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

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

Vincenzo Ambrosio

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
$\mathcal{S}_{T}$
 admits a nontrivial solution
$v(x,ξ)$
 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
$m=0$
.
Keywords: Periodic solutions, extension method, Linking Theorem, Cerami condition, superlinear problem.
Mathematics Subject Classification: Primary: 49J35, 35A15, 35R11; Secondary: 35J60, 35S05.
Citation: Vincenzo Ambrosio. Periodic solutions for a superlinear fractional problem without the Ambrosetti-Rabinowitz condition. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[4]

D. Applebaum, Lévy Processes and Stochastic Calculus Cambridge Studies in advanced mathematics, Cambridge, 2004.  Google Scholar

[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.  Google Scholar

[6]

X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, Higher Trascendental Functions McGraw-Hill vol. 1, 2, New York-Toronto-London, 1953.  Google Scholar

[13]

J. Fröhlich, B. L. G. Jonsson and E. Lenzmann, Boson stars as solitary waves, Comm. Math. Phys., 274 (2007), 1-30.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

M. Schechter and W. Zou, Superlinear problems, Pacific J. Math., 214 (2004), 145-160.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

M. Struwe, Variational methods: Application to Nonlinear Partial Differential Equations and Hamiltonian Systems Springer-Verlag, Berlin, 1990.  Google Scholar

[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.  Google Scholar

[26]

A. Zygmund, Trigonometric Series Vol. 1, 2 Cambridge University Press, Cambridge, 2002.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[4]

D. Applebaum, Lévy Processes and Stochastic Calculus Cambridge Studies in advanced mathematics, Cambridge, 2004.  Google Scholar

[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.  Google Scholar

[6]

X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

A. Erdélyi, W. Magnus, F. Oberhettinger and F. Tricomi, Higher Trascendental Functions McGraw-Hill vol. 1, 2, New York-Toronto-London, 1953.  Google Scholar

[13]

J. Fröhlich, B. L. G. Jonsson and E. Lenzmann, Boson stars as solitary waves, Comm. Math. Phys., 274 (2007), 1-30.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

M. Schechter and W. Zou, Superlinear problems, Pacific J. Math., 214 (2004), 145-160.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

M. Struwe, Variational methods: Application to Nonlinear Partial Differential Equations and Hamiltonian Systems Springer-Verlag, Berlin, 1990.  Google Scholar

[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.  Google Scholar

[26]

A. Zygmund, Trigonometric Series Vol. 1, 2 Cambridge University Press, Cambridge, 2002.  Google Scholar

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