×
Fang, Xiang-Dong; Liu, Mei-Qi

Localized solutions of higher topological type for semiclassical generalized quasilinear Schrödinger equations. (English) Zbl 1512.35200

Z. Angew. Math. Phys. 74, No. 2, Paper No. 81, 22 p. (2023).
Summary: We consider the following semiclassical generalized quasilinear Schrödinger equation \[ -\varepsilon^2\text{div} (g^2(v)\nabla v)+\varepsilon^2g(v) g'(v)|\nabla v|^2+V(x)v=f(v), \quad x\in{\mathbb{R}}^N, \] where \(V\in C^1({\mathbb{R}}^N)\) is bounded and \(f\) is odd in \(v\) and satisfies a monotonicity condition. We establish the existence of multiple localized solutions concentrating at the set of critical points of \(V\).

MSC:

35J20 Variational methods for second-order elliptic equations
35J62 Quasilinear elliptic equations
35Q55 NLS equations (nonlinear Schrödinger equations)

Keywords:

Schrödinger equation; Nehari manifold; multiplicity of localized solutions
Cite Review PDF
Full Text: DOI

References:

[1] Byeon, J.; Wang, Z-Q, Standing waves with a critical frequency for nonlinear Schrödinger equations. II, Calc. Var. Partial Differ. Equ., 18, 207-219 (2003) · Zbl 1073.35199 · doi:10.1007/s00526-002-0191-8
[2] Chen, S.; Wang, Z-Q, Localized nodal solutions of higher topological type for semiclassical nonlinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 56, 1, 26 (2017) · Zbl 1379.35074 · doi:10.1007/s00526-016-1094-4
[3] Chen, JH; Huang, XJ; Cheng, BT; Tang, XH, Existence and concentration behavior of ground state solutions for a class of generalized quasilinear Schrödinger equations in \({\mathbb{R} }^N\), Acta Math. Sci. Ser. B (Engl. Ed.), 40, 5, 1495-1524 (2020) · Zbl 1513.35232
[4] Del Pino, M.; Felmer, P., Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. Equ., 4, 121-137 (1996) · Zbl 0844.35032 · doi:10.1007/BF01189950
[5] Deng, YB; Peng, SJ; Wang, JX, Nodal soliton solutions for generalized quasilinear Schrödinger equations, J. Math. Phys., 55, 5, 051501 (2014) · Zbl 1292.81037 · doi:10.1063/1.4874108
[6] Deng, YB; Peng, SJ; Yan, SS, Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth, J. Differ. Equ., 258, 115-147 (2015) · Zbl 1302.35345 · doi:10.1016/j.jde.2014.09.006
[7] Deng, YB; Peng, SJ; Yan, SS, Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, J. Differ. Equ., 260, 1228-1262 (2016) · Zbl 1330.35099 · doi:10.1016/j.jde.2015.09.021
[8] D’Aprile, T.; Pistoia, A., On the number of sign-changing solution of a semiclassical nonlinear Schrödinger equation, Adv. Differ. Equ., 12, 737-758 (2007) · Zbl 1207.35040
[9] Fang, XD, Multiple solutions of higher topological type for semiclassical nonlinear Schrödinger equations, NoDEA Nonlinear Differ. Equ. Appl., 28, 1, 26 (2021) · Zbl 1459.35106 · doi:10.1007/s00030-021-00673-z
[10] Fang, XD; Szulkin, A., Multiple solutions for a quasilinear Schrödinger equation, J. Differ. Equ., 254, 2015-2032 (2013) · Zbl 1263.35113 · doi:10.1016/j.jde.2012.11.017
[11] Floer, A.; Weinstein, A., Nonspreading wave pockets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69, 397-408 (1986) · Zbl 0613.35076 · doi:10.1016/0022-1236(86)90096-0
[12] Gilbarg, D.; Trudinger, NS, Elliptic Partial Differential Equations of Second Order (1983), Berlin: Springer, Berlin · Zbl 0562.35001
[13] Li, QQ; Wu, X., Existence, multiplicity, and concentration of solutions for generalized quasilinear Schrödinger equations with critical growth, J. Math. Phys., 58, 4, 041501 (2017) · Zbl 1369.35085 · doi:10.1063/1.4982035
[14] Liu, JQ; Wang, YQ; Wang, ZQ, Soliton solutions for quasilinear Schrödinger equations, II, J. Differ. Equ., 187, 473-493 (2003) · Zbl 1229.35268 · doi:10.1016/S0022-0396(02)00064-5
[15] Rabinowitz, PH, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43, 270-291 (1992) · Zbl 0763.35087 · doi:10.1007/BF00946631
[16] Rabinowitz, PH, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics (1986), Providence: American Mathematical Society, Providence · Zbl 0609.58002 · doi:10.1090/cbms/065
[17] Reed, M.; Simon, B., Methods of Modern Mathematical Physics (1978), New York: Academic Press, New York · Zbl 0401.47001
[18] Shen, YT; Wang, YJ, Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal., 80, 194-201 (2013) · Zbl 1278.35233 · doi:10.1016/j.na.2012.10.005
[19] Struwe, M., Variational Methods (1996), Berlin: Springer, Berlin · Zbl 0864.49001 · doi:10.1007/978-3-662-03212-1
[20] Szulkin, A.; Weth, T.; Gao, DY; Motreanu, D., The method of Nehari manifold, Handbook of Nonconvex Analysis and Applications, 597-632 (2010), Boston: International Press, Boston · Zbl 1218.58010
[21] Wang, X., On concentration of positive bound states of nonlinear Schrödinger equations, Commun. Math. Phys., 153, 229-244 (1993) · Zbl 0795.35118 · doi:10.1007/BF02096642
[22] Willem, M., Minimax Theorems (1996), Boston: Birkhäuser, Boston · Zbl 0856.49001 · doi:10.1007/978-1-4612-4146-1
[23] Wei, J.; Weth, T., On the number of nodal solutions to a singularly perturbed Neumann problem, Manuscr. Math., 117, 3, 333-344 (2005) · Zbl 1205.35031 · doi:10.1007/s00229-005-0561-6
[24] Zhang, J.; Zhao, FK, Multiple solutions for a semiclassical Schrödinger equation, Nonlinear Anal., 75, 4, 1834-1842 (2012) · Zbl 1241.35193 · doi:10.1016/j.na.2011.09.032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.