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 and 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.

[2]

V. Ambrosio, Mountain pass solutions for the fractional Berestycki–Lions problem, Adv. Differ. Equ., 23 (2018), 455-488. 

[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.

[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.

[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.

[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.

[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.

[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.

[9]

H. BerestyckiT. 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. 

[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.

[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.

[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. 

[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.

[14]

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.

[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.

[16]

P. FelmerA. 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.

[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.

[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.
[19]

J. HirataN. 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. 

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[30]

J. Mederski, Nonradial solutions of nonlinear scalar field equations, preprint, arXiv: 1711.05711.

[31]

J. Mederski, General class of optimal Sobolev inequalities and nonlinear scalar field equations, preprint, arXiv: 1812.11451.

[32]

R. S. Palais, Lusternik–Schnirelman theory on Banach manifolds, Topology, 5 (1966), 115-132.  doi: 10.1016/0040-9383(66)90013-9.

[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.

[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.

[35]

W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162. 

[36]

M. Struwe, Multiple solutions of differential equations without the Palais-Smale condition, Math. Ann., 261 (1982), 399-412.  doi: 10.1007/BF01455458.

[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.

[38]

J. TanY. Wang and J. Yang, Nonlinear fractional field equations, Nonlinear Anal., 75 (2012), 2098-2110.  doi: 10.1016/j.na.2011.10.010.

[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.

show all references

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.

[2]

V. Ambrosio, Mountain pass solutions for the fractional Berestycki–Lions problem, Adv. Differ. Equ., 23 (2018), 455-488. 

[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.

[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.

[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.

[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.

[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.

[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.

[9]

H. BerestyckiT. 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. 

[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.

[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.

[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. 

[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.

[14]

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.

[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.

[16]

P. FelmerA. 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.

[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.

[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.
[19]

J. HirataN. 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. 

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[30]

J. Mederski, Nonradial solutions of nonlinear scalar field equations, preprint, arXiv: 1711.05711.

[31]

J. Mederski, General class of optimal Sobolev inequalities and nonlinear scalar field equations, preprint, arXiv: 1812.11451.

[32]

R. S. Palais, Lusternik–Schnirelman theory on Banach manifolds, Topology, 5 (1966), 115-132.  doi: 10.1016/0040-9383(66)90013-9.

[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.

[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.

[35]

W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162. 

[36]

M. Struwe, Multiple solutions of differential equations without the Palais-Smale condition, Math. Ann., 261 (1982), 399-412.  doi: 10.1007/BF01455458.

[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.

[38]

J. TanY. Wang and J. Yang, Nonlinear fractional field equations, Nonlinear Anal., 75 (2012), 2098-2110.  doi: 10.1016/j.na.2011.10.010.

[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.

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