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

Jianqing Chen; Qian Zhang

doi:10.3934/cpaa.2021193

Article Contents

2022, Volume 21, Issue 2: 669-686. Doi: 10.3934/cpaa.2021193

This issue Previous Article Synchronized and ground-state solutions to a coupled Schrödinger system Next Article Nonnegative solutions to a doubly degenerate nutrient taxis system

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
^*Corresponding author

Received: May 31, 2021

Early access: November 2021

Published: February 2022

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

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Abstract

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.

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References

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