Advanced Search
Article Contents
Article Contents

Positive solutions for critical quasilinear Schrödinger equations with potentials vanishing at infinity

  • * Corresponding author: Guofa Li

    * Corresponding author: Guofa Li 

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11971339, 11901345), the Yunnan Local Colleges Applied Basic Research Projects (Grant No. 202001BA070001-032) and Scientific and Technological Innovation Team of Nonlinear Analysis and Algebra with Their Applications in Universities of Yunnan Province, China (Grant No. 2020CXTD25)

Abstract Full Text(HTML) Related Papers Cited by
  • In this paper, we study the existence of positive solutions for the following quasilinear Schrödinger equations

    $ \begin{equation*} -\triangle u+V(x)u+\frac{\kappa}{2}[\triangle|u|^{2}]u = \lambda K(x)h(u)+\mu|u|^{2^*-2}u, \quad x\in\mathbb{R}^{N}, \end{equation*} $

    where $ \kappa>0 $, $ \lambda>0, \mu>0, h\in C(\mathbb{R}, \mathbb{R}) $ is superlinear at infinity, the potentials $ V(x) $ and $ K(x) $ are vanishing at infinity. In order to discuss the existence of solutions we apply minimax techniques together with careful $ L^{\infty} $-estimates. For the subcritical case ($ \mu = 0 $) we can deal with large $ \kappa>0 $. For the critical case we treat that $ \kappa>0 $ is small.

    Mathematics Subject Classification: Primary: 35J20; Secondary: 35J62.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] J. F. L. Aires and M. A. S. Souto, Equation with positive coefficient in the quasilinear term and vanishing potential, Topol. Methods Nonlinear Anal., 46 (2015), 813-833. 
    [2] C. O. Alves and M. A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity, J. Differential Equations, 254 (2013), 1977-1991.  doi: 10.1016/j.jde.2012.11.013.
    [3] C. O. AlvesY. Wang and Y. Shen, Soliton solutions for a class of quasilinear Schrödinger equations with a parameter, J. Differential Equations, 259 (2015), 318-343.  doi: 10.1016/j.jde.2015.02.030.
    [4] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations Ⅰ Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313-346.  doi: 10.1007/BF00250555.
    [5] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Commun. Pure Appl. Math., 36 (1983), 437-477.  doi: 10.1002/cpa.3160360405.
    [6] L. BrizhikA. EremkoB. Piette and W. J. Zakrzewski, Electron self-trapping in a discrete two-dimensional lattice, Physica D, 159 (2001), 71-90.  doi: 10.1016/S0167-2789(01)00332-3.
    [7] L. BrizhikA. EremkoB. Piette and W. J. Zakrzewski, Static solutions of a D-dimensional modified nonlinear Schrödinger equation, Nonlinearity, 16 (2003), 1481-1497.  doi: 10.1088/0951-7715/16/4/317.
    [8] J. ChenX. Huang and B. Cheng, Positive solutions for a class of quasilinear Schrödinger equations with superlinear condition, Appl. Math. Lett., 87 (2019), 165-171.  doi: 10.1016/j.aml.2018.07.035.
    [9] J. Chen, X. Huang, B. Cheng and C. Zhu, Some results on standing wave solutions for a class of quasilinear Schrödinger equations, J. Math. Phys., 60 (2019), 091506, 55 pp. doi: 10.1063/1.5093720.
    [10] A. de BouardN. Hayashi and J.-C. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.  doi: 10.1007/s002200050191.
    [11] Y. Deng and W. Shuai, Non-trivial solutions for a semilinear biharmonic problem with critical growth and potential vanishing at infinity, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 281-299.  doi: 10.1017/S0308210513001170.
    [12] B. Hartmann and W. J. Zakrzewski, Electrons on hexagonal lattices and applications to nanotubes, Phys. Rev. B, 68 (2003), 184302. doi: 10.1103/PhysRevB.68.184302.
    [13] C. Huang and G. Jia, Existence of positive solutions for supercritical quasilinear Schrödinger elliptic equations, J. Mathe. Anal. Appl., 472 (2019), 705-727.  doi: 10.1016/j.jmaa.2018.11.048.
    [14] L. Jeanjean and K. Tanaka, A positive solution for a nonlinear Schrödinger equation on $\mathbb{R}^{N}$, Indiana Univ. Math. J., 54 (2005), 443-464.  doi: 10.1512/iumj.2005.54.2502.
    [15] S. Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.  doi: 10.1143/JPSJ.50.3262.
    [16] Q. Li and X. Wu, Existence, multiplicity, and concentration of solutions for generalized quasilinear Schödinger equations with critical growth, J. Math. Phys., 58 (2017), 041501. doi: 10.1063/1.4982035.
    [17] Z. Liang, J. Gao and A. Li, Existence of positive solutions for a class of quasilinear Schrödinger equations with local superlinear nonlinearities, J. Math. Anal. Appl., 484 (2020), 123732. doi: 10.1016/j.jmaa.2019.123732.
    [18] P. L. Lions, The concentration-compactness principle in the calculus of variation. The locally compact case, part Ⅰ and Ⅱ, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109–145 and 223–283. doi: 10.1016/S0294-1449(16)30422-X.
    [19] U. B. Severo and G. M. de Carvalho, Quasilinear Schrödinger equations with a positive parameter and involving unbounded or decaying potentials, Appl. Anal., 100 (2021), 229-252.  doi: 10.1080/00036811.2019.1599106.
    [20] U. B. SeveroE. Gloss and E. D. da Silva, On a class of quasilinear Schrödinger equations with superlinear or asymptotically linear terms, J. Differential Equations, 263 (2017), 3550-3580.  doi: 10.1016/j.jde.2017.04.040.
    [21] Y. Shen and Y. Wang, Standing waves for a class of quasilinear Schrödinger equations, Complex Var. Elliptic Equ., 61 (2016), 817-842.  doi: 10.1080/17476933.2015.1119818.
    [22] Y. Wang, A class of quasilinear Schrödinger equations with critical or supercritical exponents, Compu. Math. Appl., 70 (2015), 562-572.  doi: 10.1016/j.camwa.2015.05.016.
    [23] Y. Wang and Z. Li, Existence of solutions to quasilinear Schrödinger equations involving critical Sobolev exponent, Taiwanese Journal of Math., 22 (2018), 401-420.  doi: 10.11650/tjm/8150.
    [24] M. Willem, Minimax Theorems, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.
    [25] M. Yang, C. A. Santos and J. Zhou, Least action nodal solutions for a quasilinear defocusing Schrödinger equation with supercritical nonlinearity, Commun. Contemp. Math., 21 (2019), 1850026, 23 pp. doi: 10.1142/S0219199718500268.
    [26] W. Zou, Sign-Changing Critical Points Theory, Springer, New York, 2008.
  • 加载中

Article Metrics

HTML views(869) PDF downloads(493) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint