\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Multi-peak positive solutions for a fractional nonlinear elliptic equation

Abstract Related Papers Cited by
  • In this paper we study the existence of positive multi-peak solutions to the semilinear equation \begin{eqnarray*} \varepsilon^{2s}(-\Delta)^{s}u + u= Q(x)u^{p-1}, \hskip0.5cm u >0, \hskip 0.2cm u\in H^{s}(\mathbb{R}^{N}) \end{eqnarray*} where $(-\Delta)^{s} $ stands for the fractional Laplacian, $s\in (0,1)$, $\varepsilon$ is a positive small parameter, $2 < p < \frac{2N}{N-2s}$, $Q(x)$ is a bounded positive continuous function. For any positive integer $k$, we prove the existence of a positive solution with $k$-peaks and concentrating near a given local minimum point of $Q$. For $s=1$ this corresponds to the result of [22].
    Mathematics Subject Classification: Primary: 35A15, 35J60; Secondary: 35B38.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    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.

    [2]

    D. Cao, E. S. Noussair and S. Yan, Existence and uniqueness results on single-peaked solutions of a semilinear problem, Ann. Inst. H. Poincaré, 15 (1998), 73-111.doi: 10.1016/S0294-1449(99)80021-3.

    [3]

    D. Cao, E. S. Noussair and S. Yan, Solutions with multiple peaks for nonlinear elliptic equations, Proc. Royal Soc. Edinburgh, 129 (1999), 235-264.doi: 10.1017/S030821050002134X.

    [4]

    D. Cao and S. Peng, Semi-classical bound states for Schröinger equations with potentials vanishing or unbounded at infinity, Comm. Partial Differential Equations, 34 (2009), 1566-1591.doi: 10.1080/03605300903346721.

    [5]

    A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some nonlocal semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.doi: 10.1080/03605302.2011.562954.

    [6]

    X. Chang and Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations, 256 (2014), 2956-2992.doi: 10.1016/j.jde.2014.01.027.

    [7]

    G. Chen, Multiple semiclassical standing waves for fractional nonlinear Schrödinger equations, arXiv:1405.4366v1.

    [8]

    G. Chen and Y. Zheng, Concentration phenomenon for fractional nonlinear Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 2359-2376.doi: 10.3934/cpaa.2014.13.2359.

    [9]

    M. Cheng, Bound state for the fractional Schrödinger equations with unbounded potential, J. Math. Phys., 53 (2012), 043507, 7 pp.doi: 10.1063/1.3701574.

    [10]

    W. Choi, S. Kim and K. A. Lee, Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian, J. Funct. Anal., 266 (2014), 6531-6598.doi: 10.1016/j.jfa.2014.02.029.

    [11]

    J. Dávila, M. Del Pino, S. Dipierro and E. Valdinoci, Concentration phenomena for the nonlocal Schrödinger equation with dirichlet datum, arXiv:1403.4435v1.

    [12]

    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.

    [13]

    S. Dipierro, G. Palatucci and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche, 68 (2013), 201-216.doi: 10.4418/2013.68.1.15.

    [14]

    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.

    [15]

    M. M. Fall and E. Valdinoci, Uniqueness and nondegeneracy of positive solutions of $ (-\Delta)^s u + u = u^p$ in $\mathbbR^N$ when $s$ close to 1, Comm. Math. Phys., 329 (2014), 383-404.doi: 10.1007/s00220-014-1919-y.

    [16]

    P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh., 142 (2012), 1237-1262.doi: 10.1017/S0308210511000746.

    [17]

    R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, arXiv:1302.2652v1.

    [18]

    N. Laskin, Fractional Schrödinger equation, Phys. Rev. E., 66 (2002), 056108, 7 pp.doi: 10.1103/PhysRevE.66.056108.

    [19]

    N. Laskin, Fractional quantum mechanics, Phys. Rev. E., 62 (2000), p. 3135.doi: 10.1103/PhysRevE.62.3135.

    [20]

    L. Lin, Z. Liu and S. Chen, Multi-bump solutions for a semilinear Schrödinger equation, Indiana Univ. Math. J., 58 (2009), 1659-1689.doi: 10.1512/iumj.2009.58.3611.

    [21]

    W. Long, S. Peng and J. Yang, Infinitely many positive solutions for nonlinear fractional Schrödinger equations, arXiv:1402.1902v1.

    [22]

    E. S. Noussair and S. Yan, On positive multipeak solutions of a nonlinear elliptic problem, J. London Math. Soc., 62 (2002), 213-227.doi: 10.1112/S002461070000898X.

    [23]

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

    [24]

    E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. S$\vec e$MA, 49 (2009), 33-44.

    [25]

    L. Wang and C. Zhao, Infinitely many solutions to a fractional nonlinear Schrödinger equation, arXiv:1403.0042v1.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(184) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return