# American Institute of Mathematical Sciences

February  2022, 21(2): 669-686. doi: 10.3934/cpaa.2021193

## Multiple non-radially symmetrical nodal solutions for the Schrödinger system with positive quasilinear term

 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 Published  February 2022 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).
Citation: Jianqing Chen, Qian Zhang. Multiple non-radially symmetrical nodal solutions for the Schrödinger system with positive quasilinear term. Communications on Pure and Applied Analysis, 2022, 21 (2) : 669-686. 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. [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. [3] H. Berestycki and P. Lion, Nonlinear Scalar field equations, Ⅰ, Arch. Ration. Mech. Anal., 82 (1983), 313-346.  doi: 10.1007/BF00250555. [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. [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. [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. [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. [8] L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations, Expos. Math., 4 (1986), 278-288. [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. [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. [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. [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. [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. [14] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001. [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. [16] S. Kurihura, Large-amplitude quasi-solitons in superfluids films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.  doi: 10.1143/JPSJ.50.3801. [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. [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. [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. [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. [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. [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. [23] P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631. [24] B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E., 50 (1994), 687-689. [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. [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. [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. [28] M. Willem, Minimax Theorems, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1. [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.

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. [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. [3] H. Berestycki and P. Lion, Nonlinear Scalar field equations, Ⅰ, Arch. Ration. Mech. Anal., 82 (1983), 313-346.  doi: 10.1007/BF00250555. [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. [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. [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. [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. [8] L. Brüll and H. Lange, Solitary waves for quasilinear Schrödinger equations, Expos. Math., 4 (1986), 278-288. [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. [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. [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. [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. [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. [14] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2001. [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. [16] S. Kurihura, Large-amplitude quasi-solitons in superfluids films, J. Phys. Soc. Japan, 50 (1981), 3262-3267.  doi: 10.1143/JPSJ.50.3801. [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. [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. [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. [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. [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. [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. [23] P. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631. [24] B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E., 50 (1994), 687-689. [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. [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. [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. [28] M. Willem, Minimax Theorems, Birkhäuser, 1996. doi: 10.1007/978-1-4612-4146-1. [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.
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