Research article

Sign-changing solutions for a class of p-Laplacian Kirchhoff-type problem with logarithmic nonlinearity

  • Received: 19 December 2019 Accepted: 12 February 2020 Published: 26 February 2020
  • MSC : 35J20, 35J65

  • In this paper, we study the existence of ground state sign-changing solutions for following $p$-Laplacian Kirchhoff-type problem with logarithmic nonlinearity $ \begin{equation*} \left\{ \renewcommand{\arraystretch}{1.25} \begin{array}{ll} -(a+ b\int _{\Omega}|\nabla u|^{p}dx)\Delta_p u = |u|^{q-2}u\ln u^2, ~x\in\Omega \\ u = 0, ~\ x\in \partial\Omega, \end{array} \right. \end{equation*} $ where $\Omega\subset \mathbb{R}^{N}$ is a smooth bounded domain, $a, b \gt 0$ are constant, $4\leq 2 p \lt q \lt p^*$ and $N \gt p$. By using constraint variational method, topological degree theory and the quantitative deformation lemma, we prove the existence of ground state sign-changing solutions with precisely two nodal domains.

    Citation: Ya-Lei Li, Da-Bin Wang, Jin-Long Zhang. Sign-changing solutions for a class of p-Laplacian Kirchhoff-type problem with logarithmic nonlinearity[J]. AIMS Mathematics, 2020, 5(3): 2100-2112. doi: 10.3934/math.2020139

    Related Papers:

  • In this paper, we study the existence of ground state sign-changing solutions for following $p$-Laplacian Kirchhoff-type problem with logarithmic nonlinearity $ \begin{equation*} \left\{ \renewcommand{\arraystretch}{1.25} \begin{array}{ll} -(a+ b\int _{\Omega}|\nabla u|^{p}dx)\Delta_p u = |u|^{q-2}u\ln u^2, ~x\in\Omega \\ u = 0, ~\ x\in \partial\Omega, \end{array} \right. \end{equation*} $ where $\Omega\subset \mathbb{R}^{N}$ is a smooth bounded domain, $a, b \gt 0$ are constant, $4\leq 2 p \lt q \lt p^*$ and $N \gt p$. By using constraint variational method, topological degree theory and the quantitative deformation lemma, we prove the existence of ground state sign-changing solutions with precisely two nodal domains.


    加载中


    [1] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
    [2] J. L. Lions, On some questions in boundary value problems of mathematical physics, In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, NorthHolland Math. Stud., North-Holland, Amsterdam, New York, (1978), 284-346.
    [3] D. Cassani, Z. Liu, C. Tarsi, Multiplicity of sign-changing solutions for Kirchhoff-type equations, Nonlinear Anal., 186 (2019), 145-161. doi: 10.1016/j.na.2019.01.025
    [4] B. T. Cheng, X. H. Tang, Ground state sign-changing solutions for asymptotically 3-linear Kirchhoff-type problems, Complex Var. Elliptic, 62 (2017), 1093-1116. doi: 10.1080/17476933.2016.1270272
    [5] Y. B. Deng, S. J. Peng, W. Shuai, Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in $\mathbb{R}^3$, J. Funct. Anal., 269 (2015), 3500-3527.
    [6] Y. B. Deng, W. Shuai, Sign-changing multi-bump solutions for Kirchhoff-type equations in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst. Ser. A, 38 (2018), 3139-3168.
    [7] G. M. Figueiredo, R. G. Nascimento, Existence of a nodal solution with minimal energy for a Kirchhoff equation, Math. Nachr., 288 (2015), 48-60. doi: 10.1002/mana.201300195
    [8] G. M. Figueiredo, J. R. Santos Júnior, Existence of a least energy nodal solution for a Schrödinger-Kirchhoff equation with potential vanishing at infinity, J. Math. Phys., 56 (2015), 051506.
    [9] X. Han, X. Ma, X. M. He, Existence of sign-changing solutions for a class of p-Laplacian Kirchhoff-type equations, Complex Var. Elliptic, 64 (2019), 181-203. doi: 10.1080/17476933.2018.1427078
    [10] W. Han, J. Yao, The sign-changing solutions for a class of p-Laplacian Kirchhoff type problem in bounded domains, Comput. Math. Appl., 76 (2018), 1779-1790. doi: 10.1016/j.camwa.2018.07.029
    [11] F. Y. Li, C. Gao, X. Zhu, Existence and concentration of sign-changing solutions to Kirchhoff-type system with Hartree-type nonlinearity, J. Math. Anal. Appl., 448 (2017), 60-80. doi: 10.1016/j.jmaa.2016.10.069
    [12] Q. Li, X. Du, Z. Zhao, Existence of sign-changing solutions for nonlocal Kirchhoff-Schrödingertype equations in $\mathbb{R}^3$, J. Math. Anal. Appl., 477 (2019), 174-186.
    [13] S. Lu, Signed and sign-changing solutions for a Kirchhoff-type equation in bounded domains, J. Math. Anal. Appl., 432 (2015), 965-982. doi: 10.1016/j.jmaa.2015.07.033
    [14] A. M. Mao, Z. T. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70 (2009), 1275-1287. doi: 10.1016/j.na.2008.02.011
    [15] A. Mao, S. Luan, Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems, J. Math. Anal. Appl., 383 (2011), 239-243. doi: 10.1016/j.jmaa.2011.05.021
    [16] M. Shao, A. Mao, M. Shao, Signed and sign-changing solutions of Kirchhoff type problems, J. Fixed Point Theory Appl., 20 (2018), 2.
    [17] W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differential Equations, 259 (2015), 1256-1274. doi: 10.1016/j.jde.2015.02.040
    [18] J. Sun, L. Li, M. Cencelj, Infinitely many sign-changing solutions for Kirchhoff type problems in $\mathbb{R}^3$, Nonlinear Anal., 186 (2019), 33-54.
    [19] X. H. Tang, B. Cheng, Ground state sign-changing solutions for Kirchhoff type problems in bounded domains, J. Differential Equations, 261 (2016), 2384-2402. doi: 10.1016/j.jde.2016.04.032
    [20] D. B. Wang, Least energy sign-changing solutions of Kirchhoff-type equation with critical growth, J. Math. Phys., 61 (2020), 011501. Available from: https://doi.org/10.1063/1.5074163.
    [21] D. B. Wang, T. Li, X. Hao, Least-energy sign-changing solutions for KirchhoffSchrödinger-Poisson systems in $\mathbb{R}^3$, Bound. Value Probl., 75 (2019). Available from: https://doi.org/10.1186/s13661-019-1183-3.
    [22] D. B. Wang, Y. Ma, W. Guan, Least energy sign-changing solutions for the fractional Schrödinger-Poisson systems in $\mathbb{R}^3$, Bound. Value Probl., 25 (2019). Available from: https://doi.org/10.1186/s13661-019-1128-x.
    [23] D. B. Wang, H. Zhang, W. Guan, Existence of least-energy sign-changing solutions for Schrödinger-Poisson system with critical growth, J. Math. Anal. Appl., 479 (2019), 2284-2301. doi: 10.1016/j.jmaa.2019.07.052
    [24] D. B. Wang, H. Zhang, Y. Ma, et al, Ground state sign-changing solutions for a class of nonlinear fractional Schrödinger-Poisson system with potential vanishing at infinity, J. Appl. Math. Comput., 61 (2019), 611-634. doi: 10.1007/s12190-019-01265-y
    [25] L. Wang, B. L. Zhang, K. Cheng, Ground state sign-changing solutions for the SchrödingerKirchhoff equation in $\mathbb{R}^3$, J. Math. Anal. Appl., 466 (2018), 1545-1569.
    [26] L. Wen, X. H. Tang, S. Chen, Ground state sign-changing solutions for Kirchhoff equations with logarithmic nonlinearity, Electron. J. Qual. Theo., 47 (2019), 1-13.
    [27] H. Ye, The existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations in $\mathbb{R}^N$, J. Math. Anal. Appl., 431 (2015), 935-954.
    [28] Z. T. Zhang, K. Perera, Sign changing solutions of Kirchhoff type problems via invariant sets of descentow, J. Math. Anal. Appl., 317 (2006), 456-463. doi: 10.1016/j.jmaa.2005.06.102
    [29] C. O. Alves, D. C. de Morais Filho, Existence and concentration of positive solutions for a Schrödinger logarithmic equation, Z. Angew. Math. Phys., 69 (2018), 144.
    [30] P. d'Avenia, E. Montefusco, M. Squassina, On the logarithmic Schrödinger equation, Commun. Contemp. Math., 16 (2014), 313-402.
    [31] P. d'Avenia, M. Squassina, M. Zenari, Fractional logarithmic Schrödinger equations, Math. Methods Appl. Sci., 38 (2015), 5207-5216. doi: 10.1002/mma.3449
    [32] C. Ji, A. Szulkin, A logarithmic Schrödinger equation with asymptotic conditions on the potential, J. Math. Anal. Appl., 437 (2016), 241-254. doi: 10.1016/j.jmaa.2015.11.071
    [33] W. Shuai, Multiple solutions for logarithmic Schrödinger equations, Nonlinearity, 32 (2019), 2201-2225. doi: 10.1088/1361-6544/ab08f4
    [34] M. Squassina, A. Szulkin, Multiple solutions to logarithmic Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 54 (2015), 585-597. doi: 10.1007/s00526-014-0796-8
    [35] K. Tanaka, C. Zhang, Multi-bump solutions for logarithmic Schrödinger equations, Calc. Var. Partial Differential Equations, 56 (2017), 33.
    [36] S. Tian, Multiple solutions for the semilinear elliptic equations with the sign-changing logarithmic nonlinearity, J. Math. Anal. Appl., 454 (2017), 816-228. doi: 10.1016/j.jmaa.2017.05.015
    [37] W. C. Troy, Uniqueness of positive ground state solutions of the logarithmic Schrödinger equation, Arch. Ration. Mech. Anal., 222 (2016), 1581-1600. doi: 10.1007/s00205-016-1028-5
    [38] C. Miranda, Un'osservazione su un teorema di Brouwer, Bol. Un. Mat. Ital., 3 (1940), 5-7.
    [39] M. Willem, Minimax Theorems, Birkhäuser, Bosten, 1996.
    [40] Klaus Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
    [41] D. J. Guo, Nonlinear Functional Analysis, Higher Education Press, Beijing, 2015.
    [42] E. Zeidler, Nonlinear functional analysis and its applications. I. Fixed point theorems, SpringerVerlag, New York, 1986.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3332) PDF downloads(438) Cited by(13)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog