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June  2021, 14(6): 1931-1944. doi: 10.3934/dcdss.2021016

Existence of nontrivial solutions to Chern-Simons-Schrödinger system with indefinite potential

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: Chunlei Tang

Received  October 2020 Revised  January 2021 Published  June 2021 Early access  January 2021

Fund Project: This work is supported by National Natural Science Foundation of China (No. 11971393)

We consider a class of Chern-Simons-Schrödinger system
$ \begin{align*} \begin{cases} -\Delta u+V(x) u+A_{0}u+\sum\limits_{j = 1}^{2} A_{j}^{2}u = g(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{cases} \end{align*} $
where
$ V $
is coercive sign-changing potential and
$ f $
satisfies some suitable conditions. Due to lack of the mountain pass geometry and the link geometry for the corresponding variational functional, we obtain the existence of nontrivial solutions via the local link theorem.
Citation: Jincai Kang, Chunlei Tang. Existence of nontrivial solutions to Chern-Simons-Schrödinger system with indefinite potential. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1931-1944. doi: 10.3934/dcdss.2021016
References:
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T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^{N}$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149.  Google Scholar

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J. ByeonH. Huh and J. Seok, Standing waves of nonlinear Schrödinger equations with the gauge field, J. Funct. Anal., 263 (2012), 1575-1608.  doi: 10.1016/j.jfa.2012.05.024.  Google Scholar

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Y. DengS. Peng and W. Shuai, Nodal standing waves for a gauged nonlinear Schrödinger equation in $\mathbb{R}^{2}$, J. Differential Equations, 264 (2018), 4006-4035.  doi: 10.1016/j.jde.2017.12.003.  Google Scholar

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G.-D. Li, Y.-Y. Li and C.-L. Tang, Existence and concentrate behavior of positive solutions for Chern-Simons-Schrödinger systems with critical growth, Complex Var. Elliptic Equ., (2020). doi: 10.1080/17476933.2020.1723564.  Google Scholar

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G. Li and X. Luo, Normalized solutions for the Chern-Simons-Schrödinger equation in $\mathbb{R}^{2}$, Ann. Acad. Sci. Fenn. Math., 42 (2017), 405-428.  doi: 10.5186/aasfm.2017.4223.  Google Scholar

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G. LiX. Luo and W. Shuai, Sign-changing solutions to a gauged nonlinear Schrödinger equation, J. Math. Anal. Appl., 455 (2017), 1559-1578.  doi: 10.1016/j.jmaa.2017.06.048.  Google Scholar

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S. J. Li and M. Willem, Applications of local linking to critical point theory, J. Math. Anal. Appl., 189 (1995), 6-32.  doi: 10.1006/jmaa.1995.1002.  Google Scholar

[21]

W. Liang and C. Zhai, Existence of bound state solutions for the generalized Chern-Simons-Schrödinger system in $H^1(\mathbb{R}^2)$, Appl. Math. Lett., 100 (2020), 106028, 7 pp. doi: 10.1016/j.aml.2019.106028.  Google Scholar

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S. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9.  doi: 10.1007/s00526-011-0447-2.  Google Scholar

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B. Liu and P. Smith, Global wellposedness of the equivariant Chern-Simons-Schrödinger equation, Rev. Mat. Iberoam., 32 (2016), 751-794.  doi: 10.4171/RMI/898.  Google Scholar

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B. Liu, P. Smith and D. Tataru, Local wellposedness of Chern-Simons-Schrödinger, Int. Math. Res. Not. IMRN, (2014), 6341–6398. doi: 10.1093/imrn/rnt161.  Google Scholar

[25]

X. Luo, Existence and stability of standing waves for a planar gauged nonlinear Schrödinger equation, Comput. Math. Appl., 76 (2018), 2701-2709.  doi: 10.1016/j.camwa.2018.09.003.  Google Scholar

[26]

A. Pomponio and D. Ruiz, Boundary concentration of a gauged nonlinear Schrödinger equation on large balls, Calc. Var. Partial Differential Equations, 53 (2015), 289-316.  doi: 10.1007/s00526-014-0749-2.  Google Scholar

[27]

A. Pomponio and D. Ruiz, A variational analysis of a gauged nonlinear Schrödinger equation, J. Eur. Math. Soc., 17 (2015), 1463-1486.  doi: 10.4171/JEMS/535.  Google Scholar

[28]

Y. Wan and J. Tan, Standing waves for the Chern-Simons-Schrödinger systems without (AR) condition, J. Math. Anal. Appl., 415 (2014), 422-434.  doi: 10.1016/j.jmaa.2014.01.084.  Google Scholar

[29]

Y. Wan and J. Tan, Concentration of semi-classical solutions to the Chern-Simons-Schrödinger systems, Nonlinear Differential Equations Appl., 24 (2017), 28, 24 pp. doi: 10.1007/s00030-017-0448-8.  Google Scholar

[30]

Y. Wan and J. Tan, The existence of nontrivial solutions to Chern-Simons-Schrödinger systems, Discrete Contin. Dyn. Syst., 37 (2017), 2765-2786.  doi: 10.3934/dcds.2017119.  Google Scholar

[31]

M. Willem, Minimax Theorems, vol. 24, Birkh$\ddot{\mbox a}$user Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[32]

J. ZhangW. Zhang and X. Xie, Infinitely many solutions for a gauged nonlinear Schrödinger equation, Appl. Math. Lett., 88 (2019), 21-27.  doi: 10.1016/j.aml.2018.08.007.  Google Scholar

show all references

References:
[1]

T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^{N}$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149.  Google Scholar

[2]

J. ByeonH. Huh and J. Seok, Standing waves of nonlinear Schrödinger equations with the gauge field, J. Funct. Anal., 263 (2012), 1575-1608.  doi: 10.1016/j.jfa.2012.05.024.  Google Scholar

[3]

J. ByeonH. Huh and J. Seok, On standing waves with a vortex point of order $N$ for the nonlinear Chern-Simons-Schrödinger equations, J. Differential Equations, 261 (2016), 1285-1316.  doi: 10.1016/j.jde.2016.04.004.  Google Scholar

[4]

D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in $\mathbb{R}^{2}$, Comm. Partial Differential Equations, 17 (1992), 407-435.  doi: 10.1080/03605309208820848.  Google Scholar

[5]

S. ChenB. Zhang and X. Tang, Existence and concentration of semiclassical ground state solutions for the generalized Chern-Simons-Schrödinger system in $H^{1}(\mathbb{R}^{2})$, Nonlinear Anal., 185 (2019), 68-96.  doi: 10.1016/j.na.2019.02.028.  Google Scholar

[6]

J. Deng, W. Long and J. F. Yang, Multi-peak solutions to Chern-Simons-Schrödinger systems with non-radial potential, arXiv: 2007.02499v1. Google Scholar

[7]

Y. DengS. Peng and W. Shuai, Nodal standing waves for a gauged nonlinear Schrödinger equation in $\mathbb{R}^{2}$, J. Differential Equations, 264 (2018), 4006-4035.  doi: 10.1016/j.jde.2017.12.003.  Google Scholar

[8]

J. M. do Ó, $N$-Laplacian equations in $\mathbb{R}^{N}$ with critical growth, Abstr. Appl. Anal., 2 (1997), 301-315.  doi: 10.1155/S1085337597000419.  Google Scholar

[9]

J. M. do ÓE. Medeiros and U. Severo, A nonhomogeneous elliptic problem involving critical growth in dimension two, J. Math. Anal. Appl., 345 (2008), 286-304.  doi: 10.1016/j.jmaa.2008.03.074.  Google Scholar

[10]

T. Gou and Z. Zhang, Normalized solutions to the Chern-Simons-Schrödinger system, J. Funct. Anal., 280 (2021), 108894. arXiv: 1903.07306. doi: 10.1016/j.jfa.2020.108894.  Google Scholar

[11]

H. Huh, Blow-up solutions of the Chern-Simons-Schrödinger equations, Nonlinearity, 22 (2009), 967-974.  doi: 10.1088/0951-7715/22/5/003.  Google Scholar

[12]

R. Jackiw and S.-Y. Pi, Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D, 42 (1990), 3500-3513.  doi: 10.1103/PhysRevD.42.3500.  Google Scholar

[13]

R. Jackiw and S.-Y. Pi, Soliton solutions to the gauged nonlinear Schrödinger equation on the plane, Phys. Rev. Lett., 64 (1990), 2969-2972.  doi: 10.1103/PhysRevLett.64.2969.  Google Scholar

[14]

J.-C. Kang, Y.-Y. Li and C.-L. Tang, Sign-changing solutions for Chern-Simons-Schrödinger equations with asymptotically 5-linear nonlinearity, Bull. Malays. Math. Sci. Soc., (2020). doi: 10.1007/s40840-020-00974-z.  Google Scholar

[15]

J. Kang and C. Tang, Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth, Commun. Pure Appl. Anal., 19 (2020), 5239-5252.  doi: 10.3934/cpaa.2020235.  Google Scholar

[16]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-472.   Google Scholar

[17]

G.-D. Li, Y.-Y. Li and C.-L. Tang, Existence and concentrate behavior of positive solutions for Chern-Simons-Schrödinger systems with critical growth, Complex Var. Elliptic Equ., (2020). doi: 10.1080/17476933.2020.1723564.  Google Scholar

[18]

G. Li and X. Luo, Normalized solutions for the Chern-Simons-Schrödinger equation in $\mathbb{R}^{2}$, Ann. Acad. Sci. Fenn. Math., 42 (2017), 405-428.  doi: 10.5186/aasfm.2017.4223.  Google Scholar

[19]

G. LiX. Luo and W. Shuai, Sign-changing solutions to a gauged nonlinear Schrödinger equation, J. Math. Anal. Appl., 455 (2017), 1559-1578.  doi: 10.1016/j.jmaa.2017.06.048.  Google Scholar

[20]

S. J. Li and M. Willem, Applications of local linking to critical point theory, J. Math. Anal. Appl., 189 (1995), 6-32.  doi: 10.1006/jmaa.1995.1002.  Google Scholar

[21]

W. Liang and C. Zhai, Existence of bound state solutions for the generalized Chern-Simons-Schrödinger system in $H^1(\mathbb{R}^2)$, Appl. Math. Lett., 100 (2020), 106028, 7 pp. doi: 10.1016/j.aml.2019.106028.  Google Scholar

[22]

S. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9.  doi: 10.1007/s00526-011-0447-2.  Google Scholar

[23]

B. Liu and P. Smith, Global wellposedness of the equivariant Chern-Simons-Schrödinger equation, Rev. Mat. Iberoam., 32 (2016), 751-794.  doi: 10.4171/RMI/898.  Google Scholar

[24]

B. Liu, P. Smith and D. Tataru, Local wellposedness of Chern-Simons-Schrödinger, Int. Math. Res. Not. IMRN, (2014), 6341–6398. doi: 10.1093/imrn/rnt161.  Google Scholar

[25]

X. Luo, Existence and stability of standing waves for a planar gauged nonlinear Schrödinger equation, Comput. Math. Appl., 76 (2018), 2701-2709.  doi: 10.1016/j.camwa.2018.09.003.  Google Scholar

[26]

A. Pomponio and D. Ruiz, Boundary concentration of a gauged nonlinear Schrödinger equation on large balls, Calc. Var. Partial Differential Equations, 53 (2015), 289-316.  doi: 10.1007/s00526-014-0749-2.  Google Scholar

[27]

A. Pomponio and D. Ruiz, A variational analysis of a gauged nonlinear Schrödinger equation, J. Eur. Math. Soc., 17 (2015), 1463-1486.  doi: 10.4171/JEMS/535.  Google Scholar

[28]

Y. Wan and J. Tan, Standing waves for the Chern-Simons-Schrödinger systems without (AR) condition, J. Math. Anal. Appl., 415 (2014), 422-434.  doi: 10.1016/j.jmaa.2014.01.084.  Google Scholar

[29]

Y. Wan and J. Tan, Concentration of semi-classical solutions to the Chern-Simons-Schrödinger systems, Nonlinear Differential Equations Appl., 24 (2017), 28, 24 pp. doi: 10.1007/s00030-017-0448-8.  Google Scholar

[30]

Y. Wan and J. Tan, The existence of nontrivial solutions to Chern-Simons-Schrödinger systems, Discrete Contin. Dyn. Syst., 37 (2017), 2765-2786.  doi: 10.3934/dcds.2017119.  Google Scholar

[31]

M. Willem, Minimax Theorems, vol. 24, Birkh$\ddot{\mbox a}$user Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[32]

J. ZhangW. Zhang and X. Xie, Infinitely many solutions for a gauged nonlinear Schrödinger equation, Appl. Math. Lett., 88 (2019), 21-27.  doi: 10.1016/j.aml.2018.08.007.  Google Scholar

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