American Institute of Mathematical Sciences

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May  2018, 17(3): 1121-1145. doi: 10.3934/cpaa.2018054

Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth

Yanfang Xue 1,2, and  Chunlei Tang 1,,

1. 

School of Mathematics and Statistics, Southwest University, Chongqing, 400700, China
2. 

School of Mathematics and Statistics, Xin-Yang Normal University, Xinyang, 464000, China

* Corresponding author: Chunlei Tang

Received  January 2017 Revised  November 2017 Published  January 2018

Fund Project: The research is supported by National Natural Science Foundation of China(No. 11471267,11601438) and Chongqing Research Program of Basic Research and Frontier Technology(No.cstc2017jcyjAX0331).

In this paper, we are concerned with the existence of ground state solutions for the following quasilinear Schrödinger equation:
$-Δ u+V(x)u-Δ (u^2)u = K(x)|u|^{22^*-2}u+g(x,u), \ \ x∈ \mathbb{R}^N\ \ \ \ \ \ \ \ \ \ \left( 1 \right)$
where $N≥ 3$, $V, \ g$ are asymptotically periodic functions in $x$. By combining variational methods and the concentration-compactness principle, we obtain a ground state solution for equation (1) under a new reformative condition which unify the asymptotic processes of $V, g $ at infinity.
Keywords: Asymptotically periodic, Sobolev critical exponent, ground state solution, Nehari manifold.
Mathematics Subject Classification: Primary: 35A15, 35B33, 35J60.
Citation: Yanfang Xue, Chunlei Tang. Ground state solutions for asymptotically periodic quasilinear Schrödinger equations with critical growth. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1121-1145. doi: 10.3934/cpaa.2018054
References:
[1]

S. Adachi and T. Watanabe, Uniqueness of the ground state solutions of quasilinear Schrödinger equations, Nonlinear Anal., 75 (2012), 819-833.   Google Scholar

[2]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.   Google Scholar

[3]

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

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M. Colin, L. Jeanjean and M. Squassina, Stability and instability results for standing waves of quasi-linear Schrödinger equations, Nonlinearity, 23 (2010), 1353-1385.   Google Scholar

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Y. B. Deng, S. J. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, Journal of Mathematical Physics, 54 (2013), 011504.   Google Scholar

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Y. B. Deng, S. J. Peng and S. S. Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, Journal of Differential Equations, 260 (2016), 1228-1262.   Google Scholar

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J.M. do Ó, O. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, Journal of Differential Equations, 248 (2010), 722-744.   Google Scholar

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X. D. Fang and A. Szulkin, Multiple solutions for a quasilinear Schrödinger equation, J. Differential Equations, 254 (2013), 2015-2032.   Google Scholar

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F. Gladiali and M. Squassina, Uniqueness of ground states for a class of quasi-linear elliptic equations, Adv. Nonlinear Anal., 1 (2012), 159-179.   Google Scholar

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Y. He and G. B. Li, Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Soblev exponents, Disctete and Continuous Dynamical Systems, 36 (2016), 731-762.   Google Scholar

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L. Jeanjean, T. J. Luo and Z. Q. Wang, Multiple normalized solutions for quasi-linear Schrödinger equations, J. Differential Equations, 259 (2015), 3894-3928.   Google Scholar

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G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.   Google Scholar

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H. F. Lins and E. A. B. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), 2890-2905.   Google Scholar

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P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅱ, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 223-283.   Google Scholar

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P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅰ, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 109-145.   Google Scholar

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J. Liu, J. F. Liao and C. L. Tang, A positive ground state solution for a class of asymptotically periodic Schrödinger equations, Comput. Math. Appl., 71 (2016), 965-976.   Google Scholar

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J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations Ⅱ, J. Differential Equations, 187 (2003), 473-493.   Google Scholar

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J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅰ, Proc. Amer. Math. Soc., 131 (2003), 441-448.   Google Scholar

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X. Q. Liu, J. Q. Liu and Z. Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263.   Google Scholar

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X. Q. Liu, J. Q. Liu and Z. Q. Wang, Ground states for quasilinear Schrödinger equations with critical growth, Calculus of Variations and Partial Differential Equations, 46 (2013), 641-669.   Google Scholar

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R. D. Marchi, Schrödinger equations with asymptotically periodic terms, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 145 (2015), 745-757.   Google Scholar

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A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\mathbb{R}^N$, Journal of Differential Equations, 229 (2006), 570-587.   Google Scholar

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M. Poppenberg, K. Schmitt and Z. Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14 (2002), 329-344.   Google Scholar

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D. Ruiz and G. Siciliano, Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity, 23 (2010), 1221-1233.   Google Scholar

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A. Selvitella, Uniqueness and nondegeneracy of the ground state for a quasilinear Schrödinger equation with a small parameter, Nonlinear Anal., 74 (2011), 1731-1737.   Google Scholar

[31]

H. X. Shi and H. B. Chen, Generalized quasilinear asymptotically periodic Schrödinger equations with critical growth, Comput. Math. Appl., 71 (2016), 849-858.   Google Scholar

[32]

E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949.   Google Scholar

[33]

E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33.   Google Scholar

[34]

X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), 715-728.   Google Scholar

[35] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.   Google Scholar
[36]

X. Wu, Multiple solutions for quasilinear Schrödinger equations with a parameter, J. Differential Equations, 256 (2014), 2619-2632.   Google Scholar

[37]

H. Zhang, J. X. Xu and F. B. Zhang, On a class of semilinear Schrödinger equations with indefinite linear part, J. Math. Anal. Appl., 414 (2014), 710-724.   Google Scholar

[38]

H. Zhang, J. X. Xu and F. B. Zhang, Ground state solutions for asymptotically periodic Schrödinger equations with indefinite linear part, Math. Methods Appl. Sci., 38 (2015), 113-122.   Google Scholar

show all references

References:
[1]

S. Adachi and T. Watanabe, Uniqueness of the ground state solutions of quasilinear Schrödinger equations, Nonlinear Anal., 75 (2012), 819-833.   Google Scholar

[2]

H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.   Google Scholar

[3]

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

[4]

M. Colin, L. Jeanjean and M. Squassina, Stability and instability results for standing waves of quasi-linear Schrödinger equations, Nonlinearity, 23 (2010), 1353-1385.   Google Scholar

[5]

Y. B. Deng, S. J. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, Journal of Mathematical Physics, 54 (2013), 011504.   Google Scholar

[6]

Y. B. Deng, S. J. Peng and S. S. Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, Journal of Differential Equations, 260 (2016), 1228-1262.   Google Scholar

[7]

J. M. do Ó and U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal., 8 (2009), 621-644.   Google Scholar

[8]

J.M. do Ó, O. H. Miyagaki and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, Journal of Differential Equations, 248 (2010), 722-744.   Google Scholar

[9]

X. D. Fang and A. Szulkin, Multiple solutions for a quasilinear Schrödinger equation, J. Differential Equations, 254 (2013), 2015-2032.   Google Scholar

[10]

F. Gladiali and M. Squassina, Uniqueness of ground states for a class of quasi-linear elliptic equations, Adv. Nonlinear Anal., 1 (2012), 159-179.   Google Scholar

[11]

Y. He and G. B. Li, Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Soblev exponents, Disctete and Continuous Dynamical Systems, 36 (2016), 731-762.   Google Scholar

[12]

L. Jeanjean, T. J. Luo and Z. Q. Wang, Multiple normalized solutions for quasi-linear Schrödinger equations, J. Differential Equations, 259 (2015), 3894-3928.   Google Scholar

[13]

G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.   Google Scholar

[14]

H. F. Lins and E. A. B. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), 2890-2905.   Google Scholar

[15]

P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅱ, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 223-283.   Google Scholar

[16]

P. L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case. Ⅰ, Ann. Inst. H. Poincare Anal. Non Lineaire, 1 (1984), 109-145.   Google Scholar

[17]

J. Liu, J. F. Liao and C. L. Tang, A positive ground state solution for a class of asymptotically periodic Schrödinger equations, Comput. Math. Appl., 71 (2016), 965-976.   Google Scholar

[18]

J. Liu, J. F. Liao and C. L. Tang, A positive ground state solution for a class of asymptotically periodic Schrödinger equations with critical exponent, Comput. Math. Appl., 72 (2016), 1851-1864.   Google Scholar

[19]

J. Q. Liu, X. Q. Liu and Z. Q. Wang, Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Comm. Partial Differential Equations, 39 (2014), 2216-2239.   Google Scholar

[20]

J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations, 29 (2004), 879-901.   Google Scholar

[21]

J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations Ⅱ, J. Differential Equations, 187 (2003), 473-493.   Google Scholar

[22]

J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅰ, Proc. Amer. Math. Soc., 131 (2003), 441-448.   Google Scholar

[23]

X. Q. Liu, J. Q. Liu and Z. Q. Wang, Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013), 253-263.   Google Scholar

[24]

X. Q. Liu, J. Q. Liu and Z. Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method, Journal of Differential Equations, 254 (2013), 102-124.   Google Scholar

[25]

X. Q. Liu, J. Q. Liu and Z. Q. Wang, Ground states for quasilinear Schrödinger equations with critical growth, Calculus of Variations and Partial Differential Equations, 46 (2013), 641-669.   Google Scholar

[26]

R. D. Marchi, Schrödinger equations with asymptotically periodic terms, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 145 (2015), 745-757.   Google Scholar

[27]

A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\mathbb{R}^N$, Journal of Differential Equations, 229 (2006), 570-587.   Google Scholar

[28]

M. Poppenberg, K. Schmitt and Z. Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14 (2002), 329-344.   Google Scholar

[29]

D. Ruiz and G. Siciliano, Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity, 23 (2010), 1221-1233.   Google Scholar

[30]

A. Selvitella, Uniqueness and nondegeneracy of the ground state for a quasilinear Schrödinger equation with a small parameter, Nonlinear Anal., 74 (2011), 1731-1737.   Google Scholar

[31]

H. X. Shi and H. B. Chen, Generalized quasilinear asymptotically periodic Schrödinger equations with critical growth, Comput. Math. Appl., 71 (2016), 849-858.   Google Scholar

[32]

E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949.   Google Scholar

[33]

E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations, 39 (2010), 1-33.   Google Scholar

[34]

X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), 715-728.   Google Scholar

[35] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.   Google Scholar
[36]

X. Wu, Multiple solutions for quasilinear Schrödinger equations with a parameter, J. Differential Equations, 256 (2014), 2619-2632.   Google Scholar

[37]

H. Zhang, J. X. Xu and F. B. Zhang, On a class of semilinear Schrödinger equations with indefinite linear part, J. Math. Anal. Appl., 414 (2014), 710-724.   Google Scholar

[38]

H. Zhang, J. X. Xu and F. B. Zhang, Ground state solutions for asymptotically periodic Schrödinger equations with indefinite linear part, Math. Methods Appl. Sci., 38 (2015), 113-122.   Google Scholar

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