New concentrated solutions for the nonlinear Schrödinger–Newton system

ABSTRACT

In this paper, we study the following nonlinear Schrödinger–Newton system $\{\begin{array}{ll}\mathrm{\Delta }u-V\left(x\right)u+\mathrm{\Psi }u=0,& x\in {\mathbb{R}}^3,\\ {\displaystyle \mathrm{\Delta }\mathrm{\Psi }+\frac{1}{2}{u}^2=0,}& x\in {\mathbb{R}}^3,\end{array}$which is a nonlinear system obtained by coupling the linear Schrödinger equation of quantum mechanics with the gravitation law of Newtonian mechanics. Assuming that $V\left(y\right)$ is radial and satisfies some algebraic decay at the infinity, we construct infinitely many non-radial positive solutions which have polygonal symmetry with respect to $y_1$ and $y_2$ and are even in $y_2$ for the system above by the Lyapunov-Schmidt reduction method. Moreover, we have overcame some new difficulties caused by the non-local term.

KEYWORDS:

• Nonlinear Schrödinger–Newton system
• infinitely many solutions
• new solutions
• finite dimensional reduction

AMS:

• 35B25
• 35J20
• 35J60

Acknowledgments

The authors would like to thank Chunhua Wang from Central China Normal University for the helpful discussions with her.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

Additional information

Funding

This paper was supported by National Natural Science Foundation of China [grant number 12071169] and the Fundamental Research Funds for the Central Universities [grant number KJ02072020- 0319].