American Institute of Mathematical Sciences

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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Anal.}, 165 (2002), 295. doi: 10.1007/s00205-002-0225-6. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] R. Cont and P. Tankov, Financial Modelling with Jump Processes,, Chapman & Hall/CRC Financial Mathematics Series, (2004). Google Scholar [21] D. 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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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. 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Google Scholar [47] Y. Li, On a singularly perturbed elliptic equation,, \emph{Adv. Differential Equations}, 2 (1997), 955. Google Scholar [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. Google Scholar [49] B. B. Mandelbrot, The Fractal Geometry of Nature,, W. H. Freeman and Co., (1982). Google Scholar [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. Google Scholar [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. Google Scholar [52] E. Nelson, Quantum Fluctuations,, Princeton Series in Physics, (1985). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [56] M. A. Shubin, Pseudodifferential Operators and Spectral Theory,, 2nd edition, (2001). doi: 10.1007/978-3-642-56579-3. Google Scholar [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. Google Scholar [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. Google Scholar [59] J. J. Stoker, Water waves: The Mathematical Theory with Applications,, Pure and Applied Mathematics, (1957). Google Scholar [60] J. F. Toland, The Peierls-Nabarro and Benjamin-Ono equations,, \emph{J. Funct. Anal.}, 145 (1997), 136. doi: 10.1006/jfan.1996.3016. Google Scholar [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. Google Scholar [62] G. B. Whitham, Linear and Nonlinear Waves,, Wiley-Interscience [John Wiley & Sons], (1974). Google Scholar

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References:
 [1] R. A. Adams, Sobolev Spaces,, Academic Press, (1975). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [8] P. W. Bates, On some nonlocal evolution equations arising in materials science,, in \emph{Nonlinear dynamics and evolution equations}, (2006), 13. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] R. Cont and P. Tankov, Financial Modelling with Jump Processes,, Chapman & Hall/CRC Financial Mathematics Series, (2004). Google Scholar [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. Google Scholar [22] W. Craig, C. Sulem and P.-L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach,, \emph{Nonlinearity}, 5 (1992), 497. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [28] J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics,, \emph{Invent. Math.}, 29 (1975), 39. Google Scholar [29] G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics,, Springer-Verlag, (1976). Google Scholar [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. Google Scholar [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. Google Scholar [32] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals,, Emended edition, (2010). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [46] N. Laskin, Fractional Schrödinger equation,, \emph{Phys. Rev. E}, 66 (2002). doi: 10.1103/PhysRevE.66.056108. Google Scholar [47] Y. Li, On a singularly perturbed elliptic equation,, \emph{Adv. Differential Equations}, 2 (1997), 955. Google Scholar [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. Google Scholar [49] B. B. Mandelbrot, The Fractal Geometry of Nature,, W. H. Freeman and Co., (1982). Google Scholar [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. Google Scholar [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. Google Scholar [52] E. Nelson, Quantum Fluctuations,, Princeton Series in Physics, (1985). Google Scholar [53] D. P. Nicholls and M. 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