| Citation: |
Liting Jiang, Guofeng Che, Haibo Chen. GROUND STATE SOLUTIONS FOR THE CHERN–SIMONS–SCHRÖDINGER SYSTEM WITH HARTREE–TYPE NONLINEARITY IN $ \mathbb{R}^{2}$[J]. Journal of Applied Analysis & Computation, 2025, 15(4): 2195-2211. doi: 10.11948/20240421
|
GROUND STATE SOLUTIONS FOR THE CHERN–SIMONS–SCHRÖDINGER SYSTEM WITH HARTREE–TYPE NONLINEARITY IN $ \mathbb{R}^{2}$
-
Abstract
In this paper, we consider the following Chern–Simons–Schrödinger system with Hartree–type nonlinearity in $\mathbb{R}^{2}$
$\left\{\begin{array}{l}-\Delta u+ (1+\mu V(x))u+ A_{0}u+ A_{1}^{2}u+ A_{2}^{2}u=\left(|x|^{-\alpha}*|u|^{p}\right)|u|^{p-2}u, \\\partial_{1}A_{0}=A_{2}u^{2}, ~\partial_{2}A_{0}=-A_{1}u^{2}, \\\partial_{1}A_{2}-\partial_{2}A_{1}=-\frac{1}{2}|u|^{2}, ~\partial_{1}A_{1}+\partial_{2}A_{2}=0, \end{array}\right.$
where $p>3$, $\alpha\in(0, 2)$, $\mu>0$ is a parameter, $V(x)$ is a nonnegative continuous potential well satisfying some conditions and $*$ is a notation for the convolution of two functions in $\mathbb{R}^{2}$. By using the Nehari manifold technique and the concentration compactness principle, we obtain the existence of ground state solutions for the above problem when the parameter $\mu$ is sufficiently large. Furthermore, the concentration behaviors of these solutions are also explored.
-
-
References
[1] N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z., 2004, 248, 423–443. [2] T. Bartsch and Z. Q. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math. Phys., 2000, 51, 366–384. doi: 10.1007/PL00001511 [3] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 1983, 88, 486–490. doi: 10.1090/S0002-9939-1983-0699419-3 [4] J. Byeon, H. Huh and J. Seok, Standing waves of nonlinear Schrödinger equations with the gauge field, J. Funct. Anal., 2012, 263, 1575–1608. doi: 10.1016/j.jfa.2012.05.024 [5] D. Chae and K. Choe, Global existence in the Cauchy problem of the relativistic Chern–Simons–Higgs theory, Nonlinearity, 2002, 15, 747–758. doi: 10.1088/0951-7715/15/3/314 [6] G. Che, Y. Su and T. F. Wu, Bound state positive solutions for a Hartree system with nonlinear couplings, Appl. Anal., 2024, 103, 1176–1214. doi: 10.1080/00036811.2023.2236641 [7] G. Che, J. Sun and T. F. Wu, Non–radial ground state solutions for fractional Schrödinger–Poisson systems in $\mathbb{R}^{2}$, Ann. Mat. Pura Appl., 2024, 203, 2863–2888. doi: 10.1007/s10231-024-01470-y [8] G. Che and T. F. Wu, Multiple positive solutions for a class of Kirchhoff type equations with indefinite nonlinearities, Adv. Nonlinear Anal., 2022, 11, 598–619. [9] G. Che and T. F. Wu, Three positive solutions for Kirchhoff problems with steep potential well and concave–convex nonlinearities, Appl. Math. Lett., 2021, 121, 107348. doi: 10.1016/j.aml.2021.107348 [10] Z. Chen, X. H. Tang and J. Zhang, Sign–changing multi–bump solutions for the Chern–Simons–Schrödinger equations in $\mathbb{R}^{2}$, Adv. Nonlinear Anal., 2020, 9, 1066–1091. [11] M. Clapp and D. Salazar, Positive and sign changing solutions to a nonlinear Choquard equation, J. Math. Anal. Appl., 2013, 407, 1–15. doi: 10.1016/j.jmaa.2013.04.081 [12] G. Dunne, Self–Dual Chern–Simons Theories, Springer, Berlin, 1995. [13] E. Gagliardo, Ulteriori proprietà di alcune classi di funzioni in più variabili, Ric. Mat., 1959, 8, 24–51. [14] T. X. Gou and Z. T. Zhang, Normalized solutions to the Chern–Simons–Schrödinger system, J. Funct. Anal., 2021, 280, 108894. doi: 10.1016/j.jfa.2020.108894 [15] J. Hong, P. Kim and P. Pac, Multivortex solutions of the abelian Chern–Simons–Higgs theory, Phys. Rev. Lett., 1990, 64, 2230–2233. doi: 10.1103/PhysRevLett.64.2230 [16] H. Huh, Local and global solutions of the Chern–Simons–Higgs system, J. Funct. Anal., 2007, 242, 526–549. doi: 10.1016/j.jfa.2006.09.009 [17] H. Huh, Standing waves of the Schrödinger equation coupled with the Chern–Simons gauge field, J. Math. Phys., 2012, 53, 063702. doi: 10.1063/1.4726192 [18] R. Jackiw and S. Y. Pi, Classical and quantal nonrelativistic Chern–Simons theory, Phys. Rev. D., 1990, 42, 3500–3513. doi: 10.1103/PhysRevD.42.3500 [19] R. Jackiw and E. Weinberg, Self–dual Chern–Simons vortices, Phys. Rev. Lett., 1990, 64, 2234–2237. doi: 10.1103/PhysRevLett.64.2234 [20] L. Jiang, G. Che and T. F. Wu, Multiple solutions for the Chern–Simons–Schrödinger equation with indefinite nonlinearities in $\mathbb{R}^{2}$, Mediterr. J. Math., 2024, 21, 197. doi: 10.1007/s00009-024-02739-5 [21] S. Jiang and S. Liu, Standing waves for 6–superlinear Chern–Simons–Schrödinger systems with indefinite potentials, Nonlinear Anal., 2023, 230, 113234. doi: 10.1016/j.na.2023.113234 [22] J. C. Kang and C. L. Tang, Ground states for Chern–Simons–Schrödinger system with nonperiodic potential, J. Fixed Point Theory Appl., 2023, 25, 37. doi: 10.1007/s11784-022-01043-7 [23] E. H. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Stud. Appl. Math., 1977, 57, 93–105. doi: 10.1002/sapm197757293 [24] E. H. Lieb and M. Loss, Analysis, 2nd Edition, Graduate Studies in Mathematics, American Mathematical Society, Providence, 2001. [25] E. H. Lieb and B. Simon, The Hartree–Fock theory for Coulomb systems, Comm. Math. Phys., 1977, 53, 185–194. doi: 10.1007/BF01609845 [26] P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 1980, 4, 1063–1072. doi: 10.1016/0362-546X(80)90016-4 [27] P. L. Lions, The concentration–compactness principle in the calculus of variations, the Locally Compact Case, Part Ⅰ, Ann. Inst. H. Poincaré–Anal. Non Linéaire, 1984, 1, 109–145. doi: 10.1016/s0294-1449(16)30428-0 [28] Z. S. Liu, Z. G. Ou and J. J. Zhang, Existence and multiplicity of sign–changing standing waves for a gagued nonlinear Schrödinger equation in $\mathbb{R}^{2}$, Nonlinearity, 2019, 32, 3082–3111. doi: 10.1088/1361-6544/ab1bc4 [29] X. Luo, Existence and stability of standing waves for a planar gauged nonlinear Schrödinger equation, Comput. Math. Appl., 2018, 76, 2701–2709. doi: 10.1016/j.camwa.2018.09.003 [30] L. Ma and L. Zhao, Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. Ration. Mech. Anal., 2010, 195, 455–467. doi: 10.1007/s00205-008-0208-3 [31] V. Moroz and J. Van Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 2013, 265, 153–184. doi: 10.1016/j.jfa.2013.04.007 [32] L. Nirenberg, Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math., 1955, 8, 648–674. doi: 10.1002/cpa.3160080414 [33] R. Penrose, On gravity's role in quantum state reduction, Gen. Relativity Gravitation, 1996, 28, 581–600. doi: 10.1007/BF02105068 [34] A. Pomponio and D. Ruiz, A variational analysis of a gauged nonlinear Schrödinger equation, J. Eur. Math. Soc., 2015, 17, 1463–1486. doi: 10.4171/jems/535 [35] A. Pomponio, L. Shen, X. Zeng and Y. Zhang, Generalized Chern–Simons–Schrödinger system with sign–changing steep potential well: Critical and subcritical exponential case, J. Geom. Anal., 2023, 33, 185. doi: 10.1007/s12220-023-01244-7 [36] T. Ricciardi and G. Tarantello, Vortices in the Maxwell–Chern–Simons theory, Comm. Pure Appl. Math., 2000, 53, 811–851. doi: 10.1002/(SICI)1097-0312(200007)53:7<811::AID-CPA2>3.0.CO;2-F [37] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 1983, 87, 567–576. doi: 10.1007/BF01208265 [38] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Boston, 1996. [39] M. Yang and Y. Wei, Existence and multiplicity of solutions for nonlinear Schrödinger equations with magnetic field and Hartree type nonlinearities, J. Math. Anal. Appl., 2013, 403, 680–694. doi: 10.1016/j.jmaa.2013.02.062 [40] S. Yao, H. Chen and J. Sun, Two normalized solutions for the Chern–Simons–Schrödinger system with exponential critical growth, J. Geom. Anal., 2023, 33, 91. doi: 10.1007/s12220-022-01142-4 [41] S. Zhu, G. Che and H. Chen, Existence and asymptotic behavior of normalized solutions for Choquard equations with Kirchhoff perturbation, Rev. Real Acad. Cienc. Exactas F., 2025, 119, 2. -
-
Export File
Citation
Liting Jiang, Guofeng Che, Haibo Chen. GROUND STATE SOLUTIONS FOR THE CHERN–SIMONS–SCHRÖDINGER SYSTEM WITH HARTREE–TYPE NONLINEARITY IN $ \mathbb{R}^{2}$[J]. Journal of Applied Analysis & Computation, 2025, 15(4): 2195-2211. doi: 10.11948/20240421
DownLoad: