Existence and Limit Behavior of Constraint Minimizers for a Varying Non-Local Kirchhoff-Type Energy Functional
Abstract
:1. Introduction and Main Results
2. Preliminaries
- The is a Sobolev space with norm .
- On any compact support set of , the denotes the essentially bounded measurable function space, and is a Hölder continuous function space.
- The , denotes a Sobolev space with norm .
- The symbol → (resp. ⇀) means the strong (resp. weak) convergence.
- The letters , , , , , and represent different positive constants.
3. Proof of Theorems 1 and 2
4. Proof of Theorem 3
- (i)
- There exist a finite ball and a constant such that
- (ii)
- The is a unique maximum of and satisfies for some as . Furthermore, the is a minimum of , that is, .
- (iii)
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Chabrowski, J. On bi-nonlocal problem for elliptic equations with Neumann boundary conditions. J. Anal. Math. 2018, 134, 303–334. [Google Scholar] [CrossRef]
- Corrêa, F.J.S.A.; Figueiredo, G.M. Existence and multiplicity of nontrivial solutions for a bi-nonlocal equation. Adv. Diff. Equ. 2013, 18, 587–608. [Google Scholar] [CrossRef]
- Mao, A.M.; Wang, W.Q. Signed and sign-changing solutions of bi-nonlocal fourth order elliptic problem. J. Math. Phys. 2019, 60, 051513. [Google Scholar] [CrossRef]
- Tian, G.Q.; Suo, H.M.; An, Y.C. Multiple positive solutions for a bi-nonlocal Kirchhoff-Schrödinger-Poisson system with critical growth. Electron. Res. Arch. 2022, 30, 4493–4506. [Google Scholar] [CrossRef]
- Ye, H.Y. The existence of normalized solutions for L2-critical constrained problems related to Kirchhoff equations. Z. Angew. Math. Phys. 2015, 66, 1483–1497. [Google Scholar] [CrossRef]
- Ye, H.Y. The sharp existence of constrained minimizers for a class of nonlinear Kirchhoff equations. Math. Methods Appl. Sci. 2015, 38, 2663–2679. [Google Scholar] [CrossRef]
- Zeng, X.Y.; Zhang, Y.M. Existence and uniqueness of normalized solutions for the Kirchhoff equation. Appl. Math. Lett. 2017, 74, 52–59. [Google Scholar] [CrossRef]
- Meng, X.Y.; Zeng, X.Y. Existence and asymptotic behavior of minimizers for the Kirchhoff functional with periodic potentials. J. Math. Anal. Appl. 2022, 507, 125727. [Google Scholar] [CrossRef]
- Guo, H.L.; Zhang, Y.M.; Zhou, H.S. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Commun. Pur. Appl. Anal. 2018, 17, 1875–1897. [Google Scholar] [CrossRef]
- Li, G.B.; Ye, H.Y. On the concentration phenomenon of L2-subcritical constrained minimizers for a class of Kirchhoff equations with potentials. J. Differ. Equ. 2019, 266, 7101–7123. [Google Scholar] [CrossRef]
- Li, Y.H.; Hao, X.C.; Shi, J.P. The existence of constrained minimizers for a class of nonlinear Kirchhoff-Schrödinger equations with doubly critical exponents in dimension four. Nonlinear Anal. 2019, 186, 99–112. [Google Scholar] [CrossRef]
- Zhu, X.C.; Wang, C.J.; Xue, Y.F. Constraint minimizers of Kirchhoff-Schrödinger energy functionals with L2-subcritical perturbation. Mediterr. J. Math. 2021, 18, 224. [Google Scholar] [CrossRef]
- Zhu, X.C.; Zhang, S.; Wang, C.J.; He, C.X. Blow-up behavior of L2-norm solutions for Kirchhoff equation in a bounded domain. Bull. Malays. Math. Sci. Soc. 2023, 46, 155. [Google Scholar] [CrossRef]
- Guo, H.L.; Zhou, H.S. Properties of the minimizers for a constrained minimization problem arising in Kirchhoff equation. Discret. Cont. Dyn. A 2021, 41, 1023–1050. [Google Scholar] [CrossRef]
- Hu, T.X.; Tang, C.L. Limiting behavior and local uniqueness of normalized solutions for mass critical Kirchhoff equations. Calc. Var. 2021, 60, 210. [Google Scholar] [CrossRef]
- Bao, W.Z.; Cai, Y.Y. Mathematical theory and numerical methods for Bose-Einstein condensation. Kinet. Relat. Model. 2013, 6, 1–135. [Google Scholar] [CrossRef]
- Dalfovo, F.; Giorgini, S.; Pitaevskii, L.P.; Stringari, S. Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys. 1999, 71, 463–512. [Google Scholar] [CrossRef]
- Guo, Y.J.; Seiringer, R. On the mass concentration for Bose-Einstein condensates with attactive interactions. Lett. Math. Phys. 2014, 104, 141–156. [Google Scholar] [CrossRef]
- Guo, Y.J.; Wang, Z.Q.; Zeng, X.Y.; Zhou, H.S. Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials. Nonlinearity 2018, 31, 957–979. [Google Scholar] [CrossRef]
- Guo, Y.J.; Zeng, X.Y.; Zhou, H.S. Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials. Ann. L’Insitut Henri Poincaré C Anal. Non Linéaire 2016, 33, 809–828. [Google Scholar]
- Wang, Q.X.; Zhao, D. Existence and mass concentration of 2D attractive Bose-Einstein condensates with periodic potentials. J. Differ. Equ. 2017, 262, 2684–2704. [Google Scholar] [CrossRef]
- Zhu, X.C.; Wang, C.J. Mass concentration behavior of attractive Bose-Einstein condensates with sinusoidal potential in a circular region. Mediterr. J. Math. 2024, 21, 12. [Google Scholar] [CrossRef]
- Guo, Y.J.; Liang, W.N.; Li, Y. Existence and uniqueness of constraint minimizers for the planar Schrödinger-Poisson system with logarithmic potentials. J. Differ. Equ. 2023, 369, 299–352. [Google Scholar] [CrossRef]
- Guo, Y.J.; Lin, C.S.; Wei, J.C. Local uniqueness and refined spike profiles of ground states for two-dimensional attractive Bose-Einstein condensates. SIAM J. Math. Anal. 2017, 49, 3671–3715. [Google Scholar] [CrossRef]
- Kwong, M.K. Uniqueness of positive solutions of Δu − u + up = 0 in . Arch. Rational Mech. Anal. 1989, 105, 243–266. [Google Scholar] [CrossRef]
- Luo, Y.; Zhu, X.C. Mass concentration behavior of Bose-Einstein condensates with attractive interactions in bounded domains. Anal. Appl. 2020, 99, 2414–2427. [Google Scholar] [CrossRef]
- Gidas, B.; Ni, W.M.; Nirenberg, L. Symmetry of positive solutions of nonlinear elliptic equations in . In Mathematical Analysis and Applications Part A, Advances in Mathematics Supplementary Studies; Academic Press: New York, NY, USA, 1981; Volume 7, pp. 369–402. [Google Scholar]
- Weinstein, M.I. Nonlinear Schrödinger equations and sharp interpolations estimates. Comm. Math. Phys. 1983, 87, 567–576. [Google Scholar] [CrossRef]
- Bartsch, T.; Wang, Z.Q. Existence and multiplicity results for some superlinear elliptic problems on . Comm. Partial. Differ. Equ. 1995, 20, 1725–1741. [Google Scholar] [CrossRef]
- Han, Q.; Lin, F.H. Elliptic Partial Differential Equations; Courant Lecture Note in Mathematics 1; Courant Institute of Mathematical Science/AMS: New York, NY, USA, 2011. [Google Scholar]
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Zhu, X.; Wu, H. Existence and Limit Behavior of Constraint Minimizers for a Varying Non-Local Kirchhoff-Type Energy Functional. Mathematics 2024, 12, 661. https://doi.org/10.3390/math12050661
Zhu X, Wu H. Existence and Limit Behavior of Constraint Minimizers for a Varying Non-Local Kirchhoff-Type Energy Functional. Mathematics. 2024; 12(5):661. https://doi.org/10.3390/math12050661
Chicago/Turabian StyleZhu, Xincai, and Hanxiao Wu. 2024. "Existence and Limit Behavior of Constraint Minimizers for a Varying Non-Local Kirchhoff-Type Energy Functional" Mathematics 12, no. 5: 661. https://doi.org/10.3390/math12050661
APA StyleZhu, X., & Wu, H. (2024). Existence and Limit Behavior of Constraint Minimizers for a Varying Non-Local Kirchhoff-Type Energy Functional. Mathematics, 12(5), 661. https://doi.org/10.3390/math12050661