November  2014, 13(6): 2359-2376. doi: 10.3934/cpaa.2014.13.2359

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

Received  October 2013 Revised  April 2014 Published  July 2014

We study the concentration phenomenon for solutions of the fractional nonlinear Schrödinger equation, which is nonlocal. We mainly use the Lyapunov-Schmidt reduction method. Precisely, consider the nonlinear equation \begin{eqnarray} (-\varepsilon^2\Delta)^sv+Vv-|v|^{\alpha}v=0\quad\mbox{in}\quad\mathbf R^n, \end{eqnarray} where $n =1, 2, 3$, $\max\{\frac{1}{2}, \frac{n}{4}\}< s < 1$, $1 \leq \alpha < \alpha_*(s,n)$, $V\in C^3_{b}(\mathbf{R}^n)$. Here the exponent $\alpha_*(s,n)=\frac{4s}{n-2s}$ for $0 < s < \frac{n}{2}$ and $\alpha_*(s,n)=\infty$ for $s \geq\frac{n}{2}$. Then for each non-degenerate critical point $z_0$ of $V$, there is a nontrivial solution of equation (1) concentrating to $z_0$ as $\varepsilon\to 0$.
Citation: Guoyuan Chen, Youquan Zheng. Concentration phenomenon for fractional nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2359-2376. doi: 10.3934/cpaa.2014.13.2359
References:
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R. A. Adams, Sobolev Spaces,, Academic Press, (1975).

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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.

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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.

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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.

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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.

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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.

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P. W. Bates, On some nonlocal evolution equations arising in materials science,, in \emph{Nonlinear dynamics and evolution equations}, (2006), 13.

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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.

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W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach,, \emph{Nonlinearity}, 5 (1992), 497.

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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.

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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.

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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.

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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.

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J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics,, \emph{Invent. Math.}, 29 (1975), 39.

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G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics,, Springer-Verlag, (1976).

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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).

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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}., ().

[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}., ().

[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}., ().

[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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