.1007/s11784-017-0427-z.  Google Scholar [23] N. Ikoma and K. Tanaka, A note on deformation argument for $L^2$ normalized solutions of nonlinear Schrödinger equations and systems, Adv. Differ. Equ., 24 (2019), 609-646.   Google Scholar [24] L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659.  doi: 10.1016/S0362-546X(96)00021-1.  Google Scholar [25] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbf{R}^N$, Proc. R. Soc. Edinb. Sect. A, 129 (1999), 787-809.  doi: 10.1017/S0308210500013147.  Google Scholar [26] L. Jeanjean and S. S. Lu, Nonlinear scalar field equations with general nonlinearity, Nonlinear Anal., 190 (2020), Art. 111604, 28 pp. doi: 10.1016/j.na.2019.111604.  Google Scholar [27] L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbf{R}^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.  doi: 10.1090/S0002-9939-02-06821-1.  Google Scholar [28] E. H. Lieb and M. Loss, Analysis, 2$^{nd}$ edition, Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar [29] 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 [30] J. Mederski, Nonradial solutions of nonlinear scalar field equations, preprint, arXiv: 1711.05711. Google Scholar [31] J. Mederski, General class of optimal Sobolev inequalities and nonlinear scalar field equations, preprint, arXiv: 1812.11451. Google Scholar [32] R. S. Palais, Lusternik–Schnirelman theory on Banach manifolds, Topology, 5 (1966), 115-132.  doi: 10.1016/0040-9383(66)90013-9.  Google Scholar [33] P. H. Rabinowitz, Minimax Methods in Critical Point Theory With Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, Vol. 65, American Mathematical Society, Providence, RI, 1986. doi: 10.1090/cbms/065.  Google Scholar [34] S. Secchi, On some nonlinear fractional equations involving the Bessel operator, J. Dyn. Differ. Equ., 29 (2017), 1173-1193.  doi: 10.1007/s10884-016-9521-y.  Google Scholar [35] W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.   Google Scholar [36] M. Struwe, Multiple solutions of differential equations without the Palais-Smale condition, Math. Ann., 261 (1982), 399-412.  doi: 10.1007/BF01455458.  Google Scholar [37] M. Struwe, A generalized Palais–Smale condition and applications, in Nonlinear Functional Analysis and Its Applications, Part 2, (Berkeley, Calif., 1983), Proc. Sympos. Pure Math., Vol. 45, Part 2, American Mathematical Society, Providence, RI, (1986), 401–411.  Google Scholar [38] J. Tan, Y. Wang and J. Yang, Nonlinear fractional field equations, Nonlinear Anal., 75 (2012), 2098-2110.  doi: 10.1016/j.na.2011.10.010.  Google Scholar [39] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, Vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

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##### References:
 [1] F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc., 2 (1989), 683-773.  doi: 10.2307/1990893.  Google Scholar [2] V. Ambrosio, Mountain pass solutions for the fractional Berestycki–Lions problem, Adv. Differ. Equ., 23 (2018), 455-488.   Google Scholar [3] V. Ambrosio, Zero mass case for a fractional Berestycki–Lions-type problem, Adv. Nonlinear Anal., 7 (2018), 365-374.  doi: 10.1515/anona-2016-0153.  Google Scholar [4] T. Bartsch and S. de Valeriola, Normalized solutions of nonlinear Schrödinger equations, Arch. Math. (Basel), 100 (2013), 75-83.  doi: 10.1007/s00013-012-0468-x.  Google Scholar [5] T. Bartsch and N. Soave, A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems, J. Funct. Anal., 272 (2017), 4998-5037.  doi: 10.1016/j.jfa.2017.01.025.  Google Scholar [6] T. Bartsch and N. Soave, Correction to: A natural constraint approach to normalized solutions of nonlinear Schrödinger equations and systems [J. Funct. Anal. 272 (2017), 4998–5037], J. Funct. Anal., 275 (2018), 516-521.  doi: 10.1016/j.jfa.2018.02.007.  Google Scholar [7] T. Bartsch and N. Soave, Multiple normalized solutions for a competing system of Schrödinger equations, Calc. Var. Partial Differ. Equ., 58 (2019), Art. 58, 22 pp. doi: 10.1007/s00526-018-1476-x.  Google Scholar [8] T. Bartsch and M. Willem, Infinitely many nonradial solutions of a Euclidean scalar field equation, J. Funct. Anal., 117 (1993), 447-460.  doi: 10.1006/jfan.1993.1133.  Google Scholar [9] H. Berestycki, T. Gallouët and O. Kavian, Équations de champs scalaires euclidiens non linéaires dans le plan, C. R. Acad. Sci. Paris. Ser. I Math., 297 (1983), 307-310.   Google Scholar [10] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar [11] H. Berestycki and P. L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal., 82 (1983), 347-375.  doi: 10.1007/BF00250556.  Google Scholar [12] H. Berestycki and P. L. Lions, Existence d'états multiples dans des équations de champs scalaires non linéaires dans le cas de masse nulle, C. R. Acad. Sci. Paris. Ser. I Math., 297 (1983), 267-270.   Google Scholar [13] X. Chang and Z. Q. Wang, Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity, Nonlinearity, 26 (2013), 479-494.  doi: 10.1088/0951-7715/26/2/479.  Google Scholar [14] 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.  Google Scholar [15] 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.  Google Scholar [16] P. Felmer, A. Quaas and J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. R. Soc. Edinb. Sect. A, 142 (2012), 1237-1262.  doi: 10.1017/S0308210511000746.  Google Scholar [17] P. Felmer and I. Vergara, Scalar field equation with non-local diffusion, NoDea-Nonlinear Differ. Equ. Appl., 22 (2015), 1411-1428.  doi: 10.1007/s00030-015-0328-z.  Google Scholar [18] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory(With appendices by David Robinson), Cambridge Tracts in Mathematics, Vol. 107, Cambridge University Press, Cambridge, 1993.  doi: 10.1017/CBO9780511551703.  Google Scholar [19] J. Hirata, N. Ikoma and K. Tanaka, Nonlinear scalar field equations in $\mathbb{R}^N$: mountain pass and symmetric mountain pass approaches, Topol. Meth. Nonlinear Anal., 35 (2010), 253-276.   Google Scholar [20] J. Hirata and K. Tanaka, Nonlinear scalar field equations with $L^2$ constraint: Mountain pass and symmetric mountain pass approaches, Adv. Nonlinear Stud., 19 (2019), 263-290.  doi: 10.1515/ans-2018-2039.  Google Scholar [21] N. Ikoma, Existence of solutions of scalar field equations with fractional operator, J. Fixed Point Theory Appl., 19 (2017), 649-690.  doi: 10.1007/s11784-016-0369-x.  Google Scholar [22] N. Ikoma, Erratum to: Existence of solutions of scalar field equations with fractional operator, J. Fixed Point Theory Appl., 19 (2017), 1649-1652.  doi: 10.1007/s11784-017-0427-z.  Google Scholar [23] N. Ikoma and K. Tanaka, A note on deformation argument for $L^2$ normalized solutions of nonlinear Schrödinger equations and systems, Adv. Differ. Equ., 24 (2019), 609-646.   Google Scholar [24] L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28 (1997), 1633-1659.  doi: 10.1016/S0362-546X(96)00021-1.  Google Scholar [25] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on $\mathbf{R}^N$, Proc. R. Soc. Edinb. Sect. A, 129 (1999), 787-809.  doi: 10.1017/S0308210500013147.  Google Scholar [26] L. Jeanjean and S. S. Lu, Nonlinear scalar field equations with general nonlinearity, Nonlinear Anal., 190 (2020), Art. 111604, 28 pp. doi: 10.1016/j.na.2019.111604.  Google Scholar [27] L. Jeanjean and K. Tanaka, A remark on least energy solutions in $\mathbf{R}^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.  doi: 10.1090/S0002-9939-02-06821-1.  Google Scholar [28] E. H. Lieb and M. Loss, Analysis, 2$^{nd}$ edition, Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar [29] 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 [30] J. Mederski, Nonradial solutions of nonlinear scalar field equations, preprint, arXiv: 1711.05711. Google Scholar [31] J. Mederski, General class of optimal Sobolev inequalities and nonlinear scalar field equations, preprint, arXiv: 1812.11451. Google Scholar [32] R. S. Palais, Lusternik–Schnirelman theory on Banach manifolds, Topology, 5 (1966), 115-132.  doi: 10.1016/0040-9383(66)90013-9.  Google Scholar [33] P. H. Rabinowitz, Minimax Methods in Critical Point Theory With Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, Vol. 65, American Mathematical Society, Providence, RI, 1986. doi: 10.1090/cbms/065.  Google Scholar [34] S. Secchi, On some nonlinear fractional equations involving the Bessel operator, J. Dyn. Differ. Equ., 29 (2017), 1173-1193.  doi: 10.1007/s10884-016-9521-y.  Google Scholar [35] W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.   Google Scholar [36] M. Struwe, Multiple solutions of differential equations without the Palais-Smale condition, Math. Ann., 261 (1982), 399-412.  doi: 10.1007/BF01455458.  Google Scholar [37] M. Struwe, A generalized Palais–Smale condition and applications, in Nonlinear Functional Analysis and Its Applications, Part 2, (Berkeley, Calif., 1983), Proc. Sympos. Pure Math., Vol. 45, Part 2, American Mathematical Society, Providence, RI, (1986), 401–411.  Google Scholar [38] J. Tan, Y. Wang and J. Yang, Nonlinear fractional field equations, Nonlinear Anal., 75 (2012), 2098-2110.  doi: 10.1016/j.na.2011.10.010.  Google Scholar [39] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, Vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar