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

  • * Corresponding author: Chunlei Tang

    * Corresponding author: Chunlei Tang
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)
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  • 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.

    Mathematics Subject Classification: Primary: 35A15, 35B33, 35J60.

    Citation:

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  •   S. Adachi  and  T. Watanabe , Uniqueness of the ground state solutions of quasilinear Schrödinger equations, Nonlinear Anal., 75 (2012) , 819-833. 
      H. Brezis  and  L. Nirenberg , Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983) , 437-477. 
      M. Colin  and  L. Jeanjean , Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal., 56 (2004) , 213-226. 
      M. Colin , L. Jeanjean  and  M. Squassina , Stability and instability results for standing waves of quasi-linear Schrödinger equations, Nonlinearity, 23 (2010) , 1353-1385. 
      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. 
      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. 
      J. M. do Ó  and  U. Severo , Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal., 8 (2009) , 621-644. 
      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. 
      X. D. Fang  and  A. Szulkin , Multiple solutions for a quasilinear Schrödinger equation, J. Differential Equations, 254 (2013) , 2015-2032. 
      F. Gladiali  and  M. Squassina , Uniqueness of ground states for a class of quasi-linear elliptic equations, Adv. Nonlinear Anal., 1 (2012) , 159-179. 
      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. 
      L. Jeanjean , T. J. Luo  and  Z. Q. Wang , Multiple normalized solutions for quasi-linear Schrödinger equations, J. Differential Equations, 259 (2015) , 3894-3928. 
      G. B. Li  and  A. Szulkin , An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002) , 763-776. 
      H. F. Lins  and  E. A. B. Silva , Quasilinear asymptotically periodic elliptic equations with critical growth, Nonlinear Anal., 71 (2009) , 2890-2905. 
      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. 
      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. 
      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. 
      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. 
      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. 
      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. 
      J. Q. Liu , Y. Q. Wang  and  Z. Q. Wang , Soliton solutions for quasilinear Schrödinger equations Ⅱ, J. Differential Equations, 187 (2003) , 473-493. 
      J. Q. Liu  and  Z. Q. Wang , Soliton solutions for quasilinear Schrödinger equations, Ⅰ, Proc. Amer. Math. Soc., 131 (2003) , 441-448. 
      X. Q. Liu , J. Q. Liu  and  Z. Q. Wang , Quasilinear elliptic equations via perturbation method, Proc. Amer. Math. Soc., 141 (2013) , 253-263. 
      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. 
      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. 
      R. D. Marchi , Schrödinger equations with asymptotically periodic terms, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 145 (2015) , 745-757. 
      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. 
      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. 
      D. Ruiz  and  G. Siciliano , Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity, 23 (2010) , 1221-1233. 
      A. Selvitella , Uniqueness and nondegeneracy of the ground state for a quasilinear Schrödinger equation with a small parameter, Nonlinear Anal., 74 (2011) , 1731-1737. 
      H. X. Shi  and  H. B. Chen , Generalized quasilinear asymptotically periodic Schrödinger equations with critical growth, Comput. Math. Appl., 71 (2016) , 849-858. 
      E. A. B. Silva  and  G. F. Vieira , Quasilinear asymptotically periodic Schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010) , 2935-2949. 
      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. 
      X. H. Tang , Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015) , 715-728. 
      M. WillemMinimax Theorems, Birkhäuser, Boston, 1996. 
      X. Wu , Multiple solutions for quasilinear Schrödinger equations with a parameter, J. Differential Equations, 256 (2014) , 2619-2632. 
      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. 
      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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