American Institute of Mathematical Sciences

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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

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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

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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

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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

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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

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