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

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

* 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.

Citation: Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure and 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.

[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.

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.

[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.

[1]

Li Wang, Yang Li, Liwei Zhang. A differential equation method for solving box constrained variational inequality problems. Journal of Industrial and Management Optimization, 2011, 7 (1) : 183-198. doi: 10.3934/jimo.2011.7.183

[2]

Chenglin Wang, Jian Zhang. Cross-constrained variational method and nonlinear Schrödinger equation with partial confinement. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021036

[3]

Helin Guo, Huan-Song Zhou. Properties of the minimizers for a constrained minimization problem arising in Kirchhoff equation. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1023-1050. doi: 10.3934/dcds.2020308

[4]

Henri Berestycki, Juncheng Wei. On least energy solutions to a semilinear elliptic equation in a strip. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1083-1099. doi: 10.3934/dcds.2010.28.1083

[5]

Filomena Feo, Pablo Raúl Stinga, Bruno Volzone. The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3269-3298. doi: 10.3934/dcds.2018142

[6]

F. D. Araruna, F. O. Matias, M. P. Matos, S. M. S. Souza. Hidden regularity for the Kirchhoff equation. Communications on Pure and Applied Analysis, 2008, 7 (5) : 1049-1056. doi: 10.3934/cpaa.2008.7.1049

[7]

Vincenzo Ferone, Bruno Volzone. Symmetrization for fractional nonlinear elliptic problems. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022076

[8]

A. El Hamidi. Multiple solutions with changing sign energy to a nonlinear elliptic equation. Communications on Pure and Applied Analysis, 2004, 3 (2) : 253-265. doi: 10.3934/cpaa.2004.3.253

[9]

Erisa Hasani, Kanishka Perera. On the compactness threshold in the critical Kirchhoff equation. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 1-19. doi: 10.3934/dcds.2021106

[10]

Hongyu Ye. Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3857-3877. doi: 10.3934/dcds.2015.35.3857

[11]

Ming Chen, Chongchao Huang. A power penalty method for a class of linearly constrained variational inequality. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1381-1396. doi: 10.3934/jimo.2018012

[12]

Yi Cao, Dong Li, Lihe Wang. The optimal weighted $W^{2, p}$ estimates of elliptic equation with non-compatible conditions. Communications on Pure and Applied Analysis, 2011, 10 (2) : 561-570. doi: 10.3934/cpaa.2011.10.561

[13]

Anran Li, Jiabao Su. Multiple nontrivial solutions to a $p$-Kirchhoff equation. Communications on Pure and Applied Analysis, 2016, 15 (1) : 91-102. doi: 10.3934/cpaa.2016.15.91

[14]

Zhijian Yang, Na Feng, Yanan Li. Robust attractors for a Kirchhoff-Boussinesq type equation. Evolution Equations and Control Theory, 2020, 9 (2) : 469-486. doi: 10.3934/eect.2020020

[15]

Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks and Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465

[16]

Shi Jin, Xu Yang, Guangwei Yuan. A domain decomposition method for a two-scale transport equation with energy flux conserved at the interface. Kinetic and Related Models, 2008, 1 (1) : 65-84. doi: 10.3934/krm.2008.1.65

[17]

Jiaxiang Cai, Juan Chen, Min Chen. Efficient linearized local energy-preserving method for the Kadomtsev-Petviashvili equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2441-2453. doi: 10.3934/dcdsb.2021139

[18]

Vincent Giovangigli, Wen-An Yong. Volume viscosity and internal energy relaxation: Symmetrization and Chapman-Enskog expansion. Kinetic and Related Models, 2015, 8 (1) : 79-116. doi: 10.3934/krm.2015.8.79

[19]

Daomin Cao, Hang Li. High energy solutions of the Choquard equation. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 3023-3032. doi: 10.3934/dcds.2018129

[20]

Kazumasa Fujiwara, Shuji Machihara, Tohru Ozawa. Remark on a semirelativistic equation in the energy space. Conference Publications, 2015, 2015 (special) : 473-478. doi: 10.3934/proc.2015.0473

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (284)
  • HTML views (213)
  • Cited by (3)

Other articles
by authors

[Back to Top]