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doi: 10.3934/cpaa.2021193
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Multiple non-radially symmetrical nodal solutions for the Schrödinger system with positive quasilinear term

Jianqing Chen 1,2, and  Qian Zhang 1,,

1. 

School of Mathematics and Statistics, Fujian Normal University, Qishan Campus, Fuzhou 350117, China
2. 

FJKLMAA and Center for Applied Mathematics of Fujian Province(FJNU), Fuzhou, 350117, China

*Corresponding author

Received  June 2021 Early access November 2021

Fund Project: This work was supported by the National Natural Science Foundation of China (11871152) and Key Project of Natural Science Foundation of Fujian (2020J02035)

This paper is concerned with the following quasilinear Schrödinger system in the entire space
$ \mathbb R^{N}(N\geq3) $
:
$ \left\{\begin{aligned} &-\Delta u+A(x)u+\frac{k}{2}\triangle(u^{2})u = \frac{2\alpha }{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},\\ &-\Delta v+Bv+\frac{k}{2}\triangle(v^{2})v = \frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,\\ & u(x)\to 0,\ \ v(x)\to 0\ \ \hbox{as}\ |x|\to \infty,\end{aligned}\right. $
where
$ \alpha,\beta>1 $
,
$ 2<\alpha+\beta<2^* = \frac{2N}{N-2} $
 and
$ k >0 $
 is a parameter. By using the principle of symmetric criticality and the moser iteration, for any given integer
$ \xi\geq2 $
, we construct a non-radially symmetrical nodal solution with its
$ 2\xi $
 nodal domains. Our results can be looked on as a generalization to results by Alves, Wang and Shen (Soliton solutions for a class of quasilinear Schrödinger equations with a parameter. J. Differ. Equ. 259 (2015) 318-343).
Keywords: Quasilinear Schrödinger system, Non-radially symmetrical nodal solutions, Moser iteration.
Mathematics Subject Classification: Primary: 35J20, 35J05, 35J60.
Citation: Jianqing Chen, Qian Zhang. Multiple non-radially symmetrical nodal solutions for the Schrödinger system with positive quasilinear term. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021193
References:
[1]

C. Alves, Y. Wang and Y. Shen, Soliton solutions for a class of quasilinear Schrödinger equations with a parameter, J. Differ. Equ., 259 (2015), 318-343.  doi: 10.1016/j.jde.2015.02.030.  Google Scholar

[2]

T. Bartsch and Z. Wang, Existence and multiplicity results for superlinear elliptic problems on $\mathbb R^{N}$, Commun. Partial Differ. Equ., 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149.  Google Scholar

[3]

H. Berestycki and P. Lion, Nonlinear Scalar field equations, Ⅰ, Arch. Ration. Mech. Anal., 82 (1983), 313-346.  doi: 10.1007/BF00250555.  Google Scholar

[4]

G. Bonnaud, H. Brandi, C. Manus, G. Mainfray and T. Lehner, Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. Ⅰ: paraxial approximation, Phys. Fluids, 5 (1993), 3539-3550.   Google Scholar

[5]

A. Borovskii and A. Galkin, Dynamical modulation of an ultrashort high-intensity laser pulse in matter, J. Exp. Theor. Phys., 77 (1993), 562-573.   Google Scholar

[6]

A. Bouard, N. Hayashi and J. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Commun. Math. Phys., 189 (1997), 73-105.  doi: 10.1007/s002200050191.  Google Scholar

[7]

H. Brézis and E. Lieb, A relation between pointwise convergence of function and convergence of functional, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[8]

L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations, Expos. Math., 4 (1986), 278-288.   Google Scholar

[9]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: a dual approach, Nonlinear Anal., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008.  Google Scholar

[10]

D. Costa and Z. Wang, Multiplicity results for a class of superlinear elliptic problems, Proc. Amer. Math. Soc., 133 (2005), 787-794.  doi: 10.1090/S0002-9939-04-07635-X.  Google Scholar

[11]

Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for generalized quasilinear Schrödinger equations, J. Math. Phys., 55 (2014), 051501.  doi: 10.1063/1.4874108.  Google Scholar

[12]

Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, J. Math. Phys., 54 (2013), 011504.  doi: 10.1063/1.4774153.  Google Scholar

[13]

J. Gamboa and J. Zhou, Antisymmetric solutions for a class of quasilinear defocusing Schrödinger equations, Electron. J. Qual. Theory Differ. Equ., 16 (2020), 1-18.  doi: 10.14232/ejqtde.2020.1.16.  Google Scholar

[14]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001.  Google Scholar

[15]

C. Huang and G. Jia, Existence of positive solutions for supercritical quasilinear Schrödinger elliptic equations, J. Math. Anal. Appl., 472 (2019), 705-727.  doi: 10.1016/j.jmaa.2018.11.048.  Google Scholar

[16]

S. Kurihura, Large-amplitude quasi-solitons in superfluids films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.  doi: 10.1143/JPSJ.50.3801.  Google Scholar

[17]

P. Lions, The concentration-compactness principle in the calculus of variations, Part 1-2, Ann. Inst. H. Poincaré., 1 (1984), 109-145 and 223–283.  Google Scholar

[18]

H. Liu, Positive solution for a quasilinear elliptic equation involving critical or supercritical exponent, J. Math. Phys., 57 (2016), 159-180.  doi: 10.1063/1.4947109.  Google Scholar

[19]

J. Liu and Z. Wang, Soliton solutions for quasilinear Schrödinger equations Ⅱ, J. Differ. Equ., 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[20]

J. Liu, Y. Wang and Z. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Commun. Partial Differ. Equ., 29 (2004), 879-901.  doi: 10.1081/PDE-120037335.  Google Scholar

[21]

V. Moroz and J. Schaftingen, Ground states of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007.  Google Scholar

[22]

M. Poppenberg, K. Schmitt and Z. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 14 (2002), 329-344.  doi: 10.1007/s005260100105.  Google Scholar

[23]

P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

[24]

B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E., 50 (1994), 687-689.   Google Scholar

[25]

E. Silva and G. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differ. Equ., 39 (2010), 1-33.  doi: 10.1007/s00526-009-0299-1.  Google Scholar

[26]

A. Szulkin and S. Waliullah, Sign-changing and symmetry-breaking solutions to singular problems, Complex Var. Theory Appl., 57 (2012), 1191-1208.  doi: 10.1080/17476933.2010.504849.  Google Scholar

[27]

Y. Wang, Multiplicity of solutions for singular quasilinear Schrödinger equations with critical exponents, J. Math. Anal. Appl., 458 (2018), 1027-1043.  doi: 10.1016/j.jmaa.2017.10.015.  Google Scholar

[28]

M. Willem, Minimax Theorems, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[29]

W. Zhang and X. Liu, Infinitely many sign-changing solutions for a quasilinear elliptic equation in $\mathbb R^{N}$, J. Math. Anal. Appl., 427 (2015), 722-740.  doi: 10.1016/j.jmaa.2015.02.070.  Google Scholar

show all references

References:
[1]

C. Alves, Y. Wang and Y. Shen, Soliton solutions for a class of quasilinear Schrödinger equations with a parameter, J. Differ. Equ., 259 (2015), 318-343.  doi: 10.1016/j.jde.2015.02.030.  Google Scholar

[2]

T. Bartsch and Z. Wang, Existence and multiplicity results for superlinear elliptic problems on $\mathbb R^{N}$, Commun. Partial Differ. Equ., 20 (1995), 1725-1741.  doi: 10.1080/03605309508821149.  Google Scholar

[3]

H. Berestycki and P. Lion, Nonlinear Scalar field equations, Ⅰ, Arch. Ration. Mech. Anal., 82 (1983), 313-346.  doi: 10.1007/BF00250555.  Google Scholar

[4]

G. Bonnaud, H. Brandi, C. Manus, G. Mainfray and T. Lehner, Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. Ⅰ: paraxial approximation, Phys. Fluids, 5 (1993), 3539-3550.   Google Scholar

[5]

A. Borovskii and A. Galkin, Dynamical modulation of an ultrashort high-intensity laser pulse in matter, J. Exp. Theor. Phys., 77 (1993), 562-573.   Google Scholar

[6]

A. Bouard, N. Hayashi and J. Saut, Global existence of small solutions to a relativistic nonlinear Schrödinger equation, Commun. Math. Phys., 189 (1997), 73-105.  doi: 10.1007/s002200050191.  Google Scholar

[7]

H. Brézis and E. Lieb, A relation between pointwise convergence of function and convergence of functional, Proc. Amer. Math. Soc., 88 (1983), 486-490.  doi: 10.2307/2044999.  Google Scholar

[8]

L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations, Expos. Math., 4 (1986), 278-288.   Google Scholar

[9]

M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: a dual approach, Nonlinear Anal., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008.  Google Scholar

[10]

D. Costa and Z. Wang, Multiplicity results for a class of superlinear elliptic problems, Proc. Amer. Math. Soc., 133 (2005), 787-794.  doi: 10.1090/S0002-9939-04-07635-X.  Google Scholar

[11]

Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for generalized quasilinear Schrödinger equations, J. Math. Phys., 55 (2014), 051501.  doi: 10.1063/1.4874108.  Google Scholar

[12]

Y. Deng, S. Peng and J. Wang, Nodal soliton solutions for quasilinear Schrödinger equations with critical exponent, J. Math. Phys., 54 (2013), 011504.  doi: 10.1063/1.4774153.  Google Scholar

[13]

J. Gamboa and J. Zhou, Antisymmetric solutions for a class of quasilinear defocusing Schrödinger equations, Electron. J. Qual. Theory Differ. Equ., 16 (2020), 1-18.  doi: 10.14232/ejqtde.2020.1.16.  Google Scholar

[14]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001.  Google Scholar

[15]

C. Huang and G. Jia, Existence of positive solutions for supercritical quasilinear Schrödinger elliptic equations, J. Math. Anal. Appl., 472 (2019), 705-727.  doi: 10.1016/j.jmaa.2018.11.048.  Google Scholar

[16]

S. Kurihura, Large-amplitude quasi-solitons in superfluids films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.  doi: 10.1143/JPSJ.50.3801.  Google Scholar

[17]

P. Lions, The concentration-compactness principle in the calculus of variations, Part 1-2, Ann. Inst. H. Poincaré., 1 (1984), 109-145 and 223–283.  Google Scholar

[18]

H. Liu, Positive solution for a quasilinear elliptic equation involving critical or supercritical exponent, J. Math. Phys., 57 (2016), 159-180.  doi: 10.1063/1.4947109.  Google Scholar

[19]

J. Liu and Z. Wang, Soliton solutions for quasilinear Schrödinger equations Ⅱ, J. Differ. Equ., 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar

[20]

J. Liu, Y. Wang and Z. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Commun. Partial Differ. Equ., 29 (2004), 879-901.  doi: 10.1081/PDE-120037335.  Google Scholar

[21]

V. Moroz and J. Schaftingen, Ground states of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.  doi: 10.1016/j.jfa.2013.04.007.  Google Scholar

[22]

M. Poppenberg, K. Schmitt and Z. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differ. Equ., 14 (2002), 329-344.  doi: 10.1007/s005260100105.  Google Scholar

[23]

P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.  Google Scholar

[24]

B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E., 50 (1994), 687-689.   Google Scholar

[25]

E. Silva and G. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. Partial Differ. Equ., 39 (2010), 1-33.  doi: 10.1007/s00526-009-0299-1.  Google Scholar

[26]

A. Szulkin and S. Waliullah, Sign-changing and symmetry-breaking solutions to singular problems, Complex Var. Theory Appl., 57 (2012), 1191-1208.  doi: 10.1080/17476933.2010.504849.  Google Scholar

[27]

Y. Wang, Multiplicity of solutions for singular quasilinear Schrödinger equations with critical exponents, J. Math. Anal. Appl., 458 (2018), 1027-1043.  doi: 10.1016/j.jmaa.2017.10.015.  Google Scholar

[28]

M. Willem, Minimax Theorems, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[29]

W. Zhang and X. Liu, Infinitely many sign-changing solutions for a quasilinear elliptic equation in $\mathbb R^{N}$, J. Math. Anal. Appl., 427 (2015), 722-740.  doi: 10.1016/j.jmaa.2015.02.070.  Google Scholar

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