January  2017, 16(1): 331-344. doi: 10.3934/cpaa.2017016

Periodic solutions for nonlocal fractional equations

1. 

Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli Federico Ⅱ, via Cinthia, 80126 Napoli, Italy

2. 

Dipartimento P.A.U. Università degli Studi Mediterranea di Reggio Calabria, Salita Melissari -Feo di Vito, 89100 Reggio Calabria, Italy

Received  June 2016 Revised  August 2016 Published  November 2016

The purpose of this paper is to study the existence of (weak) periodic solutions for nonlocal fractional equations with periodic boundary conditions. These equations have a variational structure and, by applying a critical point result coming out from a classical Pucci-Serrin theorem in addition to a local minimum result for differentiable functionals due to Ricceri, we are able to prove the existence of at least two periodic solutions for the treated problems. As far as we know, all these results are new.

Citation: Vincenzo Ambrosio, Giovanni Molica Bisci. Periodic solutions for nonlocal fractional equations. Communications on Pure & Applied Analysis, 2017, 16 (1) : 331-344. doi: 10.3934/cpaa.2017016
References:
[1]

V. Ambrosio, Periodic solutions for a pseudo-relativistic Schrödinger equation, Nonlinear Anal. TMA, 120 (2015), 262-284. doi: 10.1016/j.na.2015.03.017.

[2]

V. Ambrosio, Periodic solutions for the non-local operator pseudo-relativistic (-∆ + m2)s -m2s with m ≥ 0, Topol. Methods Nonlinear Anal. , to appear. doi: 10.1016/j.na.2015.03.017.

[3]

V. Ambrosio, Periodic solutions for a superlinear fractional problem without the AmbrosettiRabinowitz condition, Discrete Contin. Dyn. Syst. , to appear.

[4]

G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in ${\mathbb{R}.N}$, J. Differential Equations, 255 (2013), 2340-2362. doi: 10.1016/j.jde.2013.06.016.

[5]

R. Bartolo and G. Molica Bisci, Asymptotically linear fractional p-Laplacian equations, Ann. Mat. Pura Appl. , to appear.

[6]

R. Bartolo and G. Molica Bisci, A pseudo-index approach to fractional equations, Expo. Math., 33 (2015), 502-516. doi: 10.1016/j.exmath.2014.12.001.

[7]

Z. BinlinG. Molica Bisci and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity, 28 (2015), 2247-2264. doi: 10.1088/0951-7715/28/7/2247.

[8]

L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[9]

A. CapellaJ. DávilaL. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384. doi: 10.1080/03605302.2011.562954.

[10]

R. CarmonaW. 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.

[11]

D. G. Costa and C. A. Magalhaes, Variational elliptic problems which are nonquadratic at infinity, Nonlinear Anal., 23 (1994), 1401-1412. doi: 10.1016/0362-546X(94)90135-X.

[12]

M. Dabkowski, Eventual regularity of the solutions to the supercritical dissipative quasigeostrophic equation, Geom. Funct. Anal., 21 (2011), 1-13. doi: 10.1007/s00039-011-0108-9.

[13]

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.

[14]

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.

[15]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on ${\mathbb{R}.N}$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809. doi: 10.1017/S0308210500013147.

[16]

A. KiselevF. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453. doi: 10.1007/s00222-006-0020-3.

[17]

T. KuusiG. Mingione and Y. Sire, Nonlocal self-improving properties, Anal. PDE, 8 (2015), 57-114. doi: 10.2140/apde.2015.8.57.

[18]

E. H. Lieb and M. Loss, Analysis, 2nd edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.

[19]

O. Miyagaki and M. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations, 245 (2008), 3628-3638. doi: 10.1016/j.jde.2008.02.035.

[20]

G. Molica Bisci and M. Ferrara, Subelliptic and parametric equations on Carnot groups, Proc. Amer. Math. Soc., 144 (2016), 3035-3045. doi: 10.1090/proc/12948.

[21]

G. Molica Bisci and V. Rădulescu, A characterization for elliptic problems on fractal sets, Proc. Amer. Math. Soc., 143 (2015), 2959-2968. doi: 10.1090/S0002-9939-2015-12475-6.

[22]

G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems. With a Foreword by Jean Mawhin, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 162 Cambridge, 2016. doi: 10.1017/CBO9781316282397.

[23]

G. Molica Bisci and R. Servadei, A bifurcation result for nonlocal fractional equations, Anal. Appl., 13 (2015), 371-394. doi: 10.1142/S0219530514500067.

[24]

P. Pucci, Geometric description of the mountain pass critical points, Contemporary Mathematicians, Vol. 2, Birkhäuser, Basel, 2014,469-471.

[25]

P. Pucci and V. Rădulescu, The impact of the mountain pass theory in nonlinear analysis: a mathematical survey, Boll. Un. Mat. Ital. B, Ser. Ⅸ, (2010), 543-582.

[26]

P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in ${\mathbb{R}. N}$ involving nonlocal operators, Rev. Mat. Iberoam. , to appear. doi: 10.4171/RMI/879.

[27]

P. Pucci and J. Serrin, A mountain pass theorem, J. Differential Equations, 60 (1985), 142-149. doi: 10.1016/0022-0396(85)90125-1.

[28]

B. Ricceri, On a classical existence theorem for nonlinear elliptic equations, in Esperimental, constructive and nonlinar analysis, M. Théra ed. , CMS Conf. Proc. 27, Canad. Math. Soc. (2000), 275-278.

[29]

B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. , Special Issue on Fixed point theory with applications in Nonlinear Analysis, 113 (2000), 401-410. doi: 10.1016/S0377-0427(99)00269-1.

[30]

B. Ricceri, Nonlinear eigenvalue problems, in Handbook of Nonconvex Analysis and Applications (D. Y. Gao and D. Motreanu eds. ), International Press, (2010), 543-595.

[31]

B. Ricceri, A new existence and localization theorem for Dirichlet problem, Dynam. Systems Appl., 22 (2013), 317-324.

[32]

L. Roncal and P. R. Stinga, Fractional Laplacian on the torus, Commun. Contemp. Math., 18 (2016), 26 pp. doi: 10.1142/S0219199715500339.

[33]

L. Roncal and P. R. Stinga, Transference of fractional Laplacian regularity, in Special Functions, Partial Differential Equations, and Harmonic Analysis, 203-212, Springer Proc. Math. Stat. , 108, Springer, Cham. 2014. doi: 10.1007/978-3-319-10545-1_14.

[34]

M. Ryznar, Estimate of Green function for relativistic α-stable processes, Potential Analysis, 17 (2002), 1-23. doi: 10.1023/A:1015231913916.

[35]

M. Schechter and W. Zou, Superlinear problems, Pacific J. Math., 214 (2004), 145-160. doi: 10.2140/pjm.2004.214.145.

[36]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.

[37]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.

[38]

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.

[39]

P. R. Stinga and B. Volzone, Fractional semilinear Neumann problems arising from a fractional Keller-Segel model, Calc. Var. Partial Differential Equations, 54 (2015), 1009-1042. doi: 10.1007/s00526-014-0815-9.

show all references

References:
[1]

V. Ambrosio, Periodic solutions for a pseudo-relativistic Schrödinger equation, Nonlinear Anal. TMA, 120 (2015), 262-284. doi: 10.1016/j.na.2015.03.017.

[2]

V. Ambrosio, Periodic solutions for the non-local operator pseudo-relativistic (-∆ + m2)s -m2s with m ≥ 0, Topol. Methods Nonlinear Anal. , to appear. doi: 10.1016/j.na.2015.03.017.

[3]

V. Ambrosio, Periodic solutions for a superlinear fractional problem without the AmbrosettiRabinowitz condition, Discrete Contin. Dyn. Syst. , to appear.

[4]

G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in ${\mathbb{R}.N}$, J. Differential Equations, 255 (2013), 2340-2362. doi: 10.1016/j.jde.2013.06.016.

[5]

R. Bartolo and G. Molica Bisci, Asymptotically linear fractional p-Laplacian equations, Ann. Mat. Pura Appl. , to appear.

[6]

R. Bartolo and G. Molica Bisci, A pseudo-index approach to fractional equations, Expo. Math., 33 (2015), 502-516. doi: 10.1016/j.exmath.2014.12.001.

[7]

Z. BinlinG. Molica Bisci and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity, 28 (2015), 2247-2264. doi: 10.1088/0951-7715/28/7/2247.

[8]

L. A. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[9]

A. CapellaJ. DávilaL. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384. doi: 10.1080/03605302.2011.562954.

[10]

R. CarmonaW. 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.

[11]

D. G. Costa and C. A. Magalhaes, Variational elliptic problems which are nonquadratic at infinity, Nonlinear Anal., 23 (1994), 1401-1412. doi: 10.1016/0362-546X(94)90135-X.

[12]

M. Dabkowski, Eventual regularity of the solutions to the supercritical dissipative quasigeostrophic equation, Geom. Funct. Anal., 21 (2011), 1-13. doi: 10.1007/s00039-011-0108-9.

[13]

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.

[14]

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.

[15]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on ${\mathbb{R}.N}$, Proc. Roy. Soc. Edinburgh Sect. A, 129 (1999), 787-809. doi: 10.1017/S0308210500013147.

[16]

A. KiselevF. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453. doi: 10.1007/s00222-006-0020-3.

[17]

T. KuusiG. Mingione and Y. Sire, Nonlocal self-improving properties, Anal. PDE, 8 (2015), 57-114. doi: 10.2140/apde.2015.8.57.

[18]

E. H. Lieb and M. Loss, Analysis, 2nd edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.

[19]

O. Miyagaki and M. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations, 245 (2008), 3628-3638. doi: 10.1016/j.jde.2008.02.035.

[20]

G. Molica Bisci and M. Ferrara, Subelliptic and parametric equations on Carnot groups, Proc. Amer. Math. Soc., 144 (2016), 3035-3045. doi: 10.1090/proc/12948.

[21]

G. Molica Bisci and V. Rădulescu, A characterization for elliptic problems on fractal sets, Proc. Amer. Math. Soc., 143 (2015), 2959-2968. doi: 10.1090/S0002-9939-2015-12475-6.

[22]

G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems. With a Foreword by Jean Mawhin, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 162 Cambridge, 2016. doi: 10.1017/CBO9781316282397.

[23]

G. Molica Bisci and R. Servadei, A bifurcation result for nonlocal fractional equations, Anal. Appl., 13 (2015), 371-394. doi: 10.1142/S0219530514500067.

[24]

P. Pucci, Geometric description of the mountain pass critical points, Contemporary Mathematicians, Vol. 2, Birkhäuser, Basel, 2014,469-471.

[25]

P. Pucci and V. Rădulescu, The impact of the mountain pass theory in nonlinear analysis: a mathematical survey, Boll. Un. Mat. Ital. B, Ser. Ⅸ, (2010), 543-582.

[26]

P. Pucci and S. Saldi, Critical stationary Kirchhoff equations in ${\mathbb{R}. N}$ involving nonlocal operators, Rev. Mat. Iberoam. , to appear. doi: 10.4171/RMI/879.

[27]

P. Pucci and J. Serrin, A mountain pass theorem, J. Differential Equations, 60 (1985), 142-149. doi: 10.1016/0022-0396(85)90125-1.

[28]

B. Ricceri, On a classical existence theorem for nonlinear elliptic equations, in Esperimental, constructive and nonlinar analysis, M. Théra ed. , CMS Conf. Proc. 27, Canad. Math. Soc. (2000), 275-278.

[29]

B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math. , Special Issue on Fixed point theory with applications in Nonlinear Analysis, 113 (2000), 401-410. doi: 10.1016/S0377-0427(99)00269-1.

[30]

B. Ricceri, Nonlinear eigenvalue problems, in Handbook of Nonconvex Analysis and Applications (D. Y. Gao and D. Motreanu eds. ), International Press, (2010), 543-595.

[31]

B. Ricceri, A new existence and localization theorem for Dirichlet problem, Dynam. Systems Appl., 22 (2013), 317-324.

[32]

L. Roncal and P. R. Stinga, Fractional Laplacian on the torus, Commun. Contemp. Math., 18 (2016), 26 pp. doi: 10.1142/S0219199715500339.

[33]

L. Roncal and P. R. Stinga, Transference of fractional Laplacian regularity, in Special Functions, Partial Differential Equations, and Harmonic Analysis, 203-212, Springer Proc. Math. Stat. , 108, Springer, Cham. 2014. doi: 10.1007/978-3-319-10545-1_14.

[34]

M. Ryznar, Estimate of Green function for relativistic α-stable processes, Potential Analysis, 17 (2002), 1-23. doi: 10.1023/A:1015231913916.

[35]

M. Schechter and W. Zou, Superlinear problems, Pacific J. Math., 214 (2004), 145-160. doi: 10.2140/pjm.2004.214.145.

[36]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.

[37]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.

[38]

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.

[39]

P. R. Stinga and B. Volzone, Fractional semilinear Neumann problems arising from a fractional Keller-Segel model, Calc. Var. Partial Differential Equations, 54 (2015), 1009-1042. doi: 10.1007/s00526-014-0815-9.

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