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Global Carleman estimate for the Kawahara equation and its applications
Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential
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 |
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.
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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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.
|
[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. 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. |
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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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.
|
[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. 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. |
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