In this paper, we consider a class of quasilinear Schrödinger equations arising from a model of a self-trapped electrons in quadratic or hexagonal lattices. By variational methods, we prove that this problem admits a positive solution for any positive parameter.

1.
L.
Brizhik
,
A.
Eremko
,
B.
Piette
, and
W. J.
Zakrzewski
, “
Electron self-trapping in a discrete two-dimensional lattice
,”
Physica D
159
,
71
90
(
2001
).
2.
L.
Brizhik
,
A.
Eremko
,
B.
Piette
, and
W. J.
Zakrzewski
, “
Static solutions of a D-dimensional modified nonlinear Schrödinger equation
,”
Nonlinearity
16
,
1481
1497
(
2003
).
3.
B.
Hartmann
and
W. J.
Zakrzewski
, “
Electrons on hexagonal lattices and applications to nanotubes
,”
Phys. Rev. B
68
,
184302
(
2003
).
4.
Y.
Brihaye
,
B.
Hartmann
, and
W. J.
Zakrzewski
, “
Spinning solitons of a modified nonlinear Schrödinger equation
,”
Phys. Rev. D
69
,
087701
(
2004
).
5.
L.
Brüll
,
H.
Lange
, and
E.
de Jager
, “
Stationary, oscillatory and solitary wave type solution of singular nonlinear Schrödinger equations
,”
Math. Methods Appl. Sci.
8
,
559
575
(
1986
).
6.
A.
Ambrosetti
and
Z. Q.
Wang
, “
Positive solutions to a class of quasilinear elliptic equations on R
,”
Discrete Contin. Dyn. Syst. A
9
,
55
68
(
2003
).
7.
C. O.
Alves
,
Y. J.
Wang
, and
Y. T.
Shen
, “
Soliton solutions for a class of quasilinear Schrödinger equations with a parameter
,”
J. Differ. Equations
259
,
318
343
(
2015
).
8.
Y. T.
Shen
and
Y. J.
Wang
, “
Soliton solutions for generalized quasilinear Schrödinger equations
,”
Nonlinear Anal.: Theory, Methods Appl.
80
,
194
201
(
2013
).
9.
Y. T.
Shen
and
X. K.
Guo
, “
The positive solution of degenerate variational problems and degenerate elliptic equations
,”
Chinese J. Contemp. Math.
14
,
157
165
(
1993
); available at https://mathscinet.ams.org/mathscinet/article?mr=1260352.
10.
O. A.
Ladyzhenskaya
and
N. N.
Ural’tseva
,
Linear and Quasilinaer Elliptic Equations
(
Academic
,
1968
).
11.
L.
Jeanjean
and
K.
Tanaka
, “
A remark on least energy solutions in RN
,”
Proc. Am. Math. Soc.
131
,
2399
2408
(
2002
).
12.
H.
Berestycki
and
P. L.
Lions
, “
Nonlinear scalar field equations, I existence of a ground state
,”
Arch. Rational Mech. Anal.
82
,
313
345
(
1983
).
13.
M.
Willem
,
Minimax Theorems
(
Birkhäuser
,
1996
).
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