November  2020, 19(11): 5239-5252. doi: 10.3934/cpaa.2020235

Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth

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

* Corresponding author

Received  March 2020 Revised  June 2020 Published  November 2020 Early access  September 2020

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

We investigate the following gauged nonlinear Schrödinger equation
$ \begin{equation*} \begin{cases} -\Delta u+\omega u+\lambda\bigg(\dfrac{h_{u}^{2}(|x|)}{|x|^{2}}+ \int_{|x|}^{+\infty}\dfrac{h_{u}(s)}{s}u^{2}(s)ds\bigg)u = f(u) \ \ \ \ \ \mbox{in}\ \mathbb{R}^{2},\\ u\in H_r^1(\mathbb{R}^{2}), \end{cases} \end{equation*} $
where
$ \omega,\lambda>0 $
and
$ h_{u}(s) = \frac{1}{2}\int_{0}^{s}ru^{2}(r)dr $
. When
$ f $
has exponential critical growth, by using the constrained minimization method and Trudinger-Moser inequality, it is proved that the equation has a ground state radial sign-changing solution
$ u_{\lambda} $
which changes sign exactly once. Moreover, the asymptotic behavior of
$ u_{\lambda} $
as
$ \lambda\rightarrow0 $
is analyzed.
Citation: Jincai Kang, Chunlei Tang. Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5239-5252. doi: 10.3934/cpaa.2020235
References:
[1]

C. O. Alves and D. S. Pereira, Existence and nonexistence of least energy nodal solution for a class of elliptic problem in $\mathbb{R}^{2}$, Topol. Methods Nonlinear Anal., 46 (2015), 867-892. 

[2]

L. BergéA. De Bouard and J. C. Saut, Blowing up time-dependent solutions of the planar, Chern-Simons gauged nonlinear Schrödinger equation, Nonlinearity, 8 (1995), 235-253. 

[3]

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.

[4]

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

[5]

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

[6]

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

[7]

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.

[8]

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.

[9]

J. HanH. Huh and J. Seok, Chern-Simons limit of the standing wave solutions for the Schrödinger equations coupled with a neutral scalar field, J. Funct. Anal., 266 (2014), 318-342.  doi: 10.1016/j.jfa.2013.09.019.

[10]

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

[11]

R. Jackiw and S. 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.

[12]

C. Ji and F. Fang, Standing waves for the Chern-Simons-Schrödinger equation with critical exponential growth, J. Math. Anal. Appl., 450 (2017), 578-591.  doi: 10.1016/j.jmaa.2017.01.065.

[13]

G. B. 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.

[14]

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.

[15]

B. P. 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.

[16]

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

[17]

Z. S. LiuZ. G. Ouyang and J. J. Zhang, Existence and multiplicity of sign-changing standing waves for a gauged nonlinear Schrödinger equation in $\mathbb{R}^{2}$, Nonlinearity, 32 (2019), 3082-3111.  doi: 10.1088/1361-6544/ab1bc4.

[18]

C. Miranda, Un'osservazione su un teorema di Brouwer, Boll. Unione Mat. Ital., 3 (1940), 5-7. 

[19]

S. J. Oh and F. Pusateri, Decay and scattering for the Chern-Simons-Schrödinger equations, Int. Math. Res. Not. IMRN, 2015 (2015), 13122-13147.  doi: 10.1093/imrn/rnv093.

[20]

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

[21]

L. J. Shen, Ground state solutions for a class of gauged Schrödinger equations with subcritical and critical exponential growth, Math. Method Appl. Sci., 43 (2020), 536-551.  doi: 10.1007/s40840-018-0686-x.

[22]

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

[23]

W. H. Xie and C. Chen, Sign-changing solutions for the nonlinear Chern-Simons-Schrödinger equations, Appl. Anal., 99 (2020), 880-898.  doi: 10.1080/00036811.2018.1514020.

[24]

J. ZhangW. Zhang and X. L. 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]

C. O. Alves and D. S. Pereira, Existence and nonexistence of least energy nodal solution for a class of elliptic problem in $\mathbb{R}^{2}$, Topol. Methods Nonlinear Anal., 46 (2015), 867-892. 

[2]

L. BergéA. De Bouard and J. C. Saut, Blowing up time-dependent solutions of the planar, Chern-Simons gauged nonlinear Schrödinger equation, Nonlinearity, 8 (1995), 235-253. 

[3]

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.

[4]

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

[5]

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

[6]

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

[7]

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.

[8]

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.

[9]

J. HanH. Huh and J. Seok, Chern-Simons limit of the standing wave solutions for the Schrödinger equations coupled with a neutral scalar field, J. Funct. Anal., 266 (2014), 318-342.  doi: 10.1016/j.jfa.2013.09.019.

[10]

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

[11]

R. Jackiw and S. 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.

[12]

C. Ji and F. Fang, Standing waves for the Chern-Simons-Schrödinger equation with critical exponential growth, J. Math. Anal. Appl., 450 (2017), 578-591.  doi: 10.1016/j.jmaa.2017.01.065.

[13]

G. B. 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.

[14]

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.

[15]

B. P. 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.

[16]

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

[17]

Z. S. LiuZ. G. Ouyang and J. J. Zhang, Existence and multiplicity of sign-changing standing waves for a gauged nonlinear Schrödinger equation in $\mathbb{R}^{2}$, Nonlinearity, 32 (2019), 3082-3111.  doi: 10.1088/1361-6544/ab1bc4.

[18]

C. Miranda, Un'osservazione su un teorema di Brouwer, Boll. Unione Mat. Ital., 3 (1940), 5-7. 

[19]

S. J. Oh and F. Pusateri, Decay and scattering for the Chern-Simons-Schrödinger equations, Int. Math. Res. Not. IMRN, 2015 (2015), 13122-13147.  doi: 10.1093/imrn/rnv093.

[20]

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

[21]

L. J. Shen, Ground state solutions for a class of gauged Schrödinger equations with subcritical and critical exponential growth, Math. Method Appl. Sci., 43 (2020), 536-551.  doi: 10.1007/s40840-018-0686-x.

[22]

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

[23]

W. H. Xie and C. Chen, Sign-changing solutions for the nonlinear Chern-Simons-Schrödinger equations, Appl. Anal., 99 (2020), 880-898.  doi: 10.1080/00036811.2018.1514020.

[24]

J. ZhangW. Zhang and X. L. 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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