# American Institute of Mathematical Sciences

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

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

 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.
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. [2] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. [3] M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal., 56 (2004), 213-226. [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. [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. [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. [7] J. M. do Ó and U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal., 8 (2009), 621-644. [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. [9] X. D. Fang and A. Szulkin, Multiple solutions for a quasilinear Schrödinger equation, J. Differential Equations, 254 (2013), 2015-2032. [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. [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. [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. [13] G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776. [14] H. F. Lins and E. A. B. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), 2890-2905. [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. [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. [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. [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. [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. [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. [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. [22] J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅰ, Proc. Amer. Math. Soc., 131 (2003), 441-448. [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. [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. [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. [26] R. D. Marchi, Schrödinger equations with asymptotically periodic terms, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 145 (2015), 745-757. [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. [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. [29] D. Ruiz and G. Siciliano, Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity, 23 (2010), 1221-1233. [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. [31] H. X. Shi and H. B. Chen, Generalized quasilinear asymptotically periodic Schrödinger equations with critical growth, Comput. Math. Appl., 71 (2016), 849-858. [32] E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949. [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. [34] X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), 715-728. [35] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. [36] X. Wu, Multiple solutions for quasilinear Schrödinger equations with a parameter, J. Differential Equations, 256 (2014), 2619-2632. [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. [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.

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##### References:
 [1] S. Adachi and T. Watanabe, Uniqueness of the ground state solutions of quasilinear Schrödinger equations, Nonlinear Anal., 75 (2012), 819-833. [2] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477. [3] M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal., 56 (2004), 213-226. [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. [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. [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. [7] J. M. do Ó and U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal., 8 (2009), 621-644. [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. [9] X. D. Fang and A. Szulkin, Multiple solutions for a quasilinear Schrödinger equation, J. Differential Equations, 254 (2013), 2015-2032. [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. [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. [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. [13] G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776. [14] H. F. Lins and E. A. B. Silva, Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009), 2890-2905. [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. [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. [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. [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. [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. [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. [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. [22] J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅰ, Proc. Amer. Math. Soc., 131 (2003), 441-448. [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. [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. [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. [26] R. D. Marchi, Schrödinger equations with asymptotically periodic terms, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 145 (2015), 745-757. [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. [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. [29] D. Ruiz and G. Siciliano, Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity, 23 (2010), 1221-1233. [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. [31] H. X. Shi and H. B. Chen, Generalized quasilinear asymptotically periodic Schrödinger equations with critical growth, Comput. Math. Appl., 71 (2016), 849-858. [32] E. A. B. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949. [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. [34] X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), 715-728. [35] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. [36] X. Wu, Multiple solutions for quasilinear Schrödinger equations with a parameter, J. Differential Equations, 256 (2014), 2619-2632. [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. [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.
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