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September  2018, 17(5): 1875-1897. doi: 10.3934/cpaa.2018089

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

Helin Guo 1, , Yimin Zhang 2, and  Huansong Zhou 2,,

1. 

University of Chinese Academy of Sciences and Wuhan Institute of Physics and Mathematics, CAS, Wuhan 430071, China
2. 

Center for Mathematical Sciences and Department of Mathematics, Wuhan University of Technology, Wuhan, 430070, China

* Corresponding author

Received  July 2017 Revised  November 2017 Published  April 2018

Fund Project: This work was supported by NFSC Grants 11471331,11501555 and 11471330.

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.

Keywords: Constrained variational method, Kirchhoff equation, elliptic equation, Schwarz symmetrization, Energy estimates.
Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 47J30.
Citation: Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089
References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York, 2003. Google Scholar

[5]

F. Dalfovo, S. Giorgini, L. 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.  Google Scholar

[6]

P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262.   Google Scholar

[7]

A. L. Fetter, Rotating trapped Bose-Einstein condensates, Rev. Modern Phys., 81 (2009), 647-691.  doi: 10.1103/RevModPhys.81.647.  Google Scholar

[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.  Google Scholar

[9]

Y. J. Guo, X. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

Q. Han and F. H. Lin, Elliptic Partial Differential Equations, $2^{nd}$ edition, Courant Institute of Mathematical Sciences, New York, 2011. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar

[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.  Google Scholar

[17]

K. Mcleod and J. Serrin, Uniqueness of solutions of semilinear Poisson equations, Proc. Natl. Acad. Sci. USA., 78 (1981), 6592-6595.   Google Scholar

[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.  Google Scholar

[19]

M. Reed and B. Simon, Methods of Modern Mathematical Physics Ⅳ: Analysis of Operators, Academic Press, New York-London, 1978. Google Scholar

[20]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. Google Scholar

[21]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. Zhang, Stability of attractive Bose-Einstein condensates, J. Stat. Phys., 101 (2000), 731-746. doi: 10.1023/A:1026437923987.  Google Scholar

[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.  Google Scholar

[26]

X. Y. Zeng and Y. M. Zhang, Existence and asymptotic behavior for the ground state of quasilinear elliptic equation, arXiv: 1703.00183. Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, 10. New York University, Courant Institute of Mathematical Sciences, New York, 2003. Google Scholar

[5]

F. Dalfovo, S. Giorgini, L. 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.  Google Scholar

[6]

P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262.   Google Scholar

[7]

A. L. Fetter, Rotating trapped Bose-Einstein condensates, Rev. Modern Phys., 81 (2009), 647-691.  doi: 10.1103/RevModPhys.81.647.  Google Scholar

[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.  Google Scholar

[9]

Y. J. Guo, X. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

Q. Han and F. H. Lin, Elliptic Partial Differential Equations, $2^{nd}$ edition, Courant Institute of Mathematical Sciences, New York, 2011. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883. Google Scholar

[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.  Google Scholar

[17]

K. Mcleod and J. Serrin, Uniqueness of solutions of semilinear Poisson equations, Proc. Natl. Acad. Sci. USA., 78 (1981), 6592-6595.   Google Scholar

[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.  Google Scholar

[19]

M. Reed and B. Simon, Methods of Modern Mathematical Physics Ⅳ: Analysis of Operators, Academic Press, New York-London, 1978. Google Scholar

[20]

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. Google Scholar

[21]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

J. Zhang, Stability of attractive Bose-Einstein condensates, J. Stat. Phys., 101 (2000), 731-746. doi: 10.1023/A:1026437923987.  Google Scholar

[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.  Google Scholar

[26]

X. Y. Zeng and Y. M. Zhang, Existence and asymptotic behavior for the ground state of quasilinear elliptic equation, arXiv: 1703.00183. Google Scholar

[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.  Google Scholar

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