Existence and multiplicity of sign-changing solutions for supercritical quasi-linear Schrödinger equations

Xian Zhang; Chen Huang; Xian Zhang; Chen Huang

doi:10.3934/era.2023032

Electronic Research Archive

2023, Volume 31, Issue 2: 656-674. doi: 10.3934/era.2023032

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Existence and multiplicity of sign-changing solutions for supercritical quasi-linear Schrödinger equations

Xian Zhang ¹,
Chen Huang ^{2
,
,}

1.
Business School, University of Shanghai for Science and Technology, Shanghai 200093, PR China
2.
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, PR China

Received: 25 September 2022 Revised: 29 October 2022 Accepted: 03 November 2022 Published: 16 November 2022

This paper focuses on a class of supercritical, quasi-linear Schrödinger equations. Based on the methods of invariant sets, some results about the existence and multiplicity of sign-changing solutions for supercritical equations are obtained.
- Quasi-linear Schrödinger equations,
- supercritical growth,
- variational methods,
- sign-changing solutions,
- invariant sets
Citation: Xian Zhang, Chen Huang. Existence and multiplicity of sign-changing solutions for supercritical quasi-linear Schrödinger equations[J]. Electronic Research Archive, 2023, 31(2): 656-674. doi: 10.3934/era.2023032

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Abstract

This paper focuses on a class of supercritical, quasi-linear Schrödinger equations. Based on the methods of invariant sets, some results about the existence and multiplicity of sign-changing solutions for supercritical equations are obtained.

References

[1]	A. Borovskii, A. Galkin, Dynamical modulation of an ultrashort high-intensity laser pulse in matter, J. Exp. Theor. Phys., 77 (1983), 562–573.
[2]	H. Brandi, C. Manus, G. Mainfray, T. Lehner, G. Bonnaud, Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma, Phys. Fluids. B Plasma Phys., 5 (1993), 3539–3550. https://doi.org/10.1063/1.860828 doi: 10.1063/1.860828
[3]	L. Brizhik, A. Eremko, B. Piette, W. J. Zakrzewski, Static solutions of a D-dimensional modified nonlinear Schrödinger equation, Nonlinearity, 16 (2003), 1481–1497. https://doi.org/10.1088/0951-7715/16/4/317 doi: 10.1088/0951-7715/16/4/317
[4]	W. Krolikowski, O. Bang, J. Rasmussen, J. Wyller, Modulational instability in nonlocal nonlinear Kerr media, Phys. Rev. E, 64 (2001), 016612. https://doi.org/10.1103/PhysRevE.64.016612 doi: 10.1103/PhysRevE.64.016612
[5]	S. Kurihura, Large-amplitude quasi-solitons in superfluids films, J. Phys. Soc. Japan, 50 (1981), 3262–3267. https://doi.org/10.1143/JPSJ.50.3262 doi: 10.1143/JPSJ.50.3262
[6]	H. Berestycki, P. L. Lions, Nonlinear scalar field equations, Ⅰ existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983), 313–346.
[7]	B. Li, Y. Ma, A firewall effect during the rogue wave and breather interactions to the Manakov system, Nonlinear Dyn., 2022 (2022). https://doi.org/10.1007/s11071-022-07878-6 doi: 10.1007/s11071-022-07878-6
[8]	B. Li, Y. Ma, Interaction properties between rogue wave and breathers to the manakov system arising from stationary self-focusing electromagnetic systems, Chaos Solitons Fractals, 156 (2022), 111832. https://doi.org/10.1016/j.chaos.2022.111832 doi: 10.1016/j.chaos.2022.111832
[9]	B. Li, Y. Ma, Extended generalized Darboux transformation to hybrid rogue wave and breather solutions for a nonlinear Schrödinger equation, Appl. Math. Comput., 386 (2020), 125469. https://doi.org/10.1016/j.amc.2020.125469 doi: 10.1016/j.amc.2020.125469
[10]	P. L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. Part Ⅰ, Ann. Inst. Henri Poincaré C Anal. non Lineairé, 1 (1984), 109–145. https://doi.org/10.1016/S0294-1449(16)30428-0 doi: 10.1016/S0294-1449(16)30428-0
[11]	P. L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. Part Ⅱ, Ann. Inst. Henri Poincaré Anal. non Lineairé, 1 (1984), 223–283. https://doi.org/10.1016/S0294-1449(16)30422-X doi: 10.1016/S0294-1449(16)30422-X
[12]	M. Poppenberg, K. Schmitt, Z. Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differ. Equations, 14 (2002), 329–344. https://doi.org/10.1007/s005260100105 doi: 10.1007/s005260100105
[13]	J. Liu, Y. Wang, Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations Ⅱ, J. Differ. Equations, 187 (2003), 473–493. https://doi.org/10.1016/S0022-0396(02)00064-5 doi: 10.1016/S0022-0396(02)00064-5
[14]	M. Colin, L. Jeanjean, Solutions for a quasilinear Schrödinger equations: a dual approach, Nonlinear Anal. Theory Methods Appl., 56 (2004), 213–226. https://doi.org/10.1016/j.na.2003.09.008 doi: 10.1016/j.na.2003.09.008
[15]	S. Adachi, T. Watanabe, G-invariant positive solutions for a quasilinear Schrödinger equation, Adv. Differ. Equations, 16 (2011), 289–324.
[16]	X. Fang, J. Zhang, Multiplicity of positive solutions for quasilinear elliptic equations involving critical nonlinearity, Adv. Nonlinear Anal., 9 (2020), 1420–1436. https://doi.org/10.1515/anona-2020-0058 doi: 10.1515/anona-2020-0058
[17]	H. Liu, Positive solution for a quasilinear elliptic equation involving critical or supercritical exponent, J. Math. Phys., 57 (2016), 159–180. https://doi.org/10.1063/1.4947109 doi: 10.1063/1.4947109
[18]	J. Liu, Y. Wang, Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari Method, Commun. Partial Differ. Equation, 29 (2004), 879–901. https://doi.org/10.1081/PDE-120037335 doi: 10.1081/PDE-120037335
[19]	E. Medeiros, U. Severo, On the existence of signed solution for a quasilinear elliptic problem in $\mathbb{R}^{N}$, Mat. Contemp., 32 (2007), 193–205.
[20]	E. Silva, G. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differ. Equations, 39 (2010), 1–33. https://doi.org/10.1007/s00526-009-0299-1 doi: 10.1007/s00526-009-0299-1
[21]	Y. Wang, Z. Li, Existence of solutions to quasilinear Schrödinger equations involving critical Sobolev exponent, Taiwanese J. Math., 22 (2018), 401–420. https://doi.org/10.11650/tjm/8150 doi: 10.11650/tjm/8150
[22]	H. Zhang, F. Meng, J. Zhang, Nodal solutions for quasilinear schrödinger equations with asymptotically 3-Linear nonlinearity, J. Geom. Anal., 32 (2022). https://doi.org/10.1007/s12220-022-01043-6 doi: 10.1007/s12220-022-01043-6
[23]	C. Alves, Y. Wang, Y. Shen, Soliton solutions for a class of quasilinear Schrödinger equations with a parameter, J. Differ. Equations, 259 (2015), 318–343. https://doi.org/10.1016/j.jde.2015.02.030 doi: 10.1016/j.jde.2015.02.030
[24]	C. Huang, G. Jia, Existence of positive solutions for supercritical quasilinear Schrödinger elliptic equations, J. Math. Anal. Appl., 472 (2019), 705–727. https://doi.org/10.1016/j.jmaa.2018.11.048 doi: 10.1016/j.jmaa.2018.11.048
[25]	C. Huang, G. Jia, Multiple solutions for a class of quasilinear Schrödinger equations, Complex Var. Elliptic Equations, 66 (2021), 347–359. https://doi.org/10.1080/17476933.2020.1727899 doi: 10.1080/17476933.2020.1727899
[26]	D. Costa, Z. Q. Wang, Multiplicity results for a class of superlinear elliptic problems, Pro. Amer. Math. Soc., 133 (2005), 787–794. https://doi.org/10.1090/S0002-9939-04-07635-X doi: 10.1090/S0002-9939-04-07635-X
[27]	T. Bartsch, A. Pankov, Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549–569. https://doi.org/10.1142/S0219199701000494 doi: 10.1142/S0219199701000494
[28]	J. Liu, X. Liu, Z. Q. Wang, Multiple mixed states of nodal solutions for nonlinear Schrödinger systems, Calc. Var. Partial Differ. Equations, 52 (2015), 565–586. https://doi.org/10.1007/s00526-014-0724-y doi: 10.1007/s00526-014-0724-y
[29]	T. Bartsch, Z. Liu, On a superlinear elliptic p-Laplacian equation, J. Differ. Equations, 198 (2004), 149–175. https://doi.org/10.1016/j.jde.2003.08.001 doi: 10.1016/j.jde.2003.08.001
[30]	T. Bartsch, Z. Liu, T. Weth, Nodal solutions of a p-Laplacian equation, Proc. London Math. Soc., 91 (2005), 129–152. https://doi.org/10.1112/S0024611504015187 doi: 10.1112/S0024611504015187
[31]	Z. Liu, Z. Wang, J. Zhang, Infinitely many sign-changing solutions for the nonlinear Schrödinger-Poisson system, Ann. Mat. Pura Appl., 195 (2016), 775–794. https://doi.org/10.1007/s10231-015-0489-8 doi: 10.1007/s10231-015-0489-8

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