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doi: 10.3934/dcdss.2021016
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## 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 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 and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021016
##### 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. [2] J. Byeon, H. 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. [3] J. Byeon, H. 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. [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. [5] S. Chen, B. 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. [6] J. Deng, W. Long and J. F. Yang, Multi-peak solutions to Chern-Simons-Schrödinger systems with non-radial potential, arXiv: 2007.02499v1. [7] Y. Deng, S. 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. [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. [9] J. M. do Ó, E. 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. [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. [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. [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. [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. [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. [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. [16] W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-472. [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. [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. [19] G. Li, X. 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [32] J. Zhang, W. 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.

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. [2] J. Byeon, H. 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. [3] J. Byeon, H. 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. [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. [5] S. Chen, B. 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. [6] J. Deng, W. Long and J. F. Yang, Multi-peak solutions to Chern-Simons-Schrödinger systems with non-radial potential, arXiv: 2007.02499v1. [7] Y. Deng, S. 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. [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. [9] J. M. do Ó, E. 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. [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. [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. [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. [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. [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. [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. [16] W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-472. [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. [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. [19] G. Li, X. 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [32] J. Zhang, W. 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.
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