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

Positive solution for quasilinear Schrödinger equations with a parameter

Abstract Related Papers Cited by
  • In this paper, we study the following quasilinear Schrödinger equations of the form \begin{eqnarray} -\Delta u+V(x)u-[\Delta(1+u^2)^{\alpha/2}]\frac{\alpha u}{2(1+u^2)^{(2-\alpha)/2}}=\mathrm{g}(x,u), \end{eqnarray} where $1 \le \alpha \le 2$, $N \ge 3$, $V\in C(R^N, R)$ and $\mathrm{g}\in C(R^N\times R, R)$. By using a change of variables, we get new equations, whose respective associated functionals are well defined in $H^1(R^N)$ and satisfy the geometric hypotheses of the mountain pass theorem. Using the special techniques, the existence of positive solutions is studied.
    Mathematics Subject Classification: 35J20, 35J60, 35Q55.

    Citation:

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

    J. M. Bezerra do Ó, O. H. Miyagaki and S. H. M. 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.

    [2]

    J. M. Bezerra do Ó, O. H. Miyagaki and S. 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.

    [3]

    M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Analysis: Theorey, Methods $&$ Applications, 56 (2004), 213-226.doi: 10.1016/j.na.2003.09.008.

    [4]

    Y. Cheng and J. Yang, Positive solution to a class of relativistic nonlinear Schrödinger equation, J. Math. Anal. Appl., 411 (2014), 665-674.doi: 10.1016/j.jmaa.2013.10.006.

    [5]

    S. Kurihara, Large-amplitude quasi-solitons in superfluid films, Journal of the physical Society of Japan, 50 (1981), 3262-3267.

    [6]

    P. L. Lions, The concentration-compactness principle in the calculus of variations: the locally compact cases, part I and part II, Ann. Inst. H. Poincaré Anal. Non Linëaire, 1 (1984), 109-145, 223-283.

    [7]

    A. G. Litvak and A. M. Sergeev, One dimensional collapse of plasma waves, JETP Letters, 27 (1978), 517-520.

    [8]

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

    [9]

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

    [10]

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

    [11]

    J. Liu, Y. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-892.doi: 10.1081/PDE-120037335.

    [12]

    A. Nakamura, Damping and modification of exciton solitary waves, J. Phys. Soc., 42 (1977), 1823-1835.

    [13]

    J. M. do Ó and U. Secero, Solitary waves for a class of quasilinear Schrödinger equations in dimension two, Cale. Var. Partial Differential Equations, 38 (2010), 275-315.doi: 10.1007/s00526-009-0286-6.

    [14]

    M. Porkolab and M. V. Goldman, Upper hybrid solitons and oscillating two-stream instabilities, Phys. Fluids, 19 (1976), 872-881.

    [15]

    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.

    [16]

    D. Ruiz and G. Siciliano, Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity, 23 (2010), 1221-1233.doi: 10.1088/0951-7715/23/5/011.

    [17]

    Y. Shen and Y. Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Analysis: Theorem, Methods $&$ Applications, 80 (2013), 194-201.doi: 10.1016/j.na.2012.10.005.

    [18]

    E. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 722-744.doi: 10.1007/s00526-009-0299-1.

    [19]

    Xian Wu, Multiple solutions for quasilinear Schrödinger equations with a parameter, J. Differential Equations, 256 (2014), 2619-2632.doi: 10.1016/j.jde.2014.01.026.

    [20]

    M. B. Yang, Existence of solutions for a quasilinear Schrödinger equation with subcritical nonlinearities, Nonlinear Analysis, 75 (2012), 5362-5373.doi: 10.1016/j.na.2012.04.054.

    [21]

    J. Zhang, X. H. Tang and W. Zhang, Existence of infinitely many solutions for a quasilinear elliptic equation, Applied Mathematics Letters, 37 (2014), 131-135.doi: 10.1016/j.aml.2014.06.010.

    [22]

    J. Zhang, X. H. Tang and W. Zhang, Infinitely many solutions of quasilinear Schrödinger equation with sign-changing potential, Journal of Mathematical Analysis and Applications, 420 (2014), 1762-1775.doi: 10.1016/j.jmaa.2014.06.055.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(147) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return