In this paper, we investigate the generalized quasilinear Schrödinger equation: $$ -\operatorname{div}\left(g^2(u) \nabla u\right)+g(u) g^{\prime}(u)|\nabla u|^2 +u=P(\varepsilon x) |u|^{\\\alpha p-2}u, \quad x \in \mathbb{R}^N, $$ where $N>3$, $g\!\!:\mathbb{R} \rightarrow \mathbb{R}^{+}$ is a $C^1$ even function, $g(0)=1$, $g^{\prime}(s) \geq 0$ for all $s \geq 0$, $g(s)=\beta|s|^{\alpha-1}+O\left(|s|^{\gamma-1}\right)$ as $s \rightarrow \infty$ for some constants $\alpha \in[1,2]$, $\beta>0$, $\gamma<\alpha$ and $(\alpha-1) g(s) \geq g^{\prime}(s) s$ for all $s \geq 0$, $\varepsilon>0$ is a positive parameter, and $p \in\left(2,2^*\right)$. We will study the impact of the nonlinearity's coefficient $P(x)$ on the quantity of positive solutions.
[1] Ait-Mahiout K, Alves C O. Multiple solutions for a class of quasilinear problems in Orlicz-Sobolev spaces. Asymptot Anal, 2017, 104(1/2): 49-66
[2] Alves C O, De Lima R N, Nóbrega A B. Existence and multiplicity of solutions for a class of Dirac equations. J Differential Equations, 2023, 370: 66-100
[3] Alves C O, Ji C. Multiple positive solutions for a Schrödinger logarithmic equation. Discrete Contin Dyn Syst, 2020, 40(5): 2671-2685
[4] Alves C O, Wang Y, Shen Y. Soliton solutions for a class of quasilinear Schrödinger equations with a parameter. J Differential Equations, 2015, 259(1): 318-343
[5] Brüll L, Lange H. Solitary waves for quasilinear Schrödinger equations. Exposition Math, 1986, 4(3): 279-288
[6] Candela A M, Salvatore A, Sportelli C. Bounded solutions for quasilinear modified Schrödinger equations. Calc Var Partial Differential Equations, 2022, 61(6): Art 220
[7] Cao D, Noussair E S. Multiplicity of positive and nodal solutions for nonlinear elliptic problems in $\mathbb{R}^N$. Ann Inst H Poincaré C Anal Non Linéaire, 1996, 13(5): 567-588
[8] Chen J, Huang X, Qin D, Cheng B. Existence and asymptotic behavior of standing wave solutions for a class of generalized quasilinear Schrödinger equations with critical Sobolev exponents. Asymptot Anal, 2020, 120 (3/4): 199-248
[9] Colin M, Jeanjean L. Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal, 2004, 56(2): 213-226
[10] De Bouard A, Hayashi N, Naumkin P I, Saut J C. Scattering problem and asymptotics for a relativistic nonlinear Schrödinger equation. Nonlinearity, 1999, 12(5): 1415-1425
[11] De Bouard A, Hayashi N, Saut J C. Global existence of small solutions to a relativistic nonlinear Schrödinger equation. Comm Math Phys, 1997, 189(1): 73-105
[12] De Carvalho G M, Clemente R G, De Albuquerque J C. Quasilinear Schrödinger equations with unbounded or decaying potentials in dimension 2. Math Nachr, 2023, 296(9): 4357-4373
[13] Deng Y, Peng S, Yan S. Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations. J Differential Equations, 2016, 260(2): 1228-1262
[14] Fang X D, Liu M Q. Localized solutions of higher topological type for semiclassical generalized quasilinear Schrödinger equations. Z Angew Math Phys, 2023, 74(2): Art 81
[15] Guo Y, Tang Z. Ground state solutions for the quasilinear Schrödinger equation. Nonlinear Anal, 2012, 75(6): 3235-3248
[16] Hasse R W. A general method for the solution of nonlinear soliton and kink Schrödinger equations. Z Phys B, 1980, 37(1): 83-87
[17] He X, Qian A, Zou W. Existence and concentration of positive solutions for quasilinear Schrödinger equations with critical growth. Nonlinearity, 2013, 26(12): 3137-3168
[18] Hsu T, Lin H, Hu C C. Multiple positive solutions of quasilinear elliptic equations in $\Bbb R^N$. J Math Anal Appl, 2012, 388(1): 500-512
[19] Jing Y, Liu Z, Wang Z Q. Parameter-dependent multiplicity results of sign-changing solutions for quasilinear elliptic equations. Commun Contemp Math, 2023, 25(9): Art 2250039
[20] Kurihara S. Exact soliton solution for superfluid film dynamics. J Phys Soc Japan, 1981, 50(11): 3801-3805
[21] Laedke E W, Spatschek K H, Stenflo L. Evolution theorem for a class of perturbed envelope soliton solutions. J Math Phys, 1983, 24(12): 2764-2769
[22] Lange H, Poppenberg M, Teismann H. Nash-Moser methods for the solution of quasilinear Schrödinger equations. Comm Partial Differential Equations, 1999, 24(7/8): 1399-1418
[23] Li H, Zou W. Quasilinear Schrödinger equations: ground state and infinitely many normalized solutions. Pacific J Math, 2023, 322(1): 99-138
[24] Li Q, Wu X. Existence, multiplicity,concentration of solutions for generalized quasilinear Schrödinger equations with critical growth. J Math Phys, 2017, 58(4): 1-30
[25] Li Q, Zhang J, Nie J, Wang W. Semiclassical solutions of generalized quasilinear Schrödinger equations with competing potentials. Complex Var Elliptic Equ, 2023, 68(7): 1045-1076
[26] Liu J, Wang Y, Wang Z Q. Soliton solutions for quasilinear Schrödinger equations. II. J Differential Equations, 2003, 187(2): 473-493
[27] Liu S, Yin L F. Quasilinear Schrödinger equations with concave and convex nonlinearities. Calc Var Partial Differential Equations, 2023, 62(3): 1-14
[28] Liu X Q, Liu J, Wang Z Q. Quasilinear elliptic equations via perturbation method. Proc Amer Math Soc, 2013, 141(1): 253-263
[29] Meng X, Ji S. Positive ground state solutions for generalized quasilinear Schrödinger equations with critical growth. J Geom Anal, 2023, 33(12): 1-20
[30] Miyagaki O H, Soares S H M. Soliton solutions for quasilinear Schrödinger equations with critical growth. J Differential Equations, 2010, 248(4): 722-744
[31] Poppenberg M, Schmitt K, Wang Z Q. On the existence of soliton solutions to quasilinear S}chrödinger equations. Calc Var Partial Differential Equations, 2002, 14(3): 329-344
[32] Shen Y, Wang, Y. Soliton solutions for generalized quasilinear Schrödinger equations. Nonlinear Anal, 2013, 80: 194-201
[33] Wu T F. Multiplicity of positive solutions for semilinear elliptic equations in $\Bbb R^N$. Proc Roy Soc Edinburgh Sect A, 2008, 138(3): 647-670
[34] Xiang C L. Remarks on nondegeneracy of ground states for quasilinear Schrödinger equations. Discrete Contin Dyn Syst, 2016, 36(10): 5789-5800
[35] Xue Y, Zhong X, Tang C. Existence of ground state solutions for critical quasilinear Schrödinger equations with steep potential well. Adv Nonlinear Stud, 2022, 22(1): 619-634
[36] Yang M, Santos C A, Ubilla P, Zhou J. On a defocusing quasilinear Schrödinger equation with singular term. Discrete Contin Dyn Syst, 2023, 43(1): 507-536
[37] Yuan Z, Yu J. Existence and multiplicity of positive solutions for a class of quasilinear Schrödinger equations in $\Bbb R^{N}$. Discrete Contin Dyn Syst Ser S, 2014, 14(9): 3285-3303
