Existence of normalized solutions for the Chern-Simons-Schrödinger system with critical exponential growth. (English) Zbl 07920140
Summary: This paper is dedicated to studying the existence of normalized solutions for a class of Chern-Simons-Schrödinger system, where the nonlinearity possesses critical exponential growth of Trudinger-Moser type. Under some weak assumptions, we obtain several new existence results by employing more delicate estimates and analytical technical. Our results improve and complement the works of Yao et al. (2023) [25] and Yuan et al. (2022) [27].
MSC:
| 35Jxx | Elliptic equations and elliptic systems |
| 35Axx | General topics in partial differential equations |
| 35Qxx | Partial differential equations of mathematical physics and other areas of application |
Keywords:
Chern-Simons-Schrödinger system; normalized solutions; existence; critical exponential growthReferences:
| [1] | Adimurthi; Yadava, S. L., Multiplicity results for semilinear elliptic equations in a bounded domain of \(\mathbb{R}^2\) involving critical exponent, Ann. Sc. Norm. Super. Pisa, 17, 481-504, 1990 · Zbl 0732.35029 |
| [2] | Berestycki, H.; Lions, P. L., Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal., 82, 347-375, 1983 · Zbl 0556.35046 |
| [3] | Byeon, J.; Huh, H.; Seok, J., Standing waves of nonlinear Schrödinger equations with the gauge field, J. Funct. Anal., 263, 1575-1608, 2012 · Zbl 1248.35193 |
| [4] | Cao, D., Nontrivial solution of semilinear elliptic equation with critical exponent in \(\mathbb{R}^2\), Commun. Partial Differ. Equ., 17, 407-435, 1992 · Zbl 0763.35034 |
| [5] | Chen, S.; Rădulescu, V. D.; Tang, X.; Yuan, S., Normalized solutions for Schrödinger equations with critical exponential growth in \(\mathbb{R}^2\), SIAM J. Math. Anal., 55, 7704-7740, 2023 · Zbl 1534.35168 |
| [6] | Chen, S.; Tang, X., Axially symmetric solutions for the planar Schrödinger-Poisson system with critical exponential growth, J. Differ. Equ., 269, 9144-9174, 2020 · Zbl 1448.35160 |
| [7] | Chen, H.; Xie, W., Existence and multiplicity of normalized solutions for the nonlinear Chern-Simons-Schrödinger equations, Ann. Acad. Sci. Fenn., Math., 45, 429-449, 2020 · Zbl 1437.35286 |
| [8] | de Figueiredo, D. G.; Miyagaki, O. H.; Ruf, B., Elliptic equation in \(\mathbb{R}^2\) with nonlinearities in the critical growth range, Calc. Var. Partial Differ. Equ., 3, 139-153, 1995 · Zbl 0820.35060 |
| [9] | Dunne, G., Self-Dual Chern-Simons Theories, 1995, Springer · Zbl 0834.58001 |
| [10] | Gou, T.; Zhang, Z., Normalized solutions to the Chern-Simons-Schrödinger system, J. Funct. Anal., 280, Article 108894 pp., 2021 · Zbl 1455.35080 |
| [11] | Huh, H., Standing waves of the Schrödinger equation coupled with the Chern-Simons gauge field, J. Math. Phys., 53, Article 063702 pp., 2012 · Zbl 1276.81053 |
| [12] | Jackiw, R.; Pi, S., Soliton solutions to the gauged nonlinear Schrödinger equation on the plane, Phys. Rev. Lett., 64, 2969-2972, 1990 · Zbl 1050.81526 |
| [13] | Jackiw, R.; Pi, S., Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D, 42, 3500-3513, 1990 |
| [14] | Jeanjean, L., Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal., 28, 1633-1659, 1997 · Zbl 0877.35091 |
| [15] | Li, G.; Luo, X., Normalized solutions for the Chern-Simons-Schrödinger equation in \(\mathbb{R}^2\), Ann. Acad. Sci. Fenn., Math., 42, 405-428, 2017 · Zbl 1372.35100 |
| [16] | Liang, W.; Zhai, C., Existence of bound state solutions for the generalized Chern-Simons-Schrödinger system in \(H^1( \mathbb{R}^2)\), Appl. Math. Lett., 100, Article 10628 pp., 2020 · Zbl 1430.35080 |
| [17] | Luo, X., Multiple normalized solutions for a planar gauged nonlinear Schrödinger equation, Z. Angew. Math. Phys., 69, 58, 2018 · Zbl 1393.35024 |
| [18] | Luo, X., Existence and stability of standing waves for a planar gauged nonlinear Schrödinger equation, Comput. Math. Appl., 76, 2701-2709, 2018 · Zbl 1442.35426 |
| [19] | Mao, Y.; Wu, X.; Tang, C., The existence of ground state normalized solutions for Chern-Simons-Schrödinger systems, Acta Math. Sci. Ser. B, 43, 2649-2661, 2023 · Zbl 1538.35002 |
| [20] | Palais, R., The principle of symmetric criticality, Commun. Math. Phys., 69, 19-30, 1979 · Zbl 0417.58007 |
| [21] | Pomponio, A.; Shen, L.; Zeng, X.; Zhang, Y., Generalized Chern-Simons-Schrödinger system with sign-changing steep potential well: critical and subcritical exponential case, J. Geom. Anal., 33, 185, 2023 · Zbl 1514.35166 |
| [22] | Wan, Y.; Tan, J., The existence of nontrivial of solutions to Chern-Simons-Schrödinger systems, Discrete Contin. Dyn. Syst., 37, 2765-2786, 2017 · Zbl 1376.35039 |
| [23] | Weinstein, M. I., Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87, 567-576, 1982 · Zbl 0527.35023 |
| [24] | Yao, S.; Chen, H.; Sun, J., Normalized solutions to the Chern-Simons-Schrödinger system under the nonlinear combined effect, Sci. China Math., 66, 2057-2080, 2023 · Zbl 1522.35427 |
| [25] | Yao, S.; Chen, H.; Sun, J., Two normalized solutions for the Chern-Simons-Schrödinger system with exponential critical growth, J. Geom. Anal., 33, 91, 2023 · Zbl 1511.35148 |
| [26] | Yuan, J., Multiple normalized solutions of Chern-Simons-Schrödinger system, Nonlinear Differ. Equ. Appl., 22, 1801-1816, 2015 · Zbl 1328.35224 |
| [27] | Yuan, S.; Tang, X.; Chen, S., Normalized solutions of Chern-Simons-Schrödinger equations with exponential critical growth, J. Math. Anal. Appl., 516, Article 126523 pp., 2022 · Zbl 1498.35263 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
