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Soliton solutions for a quasilinear Schrödinger equation with critical exponent

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  • This paper is concerned with the existence of soliton solutions for a quasilinear Schrödinger equation in $R^N$ with critical exponent, which appears from modelling the self-channeling of a high-power ultrashort laser in matter. By working with a perturbation approach which was initially proposed in [26], we prove that the given problem has a positive ground state solution.
    Mathematics Subject Classification: Primary: 35J62; Secondary: 35B09, 35B33.

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  • [1]

    A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Func. Anal., 14 (1973), 349-381.doi: 10.1016/0022-1236(73)90051-7.

    [2]

    H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.doi: 10.1002/cpa.3160360405.

    [3]

    João M. Bezerra do Ó, Olímpio H. Miyagaki and Sérgio H.M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations, 248 (2010), 722-744.doi: 10.1016/j.jde.2009.11.030.

    [4]

    H. Brandi, C. Manus, G. Mainfray, T. Lehner and G. Bonnaud, Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma, Phys. Fluids B, 5 (1993), 3539-3550.doi: 10.1063/1.860828.

    [5]

    F.G. Bass and N.N. Nasanov, Nonlinear electromagnetic-spin waves, Phys. Rep., 189 (1990), 165-223.doi: 10.1016/0370-1573(90)90093-H.

    [6]

    L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations, Exposition. Math., 4 (1986), 279-288.

    [7]

    X.L. Chen and R.N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma, Phys. Rev. Lett., 70 (1993), 2082-2085.doi: 10.1103/PhysRevLett.70.2082.

    [8]

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

    [9]

    S. Cuccagna, On instability of excited states of the nonlinear Schrödinger equation, Physica D, 238 (2009), 38-54.doi: 10.1016/j.physd.2008.08.010.

    [10]

    A. De Bouard, N. Hayashi and J. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Commun. Math. Phys., 189 (1997), 73-105.doi: 10.1007/s002200050191.

    [11]

    Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, J. Math. Phys., 54 (2013), 011504, 27pp.doi: 10.1063/1.4774153.

    [12]

    Y. Deng, S. Peng and J. Wang, Infinitely many sign-changing solutions for quasilinear Schrödinger equations in $R^N$, Commun. Math. Sci., 9 (2011), 859-878.doi: 10.4310/CMS.2011.v9.n3.a9.

    [13]

    Y. Deng, S. Peng and S. Yan, Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth, J. Differential Equations, 258 (2015), 115-147.doi: 10.1016/j.jde.2014.09.006.

    [14]

    Y. Deng, S. Peng and S. Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, J. Differential Equations, 260 (2016), 1228-1262.doi: 10.1016/j.jde.2015.09.021.

    [15]

    Q. Han and F. Lin, Elliptic Partial Differential Equations, Courant Lecture Notes in Mathematics, vol. 1. New York University Courant Institute of Mathematical Sciences, New York, 1997.

    [16]

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

    [17]

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

    [18]

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

    [19]

    E. Laedke, K. 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.

    [20]

    P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part I}, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.

    [21]

    P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, part II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.

    [22]

    H. Lange, M. Poppenberg and H. Teismann, Nash-Moser methods for the solution of quasilinear Schrödinger equations, Comm. Partial Differential Equations, 24 (1999), 1399-1418.doi: 10.1080/03605309908821469.

    [23]

    J. Liu, Y. Wang and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. II, J. Differential Equations, 187 (2003), 473-493.doi: 10.1016/S0022-0396(02)00064-5.

    [24]

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

    [25]

    J. Liu and Z. Wang, Soliton solutions for quasilinear Schrödinger equations. I, Proc. Amer. Math. Soc., 131 (2003), 441-448.doi: 10.1090/S0002-9939-02-06783-7 .

    [26]

    X. Liu, J. Liu and Z. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263.doi: 10.1090/S0002-9939-2012-11293-6 .

    [27]

    X. Liu, J. Liu and Z. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations, 254 (2013), 102-124.doi: 10.1016/j.jde.2012.09.006.

    [28]

    X. Liu, J. Liu and Z. Wang, Ground states for quasilinear Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 46 (2013), 641-669.doi: 10.1007/s00526-012-0497-0.

    [29]

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

    [30]

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

    [31]

    P. Pucci and J. Serrin, A general variational idnetity, Indiana Univ. Math. J., 35 (1986), 681-703.doi: 10.1512/iumj.1986.35.35036.

    [32]

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

    [33]

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

    [34]

    B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E, 50 (1994), 687-689.doi: 10.1103/PhysRevE.50.R687.

    [35]

    Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal. TMA., 80 (2013), 194-201.doi: 10.1016/j.na.2012.10.005.

    [36]

    E.A.B. Silva and G.F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33.doi: 10.1007/s00526-009-0299-1.

    [37]

    J. Yang, Y. Wang and A.A. Abdelgadir, Soliton solutions for quasilinear Schrödinger equations, J. Math. Phys., 54 (2013), 071502, 19pp.doi: 10.1063/1.4811394.

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