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Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents

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  • By using a change of variable, the quasilinear Schrödinger equation is reduced to semilinear elliptic equation. Then, Mountain Pass theorem without $(PS)_c$ condition in a suitable Orlicz space is employed to prove the existence of positive standing wave solutions for a class of quasilinear Schrödinger equations involving critical Sobolev-Hardy exponents.
    Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 35Q55.

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  • [1]

    C. O. Alves, J. M. do Ó and O. Miyagaki, On nonlinear perturbations of a periodic elliptic problem in $\mathbbR^2$ involving critical growth, Nonlinear Anal., 56 (2004), 781-791.doi: 10.1016/j.na.2003.06.003.

    [2]

    M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: A dual approach, Nonlinear Anal., 56 (2004), 213-226.doi: 10.1016/j.na.2003.09.008.

    [3]

    J. M. do Ó, O. Miyagaki and S. Soares, Soliton solutions for quasilinear Schrödinger equations:the critical exponential case, Nonlinear Anal., 67 (2007), 3357-3372.doi: 10.1016/j.na.2006.10.018.

    [4]

    J. M. do Ó and U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Comm. Pure and Applied Anal., 8 (2009), 621-644.doi: 10.3934/cpaa.2009.8.621.

    [5]

    A. Floer and A. Weisntein, 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.

    [6]

    N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 352 (2000), 5703-5743.doi: 10.1090/S0002-9947-00-02560-5.

    [7]

    R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equations, Z. Phys. B, 37 (1980), 83-87.doi: 10.1007/BF01325508.

    [8]

    D. Sh. Kang, Solutions for semilinear elliptic problems with critical Sobolev-Hardy exponents in $\mathbbR^N$, Nonlinear Anal., 66 (2007), 241–252.doi: 10.1016/j.na.2005.11.028.

    [9]

    A. M. Kosevich, B. A. Ivanov and A. S. Kovalev, Magnetic solitons, Phys. Rep., 194 (1990), 117-238.doi: 10.1016/0370-1573(90)90130-T.

    [10]

    S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.doi: 10.1143/jpsj.50.3262.

    [11]

    E. W. Laedke, K. H. Spatschek and L. Stenflo, Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys., 24 (1983), 2764-2769.doi: 10.1063/1.525675.

    [12]

    A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves, JETP Lett., 27 (1978), 517-520.doi: 0021-3640778/2710-0517.

    [13]

    J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901.doi: 10.1081/PDE-120037335.

    [14]

    J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations I, Proc. Amer. Math. Soc., 131 (2002), 441-448.doi: 10.1090/S0002-9939-02-06783-7.

    [15]

    J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations II, J. Differential Equations, 187 (2003), 473-493.doi: 10.1016/s0022-0396(02)0064-5.

    [16]

    V. G. Makhankov and V. K. Fedyanin, Non-linear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep, 104 (1984), 1-86.doi: 10.1016/0370-1573(84)90106-6.

    [17]

    J. Mawhin and M. Willem, "Critical Point Theory and Hamiltonian Systems," Springer-Verlag, 1989.

    [18]

    O. H. Miyagaki, On a class of semilinear elliptic problems in $\mathbbR^N$ with critical growth, Nonlinear Anal., 29 (1997), 773-781.doi: 10.1016/s0362-546x(96)00087-9.

    [19]

    A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\mathbbR^N$, J. Differential Equations, 229 (2006), 570-587.doi: 10.1016/j.jde.2006.07.001.

    [20]

    A. Moameni, On the existence of standing wave solutions to quasilinear Schrödinger equations, Nonlinearity, 19 (2006), 937-957.doi: 10.1088/0951-7715/19/4/009.

    [21]

    M. Poppenberg, K. Schmitt and Z. Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations 14 (2002), 329-344.doi: 10.1007/s005260100105.

    [22]

    G. R. W. Quispel and H. W. Capel, Equation of motion for the Heisenberg spin chain, Physica A, 110 (1982), 41-80.doi: 10.1016/0378-4371(82)90104-2.

    [23]

    S. Takeno and S. Homma, Classical planar Heisenberg ferromagnet, complex scalar fields and nonlinear excitations, Progr. Theoret. Phys., 65 (1981), 172-189.doi: 10.1143/PTP.65.172.

    [24]

    P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z angew Math. Phy., 43 (1992), 272-291.doi: 10.1007/BF00946631.

    [25]

    Y. J. Wang, J. Yang and Y. M. Zhang, Quasilinear elliptic equations involving the N-Laplacian with critical exponential growth in $\mathbbR^N$, Nonlinear Anal., 71 (2009), 6157-6169.doi: 10.1016/j.na.2009.06.006.

    [26]

    Y. J. Wang, Y. M. Zhang and Y. T. Shen, Multiple solutions for quasilinear Schröinger equations involving critical exponent, Applied Mathematics and Computation, 216 (2010), 849-856.doi: 10.1016/j.amc.2010.01.091.

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