Sign-changing solutions for some class of elliptic system

Abstract

This paper is concerned with the existence of multiple non-radial sign-changing solutions for $\begin{array}{rl}& -\mathrm{\Delta }u+u+\alpha K\left(|x|\right)\mathrm{\Phi }\left(x\right)u=|u{|}^{p-2}u,x\in {\mathbb{R}}^3,\\ & -\mathrm{\Delta }\mathrm{\Phi }=K\left(|x|\right)u^2,x\in {\mathbb{R}}^3,\end{array}$ where 2<p<6, $\alpha <0$ can be regarded as a parameter and $K\left(r\right)(r=|x|)$ is a positive continuous function. We proved that the above equation possesses a non-radial sign-changing solutions with exactly k maximum points and k minimum points for suitable range of α.

COMMUNICATED BY:

• J. Cheng

KEYWORDS:

• Elliptic system
• non-radial sign-changing solutions
• Lyapunov–Schmidt reduction

AMS SUBJECT CLASSIFICATIONS:

• 35J10
• 35J20
• 35J60

Acknowledgements

The author thanks the anonymous referee for helpful suggestions, which helped improve the readability and quality of the paper.

Disclosure statement

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

Additional information

Funding

This paper was supported by the National Natural Science Foundation of China [grant no. 11601139]; the Natural Science Foundation of Hubei Province [grant no. 2019CFB522]; the Scientic Research and Innovation Team of Hubei Normal University [grant no. 2019CZ010].