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Concentration phenomenon for fractional nonlinear Schrödinger equations

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  • 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 $\varepsilon\to 0$.
    Mathematics Subject Classification: Primary: 35J10, 35J30; Secondary: 35J35, 35J61, 35J91.

    Citation:

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  • [1]

    R. A. Adams, Sobolev Spaces, Academic Press, New York-London, 1975, Pure and Applied Mathematics, Vol. 65.

    [2]

    G. Alberti, G. Bouchitté and P. Seppecher, Phase transition with the line-tension effect, Arch. Rational Mech. Anal., 144 (1998), 1-46.doi: 10.1007/s002050050111.

    [3]

    A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Anal., 159 (2001), 253-271,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, Comm. Math. Phys., 235 (2003), 427-466.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, Acta Math., 167 (1991), 107-126.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 $R^N$, Rev. Mat. Iberoamericana, 6 (1990), 1-15.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, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 365-413.doi: 10.1016/S0294-1449(97)80142-4.

    [8]

    P. W. Bates, On some nonlocal evolution equations arising in materials science, in Nonlinear dynamics and evolution equations, vol. 48 of Fields Inst. Commun. Amer. Math. Soc., Providence, RI, 2006, 13-52.

    [9]

    P. Biler, G. Karch and W. A. Woyczyński, 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.

    [10]

    J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 165 (2002), 295-316.doi: 10.1007/s00205-002-0225-6.

    [11]

    X. Cabré and J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732.doi: 10.1002/cpa.20093.

    [12]

    L. Caffarelli, A. Mellet and Y. Sire, Traveling waves for a boundary reaction-diffusion equation, Adv. Math., 230 (2012), 433-457,doi: 10.1016/j.aim.2012.01.020.

    [13]

    L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.doi: 10.1002/cpa.20331.

    [14]

    L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.doi: 10.1080/03605300600987306.

    [15]

    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.

    [16]

    K.-C. Chang, Infinite-dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and their Applications, 6, Birkhäuser Boston Inc., Boston, MA, 1993.doi: 10.1007/978-1-4612-0385-8.

    [17]

    S.-Y. A. Chang and M. d. M. González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432.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, 2nd edition, Springer-Verlag, Berlin, 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, Chapman & Hall/CRC, Boca Raton, FL, 2004.

    [21]

    D. Cordoba, Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation, Ann. of Math., 148 (1998), 1135-1152.doi: 10.2307/121037.

    [22]

    W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity, 5 (1992), 497-522.

    [23]

    J. Dávila, M. del Pino and J. Wei, Concentrating standing waves for the fractional nonlinear Schrödinger equation, J. Differential Equations, 256 (2014), 858-892.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, Math. Res. Lett., 16 (2009), 909-918.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}, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.doi: 10.1007/BF01189950.

    [26]

    M. del Pino, M. Kowalczyk and J.-C. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146.doi: 10.1002/cpa.20135.

    [27]

    E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.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, Invent. Math., 29 (1975), 39-79.

    [29]

    G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976, Translated from the French by C. W. John, Grundlehren der Mathematischen Wissenschaften, 219.

    [30]

    A. Farina and E. Valdinoci, Rigidity results for elliptic PDEs with uniform limits: an abstract framework with applications, Indiana Univ. Math. J., 60 (2011), 121-141.doi: 10.1512/iumj.2011.60.4433.

    [31]

    C. Fefferman and R. de la Llave, Relativistic stability of matter. I, Rev. Mat. Iberoamericana, 2 (1986), 119-213.doi: 10.4171/RMI/30.

    [32]

    R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, Emended edition, Dover Publications Inc., Mineola, NY, 2010, Emended and with a preface by Daniel F. Styer.

    [33]

    A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.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 $\BbbR$, Acta Math., 210 (2013), 261-318.doi: 10.1007/s11511-013-0095-9.

    [36]

    G. K. Gächter and M. J. Grote, Dirichlet-to-Neumann map for three-dimensional elastic waves, Wave Motion, 37 (2003), 293-311.doi: 10.1016/S0165-2125(02)00091-4.

    [37]

    A. Garroni and G. Palatucci, A singular perturbation result with a fractional norm, in Variational problems in materials science, vol. 68 of Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 2006, 111-126.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, Discrete Contin. Dyn. Syst., 32 (2012), 1255-1286.

    [39]

    M. Grossi, On the number of single-peak solutions of the nonlinear Schrödinger equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 261-280.doi: 10.1016/S0294-1449(01)00089-0.

    [40]

    M. J. Grote and C. Kirsch, Dirichlet-to-Neumann boundary conditions for multiple scattering problems, J. Comput. Phys., 201 (2004), 630-650.doi: 10.1016/j.jcp.2004.06.012.

    [41]

    M. J. W. Hall and M. Reginatto, Schrödinger equation from an exact uncertainty principle, J. Phys. A, 35 (2002), 3289-3303.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, SIAM J. Math. Anal., 37 (2005), 302-320 (electronic).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, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 853-887.doi: 10.1016/j.anihpc.2011.06.005.

    [44]

    M. Kurzke, A nonlocal singular perturbation problem with periodic well potential, ESAIM Control Optim. Calc. Var., 12 (2006), 52-63 (electronic).doi: 10.1051/cocv:2005037.

    [45]

    M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.doi: 10.1007/BF00251502.

    [46]

    N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108.doi: 10.1103/PhysRevE.66.056108.

    [47]

    Y. Li, On a singularly perturbed elliptic equation, Adv. Differential Equations, 2 (1997), 955-980.

    [48]

    A. J. Majda and E. G. Tabak, A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow, Phys. D, 98 (1996), 515-522. Nonlinear phenomena in ocean dynamics (Los Alamos, NM, 1995).doi: 10.1016/0167-2789(96)00114-5.

    [49]

    B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Co., San Francisco, Calif., 1982, Schriftenreihe für den Referenten.

    [50]

    R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), 77.doi: 10.1016/S0370-1573(00)00070-3.

    [51]

    E. Milakis and L. Silvestre, Regularity for the nonlinear Signorini problem, Adv. Math., 217 (2008), 1301-1312.doi: 10.1016/j.aim.2007.08.009.

    [52]

    E. Nelson, Quantum Fluctuations, Princeton Series in Physics, Princeton University Press, Princeton, NJ, 1985.

    [53]

    D. P. Nicholls and M. Taber, Joint analyticity and analytic continuation of Dirichlet-Neumann operators on doubly perturbed domains, J. Math. Fluid Mech., 10 (2008), 238-271.doi: 10.1007/s00021-006-0231-9.

    [54]

    O. Savin and E. Valdinoci, Elliptic PDEs with fibered nonlinearities, J. Geom. Anal., 19 (2009), 420-432.doi: 10.1007/s12220-008-9064-5.

    [55]

    S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in $\Bbb R^N$, J. Math. Phys., 54 (2013), 031501, 17.doi: 10.1063/1.4793990.

    [56]

    M. A. Shubin, Pseudodifferential Operators and Spectral Theory, 2nd edition, Springer-Verlag, Berlin, 2001. Translated from the 1978 Russian original by Stig I. Andersson.doi: 10.1007/978-3-642-56579-3.

    [57]

    L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.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, J. Funct. Anal., 256 (2009), 1842-1864.doi: 10.1016/j.jfa.2009.01.020.

    [59]

    J. J. Stoker, Water waves: The Mathematical Theory with Applications, Pure and Applied Mathematics, Vol. IV, Interscience Publishers, Inc., New York, 1957.

    [60]

    J. F. Toland, The Peierls-Nabarro and Benjamin-Ono equations, J. Funct. Anal., 145 (1997), 136-150.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, Comm. Partial Differential Equations, 12 (1987), 1133-1173.doi: 10.1080/03605308708820522.

    [62]

    G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience [John Wiley & Sons], New York, 1974, Pure and Applied Mathematics.

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