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Concentration phenomenon for fractional nonlinear Schrödinger equations
1. | School of Mathematics and Statistics, Zhejiang University of Finance and Economics, Hangzhou 310018, Zhejiang, China |
2. | School of Science, Tianjin University, Tianjin 300072, China |
References:
[1] |
R. A. Adams, Sobolev Spaces,, Academic Press, (1975).
|
[2] |
G. Alberti, G. Bouchitté and P. Seppecher, Phase transition with the line-tension effect,, \emph{Arch. Rational Mech. Anal.}, 144 (1998), 1.
doi: 10.1007/s002050050111. |
[3] |
A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials,, \emph{Arch. Ration. Mech. Anal.}, 159 (2001), 253.
doi: 10.1007/s002050100152. |
[4] |
A. Ambrosetti, A. Malchiodi and W.-M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. I,, \emph{Comm. Math. Phys.}, 235 (2003), 427.
doi: 10.1007/s00220-003-0811-y. |
[5] |
C. J. Amick and J. F. Toland, Uniqueness and related analytic properties for the Benjamin-Ono equation--a nonlinear Neumann problem in the plane,, \emph{Acta Math.}, 167 (1991), 107.
doi: 10.1007/BF02392447. |
[6] |
A. Bahri and Y. Y. Li, On a min-max procedure for the existence of a positive solution for certain scalar field equations in $R^N$,, \emph{Rev. Mat. Iberoamericana}, 6 (1990), 1.
doi: 10.4171/RMI/92. |
[7] |
A. Bahri and P.-L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 14 (1997), 365.
doi: 10.1016/S0294-1449(97)80142-4. |
[8] |
P. W. Bates, On some nonlocal evolution equations arising in materials science,, in \emph{Nonlinear dynamics and evolution equations}, (2006), 13.
|
[9] |
P. Biler, G. Karch and W. A. Woyczyński, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 18 (2001), 613.
doi: 10.1016/S0294-1449(01)00080-4. |
[10] |
J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations,, \emph{Arch. Ration. Mech. Anal.}, 165 (2002), 295.
doi: 10.1007/s00205-002-0225-6. |
[11] |
X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions,, \emph{Comm. Pure Appl. Math.}, 58 (2005), 1678.
doi: 10.1002/cpa.20093. |
[12] |
L. Caffarelli, A. Mellet and Y. Sire, Traveling waves for a boundary reaction-diffusion equation,, \emph{Adv. Math.}, 230 (2012), 433.
doi: 10.1016/j.aim.2012.01.020. |
[13] |
L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces,, \emph{Comm. Pure Appl. Math.}, 63 (2010), 1111.
doi: 10.1002/cpa.20331. |
[14] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, \emph{Comm. Partial Differential Equations}, 32 (2007), 1245.
doi: 10.1080/03605300600987306. |
[15] |
L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces,, \emph{Calc. Var. Partial Differential Equations}, 41 (2011), 203.
doi: 10.1007/s00526-010-0359-6. |
[16] |
K.-C. Chang, Infinite-dimensional Morse Theory and Multiple Solution Problems,, Progress in Nonlinear Differential Equations and their Applications, (1993).
doi: 10.1007/978-1-4612-0385-8. |
[17] |
S.-Y. A. Chang and M. d. M. González, Fractional Laplacian in conformal geometry,, \emph{Adv. Math.}, 226 (2011), 1410.
doi: 10.1016/j.aim.2010.07.016. |
[18] |
G. Chen and Y. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations,, \arxiv{1305.4426}., (). Google Scholar |
[19] |
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, vol. 93 of Applied Mathematical Sciences, (1998).
doi: 10.1007/978-3-662-03537-5. |
[20] |
R. Cont and P. Tankov, Financial Modelling with Jump Processes,, Chapman & Hall/CRC Financial Mathematics Series, (2004).
|
[21] |
D. Cordoba, Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation,, \emph{Ann. of Math.}, 148 (1998), 1135.
doi: 10.2307/121037. |
[22] |
W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach,, \emph{Nonlinearity}, 5 (1992), 497.
|
[23] |
J. Dávila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation,, \emph{J. Differential Equations}, 256 (2014), 858.
doi: 10.1016/j.jde.2013.10.006. |
[24] |
R. de la Llave and E. Valdinoci, Symmetry for a Dirichlet-Neumann problem arising in water waves,, \emph{Math. Res. Lett.}, 16 (2009), 909.
doi: 10.4310/MRL.2009.v16.n5.a13. |
[25] |
M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains},, \emph{Calc. Var. Partial Differential Equations}, 4 (1996), 121.
doi: 10.1007/BF01189950. |
[26] |
M. del Pino, M. Kowalczyk and J.-C. Wei, Concentration on curves for nonlinear Schrödinger equations,, \emph{Comm. Pure Appl. Math.}, 60 (2007), 113.
doi: 10.1002/cpa.20135. |
[27] |
E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, \emph{Bull. Sci. Math.}, 136 (2012), 521.
doi: 10.1016/j.bulsci.2011.12.004. |
[28] |
J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics,, \emph{Invent. Math.}, 29 (1975), 39.
|
[29] |
G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics,, Springer-Verlag, (1976).
|
[30] |
A. Farina and E. Valdinoci, Rigidity results for elliptic PDEs with uniform limits: an abstract framework with applications,, \emph{Indiana Univ. Math. J.}, 60 (2011), 121.
doi: 10.1512/iumj.2011.60.4433. |
[31] |
C. Fefferman and R. de la Llave, Relativistic stability of matter. I,, \emph{Rev. Mat. Iberoamericana}, 2 (1986), 119.
doi: 10.4171/RMI/30. |
[32] |
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals,, Emended edition, (2010).
|
[33] |
A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential,, \emph{J. Funct. Anal.}, 69 (1986), 397.
doi: 10.1016/0022-1236(86)90096-0. |
[34] |
R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian,, \arxiv{1302.2652}., (). Google Scholar |
[35] |
R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\BbbR$,, \emph{Acta Math.}, 210 (2013), 261.
doi: 10.1007/s11511-013-0095-9. |
[36] |
G. K. Gächter and M. J. Grote, Dirichlet-to-Neumann map for three-dimensional elastic waves,, \emph{Wave Motion}, 37 (2003), 293.
doi: 10.1016/S0165-2125(02)00091-4. |
[37] |
A. Garroni and G. Palatucci, A singular perturbation result with a fractional norm,, in \emph{Variational problems in materials science}, (2006), 111.
doi: 10.1007/3-7643-7565-5_8. |
[38] |
M. d. M. González and R. Monneau, Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one,, \emph{Discrete Contin. Dyn. Syst.}, 32 (2012), 1255.
|
[39] |
M. Grossi, On the number of single-peak solutions of the nonlinear Schrödinger equation,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 19 (2002), 261.
doi: 10.1016/S0294-1449(01)00089-0. |
[40] |
M. J. Grote and C. Kirsch, Dirichlet-to-Neumann boundary conditions for multiple scattering problems,, \emph{J. Comput. Phys.}, 201 (2004), 630.
doi: 10.1016/j.jcp.2004.06.012. |
[41] |
M. J. W. Hall and M. Reginatto, Schrödinger equation from an exact uncertainty principle,, \emph{J. Phys. A}, 35 (2002), 3289.
doi: 10.1088/0305-4470/35/14/310. |
[42] |
B. Hu and D. P. Nicholls, Analyticity of Dirichlet-Neumann operators on Hölder and Lipschitz domains,, \emph{SIAM J. Math. Anal.}, 37 (2005), 302.
doi: 10.1137/S0036141004444810. |
[43] |
C. E. Kenig, Y. Martel and L. Robbiano, Local well-posedness and blow-up in the energy space for a class of $L^2$ critical dispersion generalized Benjamin-Ono equations,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 28 (2011), 853.
doi: 10.1016/j.anihpc.2011.06.005. |
[44] |
M. Kurzke, A nonlocal singular perturbation problem with periodic well potential,, \emph{ESAIM Control Optim. Calc. Var.}, 12 (2006), 52.
doi: 10.1051/cocv:2005037. |
[45] |
M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$,, \emph{Arch. Rational Mech. Anal.}, 105 (1989), 243.
doi: 10.1007/BF00251502. |
[46] |
N. Laskin, Fractional Schrödinger equation,, \emph{Phys. Rev. E}, 66 (2002).
doi: 10.1103/PhysRevE.66.056108. |
[47] |
Y. Li, On a singularly perturbed elliptic equation,, \emph{Adv. Differential Equations}, 2 (1997), 955.
|
[48] |
A. J. Majda and E. G. Tabak, A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow,, \emph{Phys. D}, 98 (1996), 515.
doi: 10.1016/0167-2789(96)00114-5. |
[49] |
B. B. Mandelbrot, The Fractal Geometry of Nature,, W. H. Freeman and Co., (1982).
|
[50] |
R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach,, \emph{Phys. Rep.}, 339 (2000).
doi: 10.1016/S0370-1573(00)00070-3. |
[51] |
E. Milakis and L. Silvestre, Regularity for the nonlinear Signorini problem,, \emph{Adv. Math.}, 217 (2008), 1301.
doi: 10.1016/j.aim.2007.08.009. |
[52] |
E. Nelson, Quantum Fluctuations,, Princeton Series in Physics, (1985).
|
[53] |
D. P. Nicholls and M. Taber, Joint analyticity and analytic continuation of Dirichlet-Neumann operators on doubly perturbed domains,, \emph{J. Math. Fluid Mech.}, 10 (2008), 238.
doi: 10.1007/s00021-006-0231-9. |
[54] |
O. Savin and E. Valdinoci, Elliptic PDEs with fibered nonlinearities,, \emph{J. Geom. Anal.}, 19 (2009), 420.
doi: 10.1007/s12220-008-9064-5. |
[55] |
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\Bbb R^N$,, \emph{J. Math. Phys.}, 54 (2013).
doi: 10.1063/1.4793990. |
[56] |
M. A. Shubin, Pseudodifferential Operators and Spectral Theory,, 2nd edition, (2001).
doi: 10.1007/978-3-642-56579-3. |
[57] |
L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, \emph{Comm. Pure Appl. Math.}, 60 (2007), 67.
doi: 10.1002/cpa.20153. |
[58] |
Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result,, \emph{J. Funct. Anal.}, 256 (2009), 1842.
doi: 10.1016/j.jfa.2009.01.020. |
[59] |
J. J. Stoker, Water waves: The Mathematical Theory with Applications,, Pure and Applied Mathematics, (1957).
|
[60] |
J. F. Toland, The Peierls-Nabarro and Benjamin-Ono equations,, \emph{J. Funct. Anal.}, 145 (1997), 136.
doi: 10.1006/jfan.1996.3016. |
[61] |
M. I. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation,, \emph{Comm. Partial Differential Equations}, 12 (1987), 1133.
doi: 10.1080/03605308708820522. |
[62] |
G. B. Whitham, Linear and Nonlinear Waves,, Wiley-Interscience [John Wiley & Sons], (1974).
|
show all references
References:
[1] |
R. A. Adams, Sobolev Spaces,, Academic Press, (1975).
|
[2] |
G. Alberti, G. Bouchitté and P. Seppecher, Phase transition with the line-tension effect,, \emph{Arch. Rational Mech. Anal.}, 144 (1998), 1.
doi: 10.1007/s002050050111. |
[3] |
A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials,, \emph{Arch. Ration. Mech. Anal.}, 159 (2001), 253.
doi: 10.1007/s002050100152. |
[4] |
A. Ambrosetti, A. Malchiodi and W.-M. Ni, Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. I,, \emph{Comm. Math. Phys.}, 235 (2003), 427.
doi: 10.1007/s00220-003-0811-y. |
[5] |
C. J. Amick and J. F. Toland, Uniqueness and related analytic properties for the Benjamin-Ono equation--a nonlinear Neumann problem in the plane,, \emph{Acta Math.}, 167 (1991), 107.
doi: 10.1007/BF02392447. |
[6] |
A. Bahri and Y. Y. Li, On a min-max procedure for the existence of a positive solution for certain scalar field equations in $R^N$,, \emph{Rev. Mat. Iberoamericana}, 6 (1990), 1.
doi: 10.4171/RMI/92. |
[7] |
A. Bahri and P.-L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 14 (1997), 365.
doi: 10.1016/S0294-1449(97)80142-4. |
[8] |
P. W. Bates, On some nonlocal evolution equations arising in materials science,, in \emph{Nonlinear dynamics and evolution equations}, (2006), 13.
|
[9] |
P. Biler, G. Karch and W. A. Woyczyński, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 18 (2001), 613.
doi: 10.1016/S0294-1449(01)00080-4. |
[10] |
J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations,, \emph{Arch. Ration. Mech. Anal.}, 165 (2002), 295.
doi: 10.1007/s00205-002-0225-6. |
[11] |
X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions,, \emph{Comm. Pure Appl. Math.}, 58 (2005), 1678.
doi: 10.1002/cpa.20093. |
[12] |
L. Caffarelli, A. Mellet and Y. Sire, Traveling waves for a boundary reaction-diffusion equation,, \emph{Adv. Math.}, 230 (2012), 433.
doi: 10.1016/j.aim.2012.01.020. |
[13] |
L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces,, \emph{Comm. Pure Appl. Math.}, 63 (2010), 1111.
doi: 10.1002/cpa.20331. |
[14] |
L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, \emph{Comm. Partial Differential Equations}, 32 (2007), 1245.
doi: 10.1080/03605300600987306. |
[15] |
L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces,, \emph{Calc. Var. Partial Differential Equations}, 41 (2011), 203.
doi: 10.1007/s00526-010-0359-6. |
[16] |
K.-C. Chang, Infinite-dimensional Morse Theory and Multiple Solution Problems,, Progress in Nonlinear Differential Equations and their Applications, (1993).
doi: 10.1007/978-1-4612-0385-8. |
[17] |
S.-Y. A. Chang and M. d. M. González, Fractional Laplacian in conformal geometry,, \emph{Adv. Math.}, 226 (2011), 1410.
doi: 10.1016/j.aim.2010.07.016. |
[18] |
G. Chen and Y. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations,, \arxiv{1305.4426}., (). Google Scholar |
[19] |
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory,, vol. 93 of Applied Mathematical Sciences, (1998).
doi: 10.1007/978-3-662-03537-5. |
[20] |
R. Cont and P. Tankov, Financial Modelling with Jump Processes,, Chapman & Hall/CRC Financial Mathematics Series, (2004).
|
[21] |
D. Cordoba, Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation,, \emph{Ann. of Math.}, 148 (1998), 1135.
doi: 10.2307/121037. |
[22] |
W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach,, \emph{Nonlinearity}, 5 (1992), 497.
|
[23] |
J. Dávila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation,, \emph{J. Differential Equations}, 256 (2014), 858.
doi: 10.1016/j.jde.2013.10.006. |
[24] |
R. de la Llave and E. Valdinoci, Symmetry for a Dirichlet-Neumann problem arising in water waves,, \emph{Math. Res. Lett.}, 16 (2009), 909.
doi: 10.4310/MRL.2009.v16.n5.a13. |
[25] |
M. del Pino and P. L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains},, \emph{Calc. Var. Partial Differential Equations}, 4 (1996), 121.
doi: 10.1007/BF01189950. |
[26] |
M. del Pino, M. Kowalczyk and J.-C. Wei, Concentration on curves for nonlinear Schrödinger equations,, \emph{Comm. Pure Appl. Math.}, 60 (2007), 113.
doi: 10.1002/cpa.20135. |
[27] |
E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, \emph{Bull. Sci. Math.}, 136 (2012), 521.
doi: 10.1016/j.bulsci.2011.12.004. |
[28] |
J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics,, \emph{Invent. Math.}, 29 (1975), 39.
|
[29] |
G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics,, Springer-Verlag, (1976).
|
[30] |
A. Farina and E. Valdinoci, Rigidity results for elliptic PDEs with uniform limits: an abstract framework with applications,, \emph{Indiana Univ. Math. J.}, 60 (2011), 121.
doi: 10.1512/iumj.2011.60.4433. |
[31] |
C. Fefferman and R. de la Llave, Relativistic stability of matter. I,, \emph{Rev. Mat. Iberoamericana}, 2 (1986), 119.
doi: 10.4171/RMI/30. |
[32] |
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals,, Emended edition, (2010).
|
[33] |
A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential,, \emph{J. Funct. Anal.}, 69 (1986), 397.
doi: 10.1016/0022-1236(86)90096-0. |
[34] |
R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian,, \arxiv{1302.2652}., (). Google Scholar |
[35] |
R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\BbbR$,, \emph{Acta Math.}, 210 (2013), 261.
doi: 10.1007/s11511-013-0095-9. |
[36] |
G. K. Gächter and M. J. Grote, Dirichlet-to-Neumann map for three-dimensional elastic waves,, \emph{Wave Motion}, 37 (2003), 293.
doi: 10.1016/S0165-2125(02)00091-4. |
[37] |
A. Garroni and G. Palatucci, A singular perturbation result with a fractional norm,, in \emph{Variational problems in materials science}, (2006), 111.
doi: 10.1007/3-7643-7565-5_8. |
[38] |
M. d. M. González and R. Monneau, Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one,, \emph{Discrete Contin. Dyn. Syst.}, 32 (2012), 1255.
|
[39] |
M. Grossi, On the number of single-peak solutions of the nonlinear Schrödinger equation,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 19 (2002), 261.
doi: 10.1016/S0294-1449(01)00089-0. |
[40] |
M. J. Grote and C. Kirsch, Dirichlet-to-Neumann boundary conditions for multiple scattering problems,, \emph{J. Comput. Phys.}, 201 (2004), 630.
doi: 10.1016/j.jcp.2004.06.012. |
[41] |
M. J. W. Hall and M. Reginatto, Schrödinger equation from an exact uncertainty principle,, \emph{J. Phys. A}, 35 (2002), 3289.
doi: 10.1088/0305-4470/35/14/310. |
[42] |
B. Hu and D. P. Nicholls, Analyticity of Dirichlet-Neumann operators on Hölder and Lipschitz domains,, \emph{SIAM J. Math. Anal.}, 37 (2005), 302.
doi: 10.1137/S0036141004444810. |
[43] |
C. E. Kenig, Y. Martel and L. Robbiano, Local well-posedness and blow-up in the energy space for a class of $L^2$ critical dispersion generalized Benjamin-Ono equations,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 28 (2011), 853.
doi: 10.1016/j.anihpc.2011.06.005. |
[44] |
M. Kurzke, A nonlocal singular perturbation problem with periodic well potential,, \emph{ESAIM Control Optim. Calc. Var.}, 12 (2006), 52.
doi: 10.1051/cocv:2005037. |
[45] |
M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$,, \emph{Arch. Rational Mech. Anal.}, 105 (1989), 243.
doi: 10.1007/BF00251502. |
[46] |
N. Laskin, Fractional Schrödinger equation,, \emph{Phys. Rev. E}, 66 (2002).
doi: 10.1103/PhysRevE.66.056108. |
[47] |
Y. Li, On a singularly perturbed elliptic equation,, \emph{Adv. Differential Equations}, 2 (1997), 955.
|
[48] |
A. J. Majda and E. G. Tabak, A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow,, \emph{Phys. D}, 98 (1996), 515.
doi: 10.1016/0167-2789(96)00114-5. |
[49] |
B. B. Mandelbrot, The Fractal Geometry of Nature,, W. H. Freeman and Co., (1982).
|
[50] |
R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach,, \emph{Phys. Rep.}, 339 (2000).
doi: 10.1016/S0370-1573(00)00070-3. |
[51] |
E. Milakis and L. Silvestre, Regularity for the nonlinear Signorini problem,, \emph{Adv. Math.}, 217 (2008), 1301.
doi: 10.1016/j.aim.2007.08.009. |
[52] |
E. Nelson, Quantum Fluctuations,, Princeton Series in Physics, (1985).
|
[53] |
D. P. Nicholls and M. Taber, Joint analyticity and analytic continuation of Dirichlet-Neumann operators on doubly perturbed domains,, \emph{J. Math. Fluid Mech.}, 10 (2008), 238.
doi: 10.1007/s00021-006-0231-9. |
[54] |
O. Savin and E. Valdinoci, Elliptic PDEs with fibered nonlinearities,, \emph{J. Geom. Anal.}, 19 (2009), 420.
doi: 10.1007/s12220-008-9064-5. |
[55] |
S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\Bbb R^N$,, \emph{J. Math. Phys.}, 54 (2013).
doi: 10.1063/1.4793990. |
[56] |
M. A. Shubin, Pseudodifferential Operators and Spectral Theory,, 2nd edition, (2001).
doi: 10.1007/978-3-642-56579-3. |
[57] |
L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator,, \emph{Comm. Pure Appl. Math.}, 60 (2007), 67.
doi: 10.1002/cpa.20153. |
[58] |
Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result,, \emph{J. Funct. Anal.}, 256 (2009), 1842.
doi: 10.1016/j.jfa.2009.01.020. |
[59] |
J. J. Stoker, Water waves: The Mathematical Theory with Applications,, Pure and Applied Mathematics, (1957).
|
[60] |
J. F. Toland, The Peierls-Nabarro and Benjamin-Ono equations,, \emph{J. Funct. Anal.}, 145 (1997), 136.
doi: 10.1006/jfan.1996.3016. |
[61] |
M. I. Weinstein, Existence and dynamic stability of solitary wave solutions of equations arising in long wave propagation,, \emph{Comm. Partial Differential Equations}, 12 (1987), 1133.
doi: 10.1080/03605308708820522. |
[62] |
G. B. Whitham, Linear and Nonlinear Waves,, Wiley-Interscience [John Wiley & Sons], (1974).
|
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