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On uniqueness properties of solutions of the Toda and Kac-van Moerbeke hierarchies
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 |
$T$ |
$\left\{ \begin{array}{*{35}{l}} [{{(-{{\Delta }_{x}}+{{m}^{2}})}^{s}}-{{m}^{2s}}]u=f(x,u) & \text{ in }{{(0,T)}^{N}} \\ u(x+T{{e}_{i}})=u(x) & \text{for all }x\text{ }\in {{\mathbb{R}}^{N}},i=1,\ldots ,N \\\end{array} \right. \tag{1}$ |
$s∈ (0,1)$ |
$N> 2s$ |
$T>0$ |
$m> 0$ |
$f(x,u)$ |
$T$ |
$x$ |
$(-Δ_{x}+m^{2})^{s}$ |
$\mathcal{S}_{T}=(0,T)^{N}× (0,∞)$ |
$\mathcal{S}_{T}$ |
$v(x,ξ)$ |
$T$ |
$x$ |
$m\to 0$ |
$m=0$ |
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. |
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. |
[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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