In this paper we consider the generalized quasilinear Schrödinger equations
where V and h are periodic in , . By using variational methods, we prove the existence of ground state solutions, i.e., nontrivial solutions with least possible energy.
H.Berestycki and P.L.Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal.82 (1983), 313–345. doi:10.1007/BF00250555.
2.
M.Colin and L.Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonl. Anal.56 (2004), 213–226. doi:10.1016/j.na.2003.09.008.
3.
Y.B.Deng, S.J.Peng and J.X.Wang, Nodal soliton solutions for generalized quasilinear Schrödinger equations, J. Math. Phys.55(5) (2014), 051501, 16 pp.
4.
Y.B.Deng, S.J.Peng and S.S.Yan, Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth, J. Diff. Eq.258 (2015), 115–147. doi:10.1016/j.jde.2014.09.006.
5.
Y.B.Deng, S.J.Peng and S.S.Yan, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, J. Diff. Eq.260 (2016), 1228–1262. doi:10.1016/j.jde.2015.09.021.
6.
J.M.do Ò, O.Miyagaki and S.Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Diff. Eq.248 (2010), 722–744. doi:10.1016/j.jde.2009.11.030.
7.
X.D.Fang and A.Szulkin, Multiple solutions for a quasilinear Schrödinger equation, J. Diff. Eq.254 (2013), 2015–2032. doi:10.1016/j.jde.2012.11.017.
8.
L.Jeanjean, T.J.Luo and Z.Q.Wang, Multiple normalized solutions for quasi-linear Schrödinger equations, J. Diff. Eq.259 (2015), 3894–3928. doi:10.1016/j.jde.2015.05.008.
9.
S.Kurihura, Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan50 (1981), 3262–3267. doi:10.1143/JPSJ.50.3262.
10.
F.Y.Li, X.L.Zhu and Z.P.Liang, Multiple solutions to a class of generalized quasilinear Schrödinger equations with a Kirchhoff-type perturbation, J. Math. Anal. Appl.443(1) (2016), 11–38. doi:10.1016/j.jmaa.2016.05.005.
11.
Y.Q.Li, Z.Q.Wang and J.Zeng, Ground states of nonlinear Schrödinger equations with potentials, Ann. I. H. Poincare – AN23 (2006), 829–837.
12.
J.Q.Liu, Y.Q.Wang and Z.Q.Wang, Soliton solutions for quasilinear Schrödinger equations, II, J. Diff. Eq.187 (2003), 473–493. doi:10.1016/S0022-0396(02)00064-5.
13.
J.Q.Liu and Z.Q.Wang, Soliton solutions for quasilinear Schrödinger equations, I, Proc. Amer. Math. Soc.131 (2003), 441–448. doi:10.1090/S0002-9939-02-06783-7.
14.
M.Poppenberg, K.Schmitt and Z.Q.Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var.14 (2002), 329–344. doi:10.1007/s005260100105.
15.
P.H.Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys.43 (1992), 270–291. doi:10.1007/BF00946631.
16.
D.Ruiz and G.Siciliano, Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity23(5) (2010), 1221–1233. doi:10.1088/0951-7715/23/5/011.
17.
Y.T.Shen and Y.J.Wang, Soliton solutions for generalized quasilinear Schrödinger equations, Nonl. Anal.80 (2013), 194–201. doi:10.1016/j.na.2012.10.005.
18.
Y.T.Shen and Y.J.Wang, Standing waves for a relativistic quasilinear asymptotically Schrödinger equation, Appl. Anal.95 (2016), 2565–2576. doi:10.1080/00036811.2015.1102890.
19.
H.X.Shi and H.B.Chen, Generalized quasilinear asymptotically periodic Schrödinger equations with critical growth, Comput. Math. Appl.71(3) (2016), 849–858. doi:10.1016/j.camwa.2016.01.007.
20.
H.X.Shi and H.B.Chen, Positive solutions for generalized quasilinear Schrödinger equations with potential vanishing at infinity, Appl. Math. Lett.61 (2016), 137–142. doi:10.1016/j.aml.2016.06.004.
21.
E.A.B.Silva and G.F.Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var.39 (2010), 1–33. doi:10.1007/s00526-009-0299-1.
22.
W.A.Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys.55 (1977), 149–162. doi:10.1007/BF01626517.
23.
A.Szulkin and T.Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal.257 (2009), 3802–3822. doi:10.1016/j.jfa.2009.09.013.
24.
A.Szulkin and T.Weth, The method of Nehari manifold, in: Handbook of Nonconvex Analysis and Applications, D.Y.Gao and D.Motreanu, eds, International Press, Boston, 2010, pp. 597–632.
J.Yang, Y.J.Wang and A.A.Abdelgadir, Soliton solutions for quasilinear Schrödinger equations, J. Math. Physics54 (2013), 071502. doi:10.1063/1.4811394.
27.
X.L.Zhu, F.Y.Li and Z.P.Liang, Existence of ground state solutions to a generalized quasilinear Schrödinger–Maxwell system, J. Math. Phys.57(10) (2016), 101505, 15 pp.
