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Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential

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    * Corresponding author
This work was supported by NFSC Grants 11471331,11501555 and 11471330.
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  • We study a Kirchhoff type elliptic equation with trapping potential. The existence and blow-up behavior of solutions with normalized $L^{2}$-norm for this equation are discussed.

    Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 47J30.

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  • [1] A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348 (1996), 305-330.  doi: 10.1090/S0002-9947-96-01532-2.
    [2] 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.
    [3] T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb R^{N}$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149.
    [4] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York, 2003.
    [5] F. DalfovoS. GiorginiL. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Modern Phys., 71 (1999), 463-512.  doi: 10.1103/RevModPhys.71.463.
    [6] P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262. 
    [7] A. L. Fetter, Rotating trapped Bose-Einstein condensates, Rev. Modern Phys., 81 (2009), 647-691.  doi: 10.1103/RevModPhys.81.647.
    [8] Y. J. Guo and R. Seiringer, On the mass concentration for Bose-Einstein conden-sates with attractive interactions, Lett. Math. Phys., 104 (2014), 141-156.  doi: 10.1007/s11005-013-0667-9.
    [9] Y. J. GuoX. Y. Zeng and H. S. Zhou, Concentration behavior of standing waves for almost mass critical nonlinear Schrödinger equations, J. Differential Equations, 256 (2014), 2079-2100.  doi: 10.1016/j.jde.2013.12.012.
    [10] B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb R^{n}$, Math. Anal. Appl. Part A, pp. 369-402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981.
    [11] Y. He and G. B. Li, Standing waves for a class of Kirchhoff type problems in $\mathbb R^{3}$ involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3067-3106.  doi: 10.1007/s00526-015-0894-2.
    [12] Q. Han and F. H. Lin, Elliptic Partial Differential Equations, $2^{nd}$ edition, Courant Institute of Mathematical Sciences, New York, 2011.
    [13] X. M. He and W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb R^{3}$, J. Differential Equations, 252 (2012), 1813-1834.  doi: 10.1016/j.jde.2011.08.035.
    [14] J. H. Jin and X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $\mathbb R^{N}$, J. Math. Anal. Appl., 369 (2010), 564-574.  doi: 10.1016/j.jmaa.2010.03.059.
    [15] G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
    [16] O. Kavian and F. B. Weissler, Self-similar solutions of the pseudo-conformally invariant nonlinear Schrödinger equation, Michigan Math. J., 41 (1994), 151-173.  doi: 10.1307/mmj/1029004922.
    [17] K. Mcleod and J. Serrin, Uniqueness of solutions of semilinear Poisson equations, Proc. Natl. Acad. Sci. USA., 78 (1981), 6592-6595. 
    [18] G. B. Li and H. Y. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb R^{3}$, J. Differential Equations, 257 (2014), 566-600.  doi: 10.1016/j.jde.2014.04.011.
    [19] M. Reed and B. Simon, Methods of Modern Mathematical Physics Ⅳ: Analysis of Operators, Academic Press, New York-London, 1978.
    [20] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
    [21] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576. 
    [22] H. Y. Ye, The sharp existence of constrained minimizers for a class of nonlinear Kirchhoff equations, Math. Methods Appl. Sci., 38 (2015), 2663-2679.  doi: 10.1002/mma.3247.
    [23] H. Y. Ye, The existence of normalized solutions for $L^{2}$-critical constrained problems related to Kirchhoff equations, Z. Angew. Math. Phys., 66 (2015), 1483-1497.  doi: 10.1007/s00033-014-0474-x.
    [24] J. Zhang, Stability of attractive Bose-Einstein condensates, J. Stat. Phys., 101 (2000), 731-746. doi: 10.1023/A:1026437923987.
    [25] X. Y. Zeng, Asymptotic properties of standing waves for mass subcritical nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst., 37 (2017), 1749-1762.  doi: 10.3934/dcds.2017073.
    [26] X. Y. Zeng and Y. M. Zhang, Existence and asymptotic behavior for the ground state of quasilinear elliptic equation, arXiv: 1703.00183.
    [27] X. Y. Zeng and Y. M. Zhang, Existence and uniqueness of normalized solutions for the Kirchhoff equation, Appl. Math. Lett., 74 (2017), 52-59.  doi: 10.1016/j.aml.2017.05.012.
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