# American Institute of Mathematical Sciences

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.  Google Scholar [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.  Google Scholar [3] V. Ambrosio, Periodic solutions for a superlinear fractional problem without the AmbrosettiRabinowitz condition, Discrete Contin. Dyn. Syst. , to appear. Google Scholar [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.  Google Scholar [5] R. Bartolo and G. Molica Bisci, Asymptotically linear fractional p-Laplacian equations, Ann. Mat. Pura Appl. , to appear. Google Scholar [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.  Google Scholar [7] Z. Binlin, G. 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.  Google Scholar [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.  Google Scholar [9] A. Capella, J. Dávila, L. 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.  Google Scholar [10] 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] 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 [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.  Google Scholar [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.  Google Scholar [16] A. Kiselev, F. 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.  Google Scholar [17] T. Kuusi, G. Mingione and Y. Sire, Nonlocal self-improving properties, Anal. PDE, 8 (2015), 57-114. doi: 10.2140/apde.2015.8.57.  Google Scholar [18] E. H. Lieb and M. Loss, Analysis, 2nd edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] G. Molica Bisci and R. Servadei, A bifurcation result for nonlocal fractional equations, Anal. Appl., 13 (2015), 371-394. doi: 10.1142/S0219530514500067.  Google Scholar [24] P. Pucci, Geometric description of the mountain pass critical points, Contemporary Mathematicians, Vol. 2, Birkhäuser, Basel, 2014,469-471. Google Scholar [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, Ⅲ (2010), 543-582.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [31] B. Ricceri, A new existence and localization theorem for Dirichlet problem, Dynam. Systems Appl., 22 (2013), 317-324.  Google Scholar [32] L. Roncal and P. R. Stinga, Fractional Laplacian on the torus, Commun. Contemp. Math. , 18 (2016), 26 pp. doi: 10.1142/S0219199715500339.  Google Scholar [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.  Google Scholar [34] M. Ryznar, Estimate of Green function for relativistic α-stable processes, Potential Analysis, 17 (2002), 1-23. doi: 10.1023/A:1015231913916.  Google Scholar [35] M. Schechter and W. Zou, Superlinear problems, Pacific J. Math., 214 (2004), 145-160. doi: 10.2140/pjm.2004.214.145.  Google Scholar [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.  Google Scholar [37] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [3] V. Ambrosio, Periodic solutions for a superlinear fractional problem without the AmbrosettiRabinowitz condition, Discrete Contin. Dyn. Syst. , to appear. Google Scholar [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.  Google Scholar [5] R. Bartolo and G. Molica Bisci, Asymptotically linear fractional p-Laplacian equations, Ann. Mat. Pura Appl. , to appear. Google Scholar [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.  Google Scholar [7] Z. Binlin, G. 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.  Google Scholar [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.  Google Scholar [9] A. Capella, J. Dávila, L. 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.  Google Scholar [10] 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] 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 [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.  Google Scholar [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.  Google Scholar [16] A. Kiselev, F. 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.  Google Scholar [17] T. Kuusi, G. Mingione and Y. Sire, Nonlocal self-improving properties, Anal. PDE, 8 (2015), 57-114. doi: 10.2140/apde.2015.8.57.  Google Scholar [18] E. H. Lieb and M. Loss, Analysis, 2nd edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] G. Molica Bisci and R. Servadei, A bifurcation result for nonlocal fractional equations, Anal. Appl., 13 (2015), 371-394. doi: 10.1142/S0219530514500067.  Google Scholar [24] P. Pucci, Geometric description of the mountain pass critical points, Contemporary Mathematicians, Vol. 2, Birkhäuser, Basel, 2014,469-471. Google Scholar [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, Ⅲ (2010), 543-582.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [31] B. Ricceri, A new existence and localization theorem for Dirichlet problem, Dynam. Systems Appl., 22 (2013), 317-324.  Google Scholar [32] L. Roncal and P. R. Stinga, Fractional Laplacian on the torus, Commun. Contemp. Math. , 18 (2016), 26 pp. doi: 10.1142/S0219199715500339.  Google Scholar [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.  Google Scholar [34] M. Ryznar, Estimate of Green function for relativistic α-stable processes, Potential Analysis, 17 (2002), 1-23. doi: 10.1023/A:1015231913916.  Google Scholar [35] M. Schechter and W. Zou, Superlinear problems, Pacific J. Math., 214 (2004), 145-160. doi: 10.2140/pjm.2004.214.145.  Google Scholar [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.  Google Scholar [37] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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