# American Institute of Mathematical Sciences

May  2017, 37(5): 2265-2284. doi: 10.3934/dcds.2017099

## Periodic solutions for a superlinear fractional problem without the Ambrosetti-Rabinowitz condition

 Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università degli Studi di Napoli "Federico Ⅱ", via Cinthia 1, 80126 Napoli, Italy

Received  February 2016 Revised  January 2017 Published  February 2017

The purpose of this paper is to study
 $T$
-periodic solutions to
 $\left\{ \begin{array}{*{35}{l}} [{{(-{{\Delta }_{x}}+{{m}^{2}})}^{s}}-{{m}^{2s}}]u=f(x,u) & \text{ in }{{(0,T)}^{N}} \\ u(x+T{{e}_{i}})=u(x) & \text{for all }x\text{ }\in {{\mathbb{R}}^{N}},i=1,\ldots ,N \\\end{array} \right. \tag{1}$
where
 $s∈ (0,1)$
,
 $N> 2s$
,
 $T>0$
,
 $m> 0$
and
 $f(x,u)$
is a continuous function,
 $T$
-periodic in
 $x$
and satisfying a suitable growth assumption weaker than the Ambrosetti-Rabinowitz condition.
The nonlocal operator
 $(-Δ_{x}+m^{2})^{s}$
can be realized as the Dirichlet to Neumann map for a degenerate elliptic problem posed on the half-cylinder
 $\mathcal{S}_{T}=(0,T)^{N}× (0,∞)$
. By using a variant of the Linking Theorem, we show that the extended problem in
 $\mathcal{S}_{T}$
 $v(x,ξ)$
which is
 $T$
-periodic in
 $x$
. Moreover, by a procedure of limit as
 $m\to 0$
, we prove the existence of a nontrivial solution to (1) with
 $m=0$
.
Citation: Vincenzo Ambrosio. Periodic solutions for a superlinear fractional problem without the Ambrosetti-Rabinowitz condition. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2265-2284. doi: 10.3934/dcds.2017099
##### References:
 [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7. [2] 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. [3] V. Ambrosio, Periodic solutions for the non-local operator pseudo-relativistic $(-Δ+m^{2})^{s}-m^{2s}$ with $m≥q 0$, Topol. Methods Nonlinear Anal. (2016), DOI: 10.12775/TMNA.2016.063. [4] D. Applebaum, Lévy Processes and Stochastic Calculus Cambridge Studies in advanced mathematics, Cambridge, 2004. [5] P. Biler, G. Karch and W. A. Woyczynski, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 613-637.  doi: 10.1016/S0294-1449(01)00080-4. [6] X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732. [7] 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. [8] L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240.  doi: 10.1007/s00526-010-0359-6. [9] G. Cerami, Un criterio di esistenza per i punti critici su variet{{á}} illimitate, Rend. Acad. Sci. Let. Ist. Lombardo, 112 (1978), 332-336. [10] R. Cont and P. Tankov, Financial Modelling with Jump Processes Chapman and Hall/CRC Financ. Math. Ser. , Chapman and Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203485217. [11] D. G. Costa and C. A. Magalhães, Variational elliptic problems which are nonquadratic at infinity, Nonlinear Anal., 23 (1994), 1401-1412.  doi: 10.1016/0362-546X(94)90135-X. [12] A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, Higher Trascendental Functions McGraw-Hill vol. 1, 2, New York-Toronto-London, 1953. [13] J. Fröhlich, B. L. G. Jonsson and E. Lenzmann, Boson stars as solitary waves, Comm. Math. Phys., 274 (2007), 1-30. [14] 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. [15] G. Li and C. Wang, The existence of a nontrivial solution to a nonlinear elliptic problem of Linking type without the Ambrosetti-Rabinowitz condition, Ann. Acad. Sci. Fenn. Math., 36 (2011), 461-480.  doi: 10.5186/aasfm.2011.3627. [16] 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. [17] S. B. Liu, On superlinear problems without Ambrosetti-Rabinowitz condition, Nonlinear Anal., 73 (2010), 788-795.  doi: 10.1016/j.na.2010.04.016. [18] 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. [19] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations CBMS Regional Conference Series in Mathematics, 1986. doi: 10.1090/cbms/065. [20] M. Schechter and W. Zou, Superlinear problems, Pacific J. Math., 214 (2004), 145-160. [21] L. Silvestre, Regularity of the obstacle problem for a fractional power of the {L}aplace operator, Comm. Pure Appl. Math., 60 (2006), 67-112.  doi: 10.1002/cpa.20153. [22] Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.  doi: 10.1016/j.jfa.2009.01.020. [23] J. J. Stoker, Water Waves: The Mathematical Theory with Applications Pure Appl. Math. , vol. IV, Interscience Publishers, Inc. , New York, 1957. doi: 10.1002/9781118033159. [24] M. Struwe, Variational methods: Application to Nonlinear Partial Differential Equations and Hamiltonian Systems Springer-Verlag, Berlin, 1990. [25] M. Willem, Minimax Theorems Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc. , Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. [26] A. Zygmund, Trigonometric Series Vol. 1, 2 Cambridge University Press, Cambridge, 2002.

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##### References:
 [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7. [2] 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. [3] V. Ambrosio, Periodic solutions for the non-local operator pseudo-relativistic $(-Δ+m^{2})^{s}-m^{2s}$ with $m≥q 0$, Topol. Methods Nonlinear Anal. (2016), DOI: 10.12775/TMNA.2016.063. [4] D. Applebaum, Lévy Processes and Stochastic Calculus Cambridge Studies in advanced mathematics, Cambridge, 2004. [5] P. Biler, G. Karch and W. A. Woyczynski, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 613-637.  doi: 10.1016/S0294-1449(01)00080-4. [6] X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732. [7] 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. [8] L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240.  doi: 10.1007/s00526-010-0359-6. [9] G. Cerami, Un criterio di esistenza per i punti critici su variet{{á}} illimitate, Rend. Acad. Sci. Let. Ist. Lombardo, 112 (1978), 332-336. [10] R. Cont and P. Tankov, Financial Modelling with Jump Processes Chapman and Hall/CRC Financ. Math. Ser. , Chapman and Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203485217. [11] D. G. Costa and C. A. Magalhães, Variational elliptic problems which are nonquadratic at infinity, Nonlinear Anal., 23 (1994), 1401-1412.  doi: 10.1016/0362-546X(94)90135-X. [12] A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, Higher Trascendental Functions McGraw-Hill vol. 1, 2, New York-Toronto-London, 1953. [13] J. Fröhlich, B. L. G. Jonsson and E. Lenzmann, Boson stars as solitary waves, Comm. Math. Phys., 274 (2007), 1-30. [14] 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. [15] G. Li and C. Wang, The existence of a nontrivial solution to a nonlinear elliptic problem of Linking type without the Ambrosetti-Rabinowitz condition, Ann. Acad. Sci. Fenn. Math., 36 (2011), 461-480.  doi: 10.5186/aasfm.2011.3627. [16] 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. [17] S. B. Liu, On superlinear problems without Ambrosetti-Rabinowitz condition, Nonlinear Anal., 73 (2010), 788-795.  doi: 10.1016/j.na.2010.04.016. [18] 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. [19] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations CBMS Regional Conference Series in Mathematics, 1986. doi: 10.1090/cbms/065. [20] M. Schechter and W. Zou, Superlinear problems, Pacific J. Math., 214 (2004), 145-160. [21] L. Silvestre, Regularity of the obstacle problem for a fractional power of the {L}aplace operator, Comm. Pure Appl. Math., 60 (2006), 67-112.  doi: 10.1002/cpa.20153. [22] Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.  doi: 10.1016/j.jfa.2009.01.020. [23] J. J. Stoker, Water Waves: The Mathematical Theory with Applications Pure Appl. Math. , vol. IV, Interscience Publishers, Inc. , New York, 1957. doi: 10.1002/9781118033159. [24] M. Struwe, Variational methods: Application to Nonlinear Partial Differential Equations and Hamiltonian Systems Springer-Verlag, Berlin, 1990. [25] M. Willem, Minimax Theorems Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc. , Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1. [26] A. Zygmund, Trigonometric Series Vol. 1, 2 Cambridge University Press, Cambridge, 2002.
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