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  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, 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.  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

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

[1]

Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021030

[2]

Kazuhiro Kurata, Yuki Osada. Variational problems associated with a system of nonlinear Schrödinger equations with three wave interaction. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021100

[3]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267

[4]

Daniele Cassani, Luca Vilasi, Jianjun Zhang. Concentration phenomena at saddle points of potential for Schrödinger-Poisson systems. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021039

[5]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3651-3682. doi: 10.3934/dcds.2021011

[6]

Mengyao Chen, Qi Li, Shuangjie Peng. Bound states for fractional Schrödinger-Poisson system with critical exponent. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021038

[7]

Umberto Biccari. Internal control for a non-local Schrödinger equation involving the fractional Laplace operator. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021014

[8]

Tianyu Liao. The regularity lifting methods for nonnegative solutions of Lane-Emden system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021036

[9]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[10]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037

[11]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447

[12]

Siting Liu, Levon Nurbekyan. Splitting methods for a class of non-potential mean field games. Journal of Dynamics & Games, 2021  doi: 10.3934/jdg.2021014

[13]

Yingte Sun. Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021047

[14]

Fangyi Qin, Jun Wang, Jing Yang. Infinitely many positive solutions for Schrödinger-poisson systems with nonsymmetry potentials. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021054

[15]

Maoding Zhen, Binlin Zhang, Xiumei Han. A new approach to get solutions for Kirchhoff-type fractional Schrödinger systems involving critical exponents. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021115

[16]

Norman Noguera, Ademir Pastor. Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3817-3836. doi: 10.3934/dcds.2021018

[17]

Guanwei Chen, Martin Schechter. Multiple solutions for Schrödinger lattice systems with asymptotically linear terms and perturbed terms. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021124

[18]

Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart. Electronic Research Archive, , () : -. doi: 10.3934/era.2021016

[19]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[20]

Chenjie Fan, Zehua Zhao. Decay estimates for nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3973-3984. doi: 10.3934/dcds.2021024

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (46)
  • HTML views (107)
  • Cited by (0)

Other articles
by authors

[Back to Top]