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

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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 and 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.
[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.
[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.
[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.
[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, Nonlinear perturbations of the Kirchhoff equation, Comm. Pure Appl. Math., 47 (1994), 1005-1029. doi: 10.1002/cpa.3160470705.
[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.
[8]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, "Nauka", Moscow, 1989.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	Q. Han and F. Lin, Elliptic Partial Differential Equations, $2^nd$ edition, Courant Institute of Mathematical Sciences, New York, 2011.
[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.
[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.
[18]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
[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.
[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.
[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.
[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.
[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.
[24]	M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York-London, 1978.
[25]	M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576.
[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.
[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.
[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.
[29]	J. Zhang, Stability of attractive Bose-Einstein condensates, J. Stat. Phys., 101 (2000), 731-746. doi: 10.1023/A:1026437923987.

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]	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.
[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.
[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.
[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, Nonlinear perturbations of the Kirchhoff equation, Comm. Pure Appl. Math., 47 (1994), 1005-1029. doi: 10.1002/cpa.3160470705.
[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.
[8]	D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, "Nauka", Moscow, 1989.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	Q. Han and F. Lin, Elliptic Partial Differential Equations, $2^nd$ edition, Courant Institute of Mathematical Sciences, New York, 2011.
[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.
[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.
[18]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
[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.
[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.
[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.
[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.
[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.
[24]	M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York-London, 1978.
[25]	M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1982/83), 567-576.
[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.
[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.
[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.
[29]	J. Zhang, Stability of attractive Bose-Einstein condensates, J. Stat. Phys., 101 (2000), 731-746. doi: 10.1023/A:1026437923987.

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