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March  2021, 41(3): 1023-1050. doi: 10.3934/dcds.2020308

Properties of the minimizers for a constrained minimization problem arising in Kirchhoff equation

Helin Guo and  Huan-Song Zhou ,

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

* Corresponding author: Huan-Song Zhou

Received  January 2020 Revised  June 2020 Published  March 2021 Early access  August 2020

Fund Project: This work was supported by NFSC (Grant Nos. 11871387, 11931012) and partially by the fundamental Research Funds for the Central Universities (WUT: 2019IVA106, 2019IB009)

Let
$ a>0,b>0 $
 and
$ V(x)\geq0 $
 be a coercive function in
$ \mathbb R^2 $
. We study the following constrained minimization problem on a suitable weighted Sobolev space
$ \mathcal{H} $
:
$ \begin{equation*} e_{a}(b): = \inf\left\{E_{a}^{b}(u):u\in\mathcal{H}\ \mbox{and}\ \int_{\mathbb R^{2}}|u|^{2}dx = 1\right\}, \end{equation*} $
where
$ E_{a}^{b}(u) $
 is a Kirchhoff type energy functional defined on
$ \mathcal{H} $
 by
$ \begin{equation*} E_{a}^{b}(u) = \frac{1}{2}\int_{\mathbb R^{2}}[|\nabla u|^{2}+V(x)u^{2}]dx+\frac{b}{4}\left(\int_{\mathbb R^{2}}|\nabla u|^{2}dx\right)^{2}-\frac{a}{4}\int_{\mathbb R^{2}}|u|^{4}dx. \end{equation*} $
It is known that, for some
$ a^{\ast}>0 $
,
$ e_{a}(b) $
 has no minimizer if
$ b = 0 $
 and
$ a\geq a^{\ast} $
, but
$ e_{a}(b) $
 has always a minimizer for any
$ a\geq0 $
 if
$ b>0 $
. The aim of this paper is to investigate the limit behaviors of the minimizers of
$ e_{a}(b) $
 as
$ b\rightarrow0^{+} $
. Moreover, the uniqueness of the minimizers of
$ e_{a}(b) $
 is also discussed for
$ b $
 close to 0.
Keywords: Kirchhoff type equation, constrained variational problem, energy estimates, mass concentration, uniqueness.
Mathematics Subject Classification: Primary:35J20, 35J60;Secondary:35A02.
Citation: Helin Guo, Huan-Song Zhou. Properties of the minimizers for a constrained minimization problem arising in Kirchhoff equation. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1023-1050. doi: 10.3934/dcds.2020308
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]

W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6 (2013), 1-135.  doi: 10.3934/krm.2013.6.1.  Google Scholar

[3]

T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $R^N$: Existence and multiplicity results, Comm. Partial Differential Equations, 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149.  Google Scholar

[4]

D. Cao, S. Li and P. Luo, Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 4037-4063.  doi: 10.1007/s00526-015-0930-2.  Google Scholar

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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, Nonlinear perturbations of the Kirchhoff equation, Comm. Pure Appl. Math., 47 (1994), 1005-1029.  doi: 10.1002/cpa.3160470705.  Google Scholar

[7]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in Rn, Mathematical Analysis and Applications, Part A, pp. 369–402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981.  Google Scholar

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D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, "Nauka", Moscow, 1989.  Google Scholar

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Y. 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

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Y. Guo, C. Lin and J. Wei, Local uniqueness and refined spike profiles of ground states for two-dimensional attractive Bose-Einstein condensates, SIMA J. Math. Anal., 49 (2017), 3671-3715.  doi: 10.1137/16M1100290.  Google Scholar

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Y. Guo, X. 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

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Y. Guo, X. Zeng and H.-S. Zhou, Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 809-828.  doi: 10.1016/j.anihpc.2015.01.005.  Google Scholar

[13]

H. Guo, Y. Zhang and H.-S. Zhou, Blow-up solutions for a Kirchhoff type elliptic equations with trapping potential, Commun. Pure Appl. Anal., 17 (2018), 1875-1897.  doi: 10.3934/cpaa.2018089.  Google Scholar

[14]

H. Guo and H.-S. Zhou, A constrained variational problem arising in attractive Bose-Einstein condensate with ellipse-shaped potential, Appl. Math. Lett., 87 (2019), 35-41.  doi: 10.1016/j.aml.2018.07.023.  Google Scholar

[15]

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

[16]

Y. He and G. Li, Standing waves for a class of Kirchhoff type problems in $\Bbb{R}^3$ involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3067-3106.  doi: 10.1007/s00526-015-0894-2.  Google Scholar

[17]

X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\Bbb R^3$, J. Differential Equations, 252 (2012), 1813-1834.  doi: 10.1016/j.jde.2011.08.035.  Google Scholar

[18]

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

[19]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^{p} = 0$ in $\mathbb R^{N}$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar

[20]

G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\Bbb R^3$, J. Differential Equations, 257 (2014), 566-600.  doi: 10.1016/j.jde.2014.04.011.  Google Scholar

[21]

P. Luo, S. Peng and C. Wang, Uniqueness of positive solutions with concentration for the Schrödinger-Newton problem, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 60, 41 pp. doi: 10.1007/s00526-020-1726-6.  Google Scholar

[22]

K. Mcleod and J. Serrin, Uniqueness of solutions of semilinear Poisson equations, Proc. Natl. Acad. Sci. USA, 78 (1981), 6592-6595.  doi: 10.1073/pnas.78.11.6592.  Google Scholar

[23]

W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.  doi: 10.1002/cpa.3160440705.  Google Scholar

[24] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York-London, 1978.   Google Scholar
[25]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576.   Google Scholar

[26]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[27]

X. 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

[28]

X. Zeng and Y. 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

[29]

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

W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6 (2013), 1-135.  doi: 10.3934/krm.2013.6.1.  Google Scholar

[3]

T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $R^N$: Existence and multiplicity results, Comm. Partial Differential Equations, 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149.  Google Scholar

[4]

D. Cao, S. Li and P. Luo, Uniqueness of positive bound states with multi-bump for nonlinear Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 4037-4063.  doi: 10.1007/s00526-015-0930-2.  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, Nonlinear perturbations of the Kirchhoff equation, Comm. Pure Appl. Math., 47 (1994), 1005-1029.  doi: 10.1002/cpa.3160470705.  Google Scholar

[7]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in Rn, Mathematical Analysis and Applications, Part A, pp. 369–402, Adv. in Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981.  Google Scholar

[8]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, "Nauka", Moscow, 1989.  Google Scholar

[9]

Y. 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

[10]

Y. Guo, C. Lin and J. Wei, Local uniqueness and refined spike profiles of ground states for two-dimensional attractive Bose-Einstein condensates, SIMA J. Math. Anal., 49 (2017), 3671-3715.  doi: 10.1137/16M1100290.  Google Scholar

[11]

Y. Guo, X. 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

[12]

Y. Guo, X. Zeng and H.-S. Zhou, Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 809-828.  doi: 10.1016/j.anihpc.2015.01.005.  Google Scholar

[13]

H. Guo, Y. Zhang and H.-S. Zhou, Blow-up solutions for a Kirchhoff type elliptic equations with trapping potential, Commun. Pure Appl. Anal., 17 (2018), 1875-1897.  doi: 10.3934/cpaa.2018089.  Google Scholar

[14]

H. Guo and H.-S. Zhou, A constrained variational problem arising in attractive Bose-Einstein condensate with ellipse-shaped potential, Appl. Math. Lett., 87 (2019), 35-41.  doi: 10.1016/j.aml.2018.07.023.  Google Scholar

[15]

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

[16]

Y. He and G. Li, Standing waves for a class of Kirchhoff type problems in $\Bbb{R}^3$ involving critical Sobolev exponents, Calc. Var. Partial Differential Equations, 54 (2015), 3067-3106.  doi: 10.1007/s00526-015-0894-2.  Google Scholar

[17]

X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\Bbb R^3$, J. Differential Equations, 252 (2012), 1813-1834.  doi: 10.1016/j.jde.2011.08.035.  Google Scholar

[18]

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

[19]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^{p} = 0$ in $\mathbb R^{N}$, Arch. Ration. Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar

[20]

G. Li and H. Ye, Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\Bbb R^3$, J. Differential Equations, 257 (2014), 566-600.  doi: 10.1016/j.jde.2014.04.011.  Google Scholar

[21]

P. Luo, S. Peng and C. Wang, Uniqueness of positive solutions with concentration for the Schrödinger-Newton problem, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 60, 41 pp. doi: 10.1007/s00526-020-1726-6.  Google Scholar

[22]

K. Mcleod and J. Serrin, Uniqueness of solutions of semilinear Poisson equations, Proc. Natl. Acad. Sci. USA, 78 (1981), 6592-6595.  doi: 10.1073/pnas.78.11.6592.  Google Scholar

[23]

W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991), 819-851.  doi: 10.1002/cpa.3160440705.  Google Scholar

[24] M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York-London, 1978.   Google Scholar
[25]

M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576.   Google Scholar

[26]

M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[27]

X. 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

[28]

X. Zeng and Y. 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

[29]

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

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