Existence and limit behavior of constraint minimizers for elliptic equations with two nonlocal terms

Xincai Zhu; Yajie Zhu; Xincai Zhu; Yajie Zhu

doi:10.3934/era.2024230

Electronic Research Archive

2024, Volume 32, Issue 8: 4991-5009. doi: 10.3934/era.2024230

Previous Article Next Article

Research article

Existence and limit behavior of constraint minimizers for elliptic equations with two nonlocal terms

Xincai Zhu ^,,
Yajie Zhu

School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China

Received: 10 April 2024 Revised: 15 July 2024 Accepted: 31 July 2024 Published: 19 August 2024

This paper is devoted to studying constraint minimizers for a class of elliptic equations with two nonlocal terms. Using the methods of constrained variation and energy estimation, we analyze the existence, non-existence, and limit behavior of minimizers for the related minimization problem. Our work extends and enriches the study of bi-nonlocal problems.
- elliptic equation,
- bi-nonlocal problems,
- constraint minimizer,
- limit behavior
Citation: Xincai Zhu, Yajie Zhu. Existence and limit behavior of constraint minimizers for elliptic equations with two nonlocal terms[J]. Electronic Research Archive, 2024, 32(8): 4991-5009. doi: 10.3934/era.2024230

Related Papers:

Abstract

This paper is devoted to studying constraint minimizers for a class of elliptic equations with two nonlocal terms. Using the methods of constrained variation and energy estimation, we analyze the existence, non-existence, and limit behavior of minimizers for the related minimization problem. Our work extends and enriches the study of bi-nonlocal problems.

References

[1]	G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
[2]	C. Alves, F. Corrêa, T. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49 (2005), 85–93. https://doi.org/10.1016/j.camwa.2005.01.008 doi: 10.1016/j.camwa.2005.01.008
[3]	J. Bebernos, A. Lacey, Global existence and finite time blow-up for a class of nonlocal parabolic problems, Adv. Differ. Equations, 2 (1997), 927–953. https://doi.org/10.57262/ade/1366638678 doi: 10.57262/ade/1366638678
[4]	E. Caglioti, P. Lions, C. Maichiori, M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, Commun. Math. Phys., 143 (1992), 501–525. https://doi.org/10.1007/BF02099262 doi: 10.1007/BF02099262
[5]	G. Carrier, On the non-linear vibration problem of the elastic string, Quart. Appl. Math., 3 (1945), 157–165. https://doi.org/10.1090/qam/12351 doi: 10.1090/qam/12351
[6]	J. Carrillo, On a nonlocal elliptic equation with decreasing nonlinearity arising in plasma physics and heat conduction, Nonlinear Anal. Theory Methods Appl., 32 (1998), 97–115. https://doi.org/10.1016/S0362-546X(97)00455-0 doi: 10.1016/S0362-546X(97)00455-0
[7]	J. Chabrowski, On bi-nonlocal problem for elliptic equations with Neumann boundary conditions, J. Anal. Math., 134 (2018), 303–334. https://doi.org/10.1007/s11854-018-0011-5 doi: 10.1007/s11854-018-0011-5
[8]	G. Tian, H. Suo, Y. An, Multiple positive solutions for a bi-nonlocal Kirchhoff-Schrödinger-Poisson system with critical growth, Electron. Res. Arch., 30 (2022), 4493–4506. https://doi.org/10.3934/era.2022228 doi: 10.3934/era.2022228
[9]	M. Xiang, B. Zhang, V. Rǎdulescu, Existence of solutions for a bi-nonlocal fractional $p$-Kirchhoff type problem, Comput. Math. Appl., 71 (2016), 255–266. https://doi.org/10.1016/j.camwa.2015.11.017 doi: 10.1016/j.camwa.2015.11.017
[10]	F. Júlio, S. A. Corrêa, G. Figueiredo, On an elliptic equation of $p$-Kirchhoff type via variational methods, Bull. Aust. Math. Soc., 74 (2006), 263–277. https://doi.org/10.1017/S000497270003570X doi: 10.1017/S000497270003570X
[11]	M. Hamdani, L. Mbarki, M. Allaoui, O. Darhouche, D. Repovš, Existence and multiplicity of solutions involving the $p(x)$-Laplacian equations: On the effect of two nonlocal terms, Discrete Contin. Dyn. Syst. Ser. S, 16 (2023), 1452–1467. https://doi.org/10.3934/dcdss.2022129 doi: 10.3934/dcdss.2022129
[12]	A. Mao, W. Q. Wang, Signed and sign-changing solutions of bi-nonlocal fourth order elliptic problem, J. Math. Phys., 60 (2019), 051513. https://doi.org/10.1063/1.5093461 doi: 10.1063/1.5093461
[13]	F. Dalfovo, S. Giorgini, L. Pitaevskii, S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463–512. https://doi.org/10.1103/RevModPhys.71.463 doi: 10.1103/RevModPhys.71.463
[14]	E. Gross, Hydrodynamics of a superfluid condensate, J. Math. Phys., 4 (1963), 195–207. https://doi.org/10.1063/1.1703944 doi: 10.1063/1.1703944
[15]	Y. Guo, R. Seiringer, On the mass concentration for Bose-Einstein condensates with attactive interactions, Lett. Math. Phys., 104 (2014), 141–156. https://doi.org/10.1007/s11005-013-0667-9 doi: 10.1007/s11005-013-0667-9
[16]	Y. Guo, Z. Wang, X. Zeng, H. Zhou, Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials, Nonlinearity, 31 (2018), 957–979. https://doi.org/10.1088/1361-6544/aa99a8 doi: 10.1088/1361-6544/aa99a8
[17]	H. Zhou, Y. Guo, X. Zeng, Energy estimates and symmetry breaking in attactive Bose-Einstein condensates with ring-shaped potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 809–828. https://doi.org/10.1016/j.anihpc.2015.01.005 doi: 10.1016/j.anihpc.2015.01.005
[18]	Q. Wang, D. Zhao, Existence and mass concentration of 2D attractive Bose-Einstein condensates with periodic potentials, J. Differ. Equations, 262 (2017), 2684–2704. https://doi.org/10.1016/j.jde.2016.11.004 doi: 10.1016/j.jde.2016.11.004
[19]	Y. Guo, W. Liang, Y. Li, Existence and uniqueness of constraint minimizers for the planar Schrödinger-Poisson system with logarithmic potentials, J. Differ. Equations, 369 (2023), 299–352. https://doi.org/10.1016/j.jde.2023.06.007 doi: 10.1016/j.jde.2023.06.007
[20]	Y. Guo, C. Lin, J. Wei, Local uniqueness and refined spike profiles of ground states for two-dimensional attractive Bose-Einstein condensates, SIAM J. Math. Anal., 49 (2017), 3671–3715. https://doi.org/10.1137/16M1100290 doi: 10.1137/16M1100290
[21]	H. Ye, The existence of normalized solutions for $L^{2}$-critical constrained problems related to Kirchhoff equations, Z. Angew. Math. Phys., 66 (2015), 1483–1497. https://doi.org/10.1007/s00033-014-0474-x doi: 10.1007/s00033-014-0474-x
[22]	H. Ye, The sharp existence of constrained minimizers for a class of nonlinear Kirchhoff equations, Math. Methods Appl. Sci., 38 (2015), 2663–2679. https://doi.org/10.1002/mma.3247 doi: 10.1002/mma.3247
[23]	X. Meng, X. Zeng, Existence and asymptotic behavior of minimizers for the Kirchhoff functional with periodic potentials, J. Math. Anal. Appl., 507 (2022), 125727. https://doi.org/10.1016/j.jmaa.2021.125727 doi: 10.1016/j.jmaa.2021.125727
[24]	H. Guo, Y. Zhang, H. Zhou, Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential, Commun. Pure Appl. Anal., 17 (2018), 1875–1897. https://doi.org/10.3934/cpaa.2018089 doi: 10.3934/cpaa.2018089
[25]	X. He, W. M. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb{R}^{3}$, J. Differ. Equations, 2 (2012), 1813–1834. https://doi.org/10.1016/j.jde.2011.08.035 doi: 10.1016/j.jde.2011.08.035
[26]	Y. Li, X. Hao, J. Shi, The existence of constrained minimizers for a class of nonlinear Kirchhoff-Schrödinger equations with doubly critical exponents in dimension four, Nonlinear Anal., 186 (2019), 99–112. https://doi.org/10.1016/j.na.2018.12.010 doi: 10.1016/j.na.2018.12.010
[27]	G. Li, H. Ye, On the concentration phenomenon of L$^{2}$-subcritical constrained minimizers for a class of Kirchhoff equations with potentials, J. Differ. Equations, 266 (2019), 7101–7123. https://doi.org/10.1016/j.jde.2018.11.024 doi: 10.1016/j.jde.2018.11.024
[28]	X. Zhu, C. Wang, Y. Xue, Constraint minimizers of Kirchhoff-Schrödinger energy functionals with $L^{2}$-subcritical perturbation, Mediterr. J. Math., 18 (2021), 224. https://doi.org/10.1007/s00009-021-01835-0 doi: 10.1007/s00009-021-01835-0
[29]	T. Hu, C. Tang, Limiting behavior and local uniqueness of normalized solutions for mass critical Kirchhoff equations, Calc. Var., 60 (2021), 210. https://doi.org/10.1007/s00526-021-02018-1 doi: 10.1007/s00526-021-02018-1
[30]	X. Zeng, Y. Zhang, Existence and uniqueness of normalized solutions for the Kirchhoff equation, Appl. Math. Lett., 74 (2017), 52–59. https://doi.org/10.1016/j.aml.2017.05.012 doi: 10.1016/j.aml.2017.05.012
[31]	M. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in $\mathbb{R}^n$, Arch. Rational Mech. Anal., 105 (1989), 243–266. https://doi.org/10.1007/BF00251502 doi: 10.1007/BF00251502
[32]	B. Gidas, W. Ni, L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $\mathbb{R}^{n}$, Math. Anal. Appl. Part A: Adv. Math. Suppl. Stud., 7 (1981), 369–402.
[33]	Y. Luo, X. Zhu, Mass concentration behavior of Bose-Einstein condensates with attractive interactions in bounded domains, Anal. Appl., 99 (2020), 2414–2427. https://doi.org/10.1080/00036811.2019.1566529 doi: 10.1080/00036811.2019.1566529
[34]	B. Noris, H. Tavares, G. Verzini, Existence and orbital stability of the ground states with prescribed mass for the $L^2$-critical and supercritical NLS on bounded domains, Analysis $ & $ PDE, 7 (2014), 1807–1838. https://doi.org/10.2140/apde.2014.7.1807 doi: 10.2140/apde.2014.7.1807
[35]	M. Willem, Minimax Theorems, Birkhäuser Boston Inc, Boston, 1996. https://doi.org/10.1007/978-1-4612-4146-1
[36]	Q. Han, F. Lin, Elliptic Partial Differential Equations, American Mathematical Soc., 2011.
[37]	M. Esteban, P. Lions, Existence and non-existence results for semilinear elliptic problems in unbounded domains, Proc. R. Soc. Edinburgh Sect. A: Math., 93 (1982), 1–14. https://doi.org/10.1017/S0308210500031607 doi: 10.1017/S0308210500031607
[38]	W. Ni, I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Commun. Pure Appl. Math., 44 (1991), 819–851. https://doi.org/10.1002/cpa.3160440705 doi: 10.1002/cpa.3160440705

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)