American Institute of Mathematical Sciences

Advanced
September  2019, 18(5): 2693-2715. doi: 10.3934/cpaa.2019120

Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents

Yongpeng Chen 1, , Yuxia Guo 2,, and  Zhongwei Tang 1,

1. 

School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China
2. 

Department of Mathematics, Tsinghua University, Beijing, 100084, China

* Corresponding author: yguo@math.tsinghua.edu.cn

Received  September 2018 Revised  September 2018 Published  April 2019

Fund Project: The second author is supported by Supported by National Science Foundation of China (11571040, 11771235, 11331010).
The third author is supported by National Science Foundation of China (11571040).

This paper is concerned with the critical quasilinear Schrödinger systems in
$ {\Bbb R}^N: $
$ \left\{\begin{array}{ll}-\Delta w+(\lambda a(x)+1)w-(\Delta|w|^2)w = \frac{p}{p+q}|w|^{p-2}w|z|^q+\frac{\alpha}{\alpha+\beta}|w|^{\alpha-2}w|z|^\beta\\\ -\Delta z+(\lambda b(x)+1)z-(\Delta|z|^2)z = \frac{q}{p+q}|w|^p|z|^{q-2}z+\frac{\beta}{\alpha+\beta}|w|^\alpha|z|^{\beta-2}z, \ \end{array}\right. $
where
$ \lambda>0 $
 is a parameter,
$ p>2, q>2, \alpha>2, \beta>2, $
$ 2\cdot(2^*-1) < p+q<2\cdot2^* $
 and
$ \alpha+ \beta = 2\cdot2^*. $
 By using variational method, we prove the existence of positive ground state solutions which localize near the set
$ \Omega = int \left\{a^{-1}(0)\right\}\cap int \left\{b^{-1}(0)\right\} $
 for
$ \lambda $
 large enough.
Keywords: Quasilinear Schrödinger systems, critical exponent, concentration solution.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35J655.
Citation: Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120
References:
[1]

S. Adachi, M. Shibata and T. Watanabe, Asymptotic behavior of positive solutions for a class of quasilinear elliptic equations with general nonlinearities, Comm. Pure. Appl. Anal., 13 (2014), 97-118.  doi: 10.3934/cpaa.2014.13.97.  Google Scholar

[2]

C. Alves, D. Filho and M. Sonto, On systems of elliptic equations involving subcritical or critical Sobolev eponents, Nonlinear Anal., 42 (2000), 771-787.  doi: 10.1016/S0362-546X(99)00121-2.  Google Scholar

[3]

J. Bezerra do Ó, O. Miyagaki and S. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Diff. Equat., 248 (2010), 722-744.  doi: 10.1016/j.jde.2009.11.030.  Google Scholar

[4]

A. Bouard, N. Hayashi and J. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.  doi: 10.1007/s002200050191.  Google Scholar

[5]

H. S. Brandi, C. Manus, G. mainfray, T. Lehner and G. Bonnaud, Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma, Phys. Fluids. B., 5 (1993), 3539-3550.   Google Scholar

[6]

X. L. Chen and R. N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse, Phys. Rev. Lett., 70 (1993), 2082-2085.   Google Scholar

[7]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008.  Google Scholar

[8]

Y. Deng, S. Peng and S. Yan, Positive solition solutions for generalized quasilinear Schrödinger equations with critical growth, J. Diff. Equat., 37 (2017), 4213-4230.  doi: 10.1016/j.jde.2014.09.006.  Google Scholar

[9]

Y. Guo and B. Li, Solutions for qusilinear Schrödinger systems with critical exponents, Z. Angew. Math. Phys., 66 (2015), 517-546.  doi: 10.1007/s00033-014-0416-7.  Google Scholar

[10]

Y. Guo, X. Liu and F. Zhao, Positive solutions for quasilinear systems with critical growth, Advan. Nonl. Studies., 13 (2013), 893-919.  doi: 10.1515/ans-2013-0408.  Google Scholar

[11]

Y. Guo and J. Nie, Infinitely many solutions for quasilinear Schrödinger systems with finite and sign-changing potentials, Z. Angew. Math. Phys., 67 (2016), 1-30.  doi: 10.1007/s00033-016-0621-7.  Google Scholar

[12]

Y. Guo and Z. Tang, Ground state solutions for the quasilinear Schrödinger systems, J. Math. Anal. Appl., 389 (2012), 322-339.  doi: 10.1016/j.jmaa.2011.11.064.  Google Scholar

[13]

X. He, A. Qian and W. Zou, Existence and concentration of positive solutions for quasilinear Schrö dinger equations with critical growth, Nonlinearity, 26 (2013), 3137-3168.  doi: 10.1088/0951-7715/26/12/3137.  Google Scholar

[14]

Y. He and G. Li, Concentrating solition solutions for quasilinear Schrödinger equations involving critical sobolev exponents, Disc. Cont. Dyna. Syst., 36 (2016), 731-762.  doi: 10.3934/dcds.2016.36.731.  Google Scholar

[15]

L. Jeanjean, T. Lou and Z. Q. Wang, Multiple normalized solutions for quasi-linear Schrödinger equations, J. Diff. Equat., 259 (2015), 3894-3928.  doi: 10.1016/j.jde.2015.05.008.  Google Scholar

[16]

J. Liu, X. Liu and Z. Q. Wang, Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Comm. Part. Diff. Equat., 39 (2014), 2216-2239.  doi: 10.1080/03605302.2014.942738.  Google Scholar

[17]

J. Liu, X. Liu and Z. Q. Wang, Multibump solutions for quasilinear elliptic equations, Jour. Func. Anal., 262 (2012), 4040-4102.  doi: 10.1016/j.jfa.2012.02.009.  Google Scholar

[18]

J. Liu, X. Liu and Z. Q. Wang, Existence theory for quasilinear elliptic equations via regularization approach, Topo. Meth. Nonlinear Anal., 50 (2017), 469-487.  doi: 10.12775/tmna.2017.008.  Google Scholar

[19]

J. Liu, Y. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrö dinger equation, Ⅱ, J. Diff. Equat., 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[20]

J. Liu, Y. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equation via Nehari method, Comm. Part. Diff. Equat., 29 (2004), 879-901.  doi: 10.1081/PDE-120037335.  Google Scholar

[21]

X. Liu, J. Liu and Z. Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263.  doi: 10.1090/S0002-9939-2012-11293-6.  Google Scholar

[22]

X. Liu, J. Liu and Z. Q. Wang, Quasilinear elliptic equation with critical growth via perturbation method, J. Diff. Equat., 254 (2013), 102-124.  doi: 10.1016/j.jde.2012.09.006.  Google Scholar

[23]

B. Ritchie, Relativistic self-focusing channel formation in laser-plasma interactions, Phys. Rev. E., 50 (1994), 687-689.   Google Scholar

[24]

E. Silva and G. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var., 39 (2010), 1-33.  doi: 10.1007/s00526-009-0299-1.  Google Scholar

[25]

Y. Wang and Y. Shen, Existence and asymptotic behavior of positive solutions for a class of quasilinear Schrödinger equations, Advan. Nonl. Studies., 1 (2018), 131-150.  doi: 10.1515/ans-2017-6026.  Google Scholar

[26]

Y. Wang, Y. Zhang and Y. Shen, Multiple solutions for quasilinear Schrödinger equations involving critical exponent, Appl. Math. Comp., 216 (2010), 849-856.  doi: 10.1016/j.amc.2010.01.091.  Google Scholar

[27]

Y. Wang and W. Zou, Bound states to critical quasilinear Schrödinger equations, Non. Diff. Equat. App., 19 (2012), 19-47.  doi: 10.1007/s00030-011-0116-3.  Google Scholar

[28]

M. Willem, Minimax Theorem, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[29]

K. Wu, Positive solutons of quasilinear Schrödinger equations critical growth, Appl. Math. Letters., 45 (2015), 52-57.  doi: 10.1016/j.aml.2015.01.005.  Google Scholar

[30]

X. Zeng, Y. Zhang and H. Zhou, Positive solutions for a quasilinear Schrödinger equation involving Hardy potential and critical exponent, Comm. Cont. Math., 16 (2014), 1-32.  doi: 10.1142/S0219199714500345.  Google Scholar

show all references

References:
[1]

S. Adachi, M. Shibata and T. Watanabe, Asymptotic behavior of positive solutions for a class of quasilinear elliptic equations with general nonlinearities, Comm. Pure. Appl. Anal., 13 (2014), 97-118.  doi: 10.3934/cpaa.2014.13.97.  Google Scholar

[2]

C. Alves, D. Filho and M. Sonto, On systems of elliptic equations involving subcritical or critical Sobolev eponents, Nonlinear Anal., 42 (2000), 771-787.  doi: 10.1016/S0362-546X(99)00121-2.  Google Scholar

[3]

J. Bezerra do Ó, O. Miyagaki and S. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Diff. Equat., 248 (2010), 722-744.  doi: 10.1016/j.jde.2009.11.030.  Google Scholar

[4]

A. Bouard, N. Hayashi and J. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73-105.  doi: 10.1007/s002200050191.  Google Scholar

[5]

H. S. Brandi, C. Manus, G. mainfray, T. Lehner and G. Bonnaud, Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma, Phys. Fluids. B., 5 (1993), 3539-3550.   Google Scholar

[6]

X. L. Chen and R. N. Sudan, Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse, Phys. Rev. Lett., 70 (1993), 2082-2085.   Google Scholar

[7]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008.  Google Scholar

[8]

Y. Deng, S. Peng and S. Yan, Positive solition solutions for generalized quasilinear Schrödinger equations with critical growth, J. Diff. Equat., 37 (2017), 4213-4230.  doi: 10.1016/j.jde.2014.09.006.  Google Scholar

[9]

Y. Guo and B. Li, Solutions for qusilinear Schrödinger systems with critical exponents, Z. Angew. Math. Phys., 66 (2015), 517-546.  doi: 10.1007/s00033-014-0416-7.  Google Scholar

[10]

Y. Guo, X. Liu and F. Zhao, Positive solutions for quasilinear systems with critical growth, Advan. Nonl. Studies., 13 (2013), 893-919.  doi: 10.1515/ans-2013-0408.  Google Scholar

[11]

Y. Guo and J. Nie, Infinitely many solutions for quasilinear Schrödinger systems with finite and sign-changing potentials, Z. Angew. Math. Phys., 67 (2016), 1-30.  doi: 10.1007/s00033-016-0621-7.  Google Scholar

[12]

Y. Guo and Z. Tang, Ground state solutions for the quasilinear Schrödinger systems, J. Math. Anal. Appl., 389 (2012), 322-339.  doi: 10.1016/j.jmaa.2011.11.064.  Google Scholar

[13]

X. He, A. Qian and W. Zou, Existence and concentration of positive solutions for quasilinear Schrö dinger equations with critical growth, Nonlinearity, 26 (2013), 3137-3168.  doi: 10.1088/0951-7715/26/12/3137.  Google Scholar

[14]

Y. He and G. Li, Concentrating solition solutions for quasilinear Schrödinger equations involving critical sobolev exponents, Disc. Cont. Dyna. Syst., 36 (2016), 731-762.  doi: 10.3934/dcds.2016.36.731.  Google Scholar

[15]

L. Jeanjean, T. Lou and Z. Q. Wang, Multiple normalized solutions for quasi-linear Schrödinger equations, J. Diff. Equat., 259 (2015), 3894-3928.  doi: 10.1016/j.jde.2015.05.008.  Google Scholar

[16]

J. Liu, X. Liu and Z. Q. Wang, Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Comm. Part. Diff. Equat., 39 (2014), 2216-2239.  doi: 10.1080/03605302.2014.942738.  Google Scholar

[17]

J. Liu, X. Liu and Z. Q. Wang, Multibump solutions for quasilinear elliptic equations, Jour. Func. Anal., 262 (2012), 4040-4102.  doi: 10.1016/j.jfa.2012.02.009.  Google Scholar

[18]

J. Liu, X. Liu and Z. Q. Wang, Existence theory for quasilinear elliptic equations via regularization approach, Topo. Meth. Nonlinear Anal., 50 (2017), 469-487.  doi: 10.12775/tmna.2017.008.  Google Scholar

[19]

J. Liu, Y. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrö dinger equation, Ⅱ, J. Diff. Equat., 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[20]

J. Liu, Y. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equation via Nehari method, Comm. Part. Diff. Equat., 29 (2004), 879-901.  doi: 10.1081/PDE-120037335.  Google Scholar

[21]

X. Liu, J. Liu and Z. Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263.  doi: 10.1090/S0002-9939-2012-11293-6.  Google Scholar

[22]

X. Liu, J. Liu and Z. Q. Wang, Quasilinear elliptic equation with critical growth via perturbation method, J. Diff. Equat., 254 (2013), 102-124.  doi: 10.1016/j.jde.2012.09.006.  Google Scholar

[23]

B. Ritchie, Relativistic self-focusing channel formation in laser-plasma interactions, Phys. Rev. E., 50 (1994), 687-689.   Google Scholar

[24]

E. Silva and G. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var., 39 (2010), 1-33.  doi: 10.1007/s00526-009-0299-1.  Google Scholar

[25]

Y. Wang and Y. Shen, Existence and asymptotic behavior of positive solutions for a class of quasilinear Schrödinger equations, Advan. Nonl. Studies., 1 (2018), 131-150.  doi: 10.1515/ans-2017-6026.  Google Scholar

[26]

Y. Wang, Y. Zhang and Y. Shen, Multiple solutions for quasilinear Schrödinger equations involving critical exponent, Appl. Math. Comp., 216 (2010), 849-856.  doi: 10.1016/j.amc.2010.01.091.  Google Scholar

[27]

Y. Wang and W. Zou, Bound states to critical quasilinear Schrödinger equations, Non. Diff. Equat. App., 19 (2012), 19-47.  doi: 10.1007/s00030-011-0116-3.  Google Scholar

[28]

M. Willem, Minimax Theorem, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[29]

K. Wu, Positive solutons of quasilinear Schrödinger equations critical growth, Appl. Math. Letters., 45 (2015), 52-57.  doi: 10.1016/j.aml.2015.01.005.  Google Scholar

[30]

X. Zeng, Y. Zhang and H. Zhou, Positive solutions for a quasilinear Schrödinger equation involving Hardy potential and critical exponent, Comm. Cont. Math., 16 (2014), 1-32.  doi: 10.1142/S0219199714500345.  Google Scholar

[1]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461
[2]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253
[3]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447
[4]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121
[5]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436
[6]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450
[7]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469
[8]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247
[9]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376
[10]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276
[11]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456
[12]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264
[13]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454
[14]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119
[15]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216
[16]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348
[17]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440
[18]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118
[19]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434
[20]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (120)
  • HTML views (219)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences