# American Institute of Mathematical Sciences

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

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

 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.
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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [23] B. Ritchie, Relativistic self-focusing channel formation in laser-plasma interactions, Phys. Rev. E., 50 (1994), 687-689. [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. [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. [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. [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. [28] M. Willem, Minimax Theorem, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [23] B. Ritchie, Relativistic self-focusing channel formation in laser-plasma interactions, Phys. Rev. E., 50 (1994), 687-689. [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. [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. [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. [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. [28] M. Willem, Minimax Theorem, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. [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. [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.
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