Sign-changing solutions for fractional Schrödinger–Poisson system in ℝ³

ABSTRACT

We consider the existence of sign-changing solutions for the following fractional Schrödinger–Poisson system $$\begin{array}{c}\hfill \left\{\begin{array}{c}{(-\mathrm{\Delta })}^{s}u+V(x)u+\lambda \varphi (x)u=f(u),\&x\in {\mathbb{R}}^3,\hfill \\ {(-\mathrm{\Delta })}^t\varphi =u^2,\&x\in {\mathbb{R}}^3,\hfill \end{array}\right.\end{array}$$

where $s,t\in (0,1)$, $\lambda$ is a positive parameter, $V(x):{\mathbb{R}}^3\to {\mathbb{R}}^{+}$ is a continuous potential function. Since multiple nonlocal terms are involving in the system, some new techniques are applied to prove the existence of a least energy sign-changing solution $u_{\lambda }$. Moreover, we prove that the energy of any sign-changing solution to this system is strictly larger than twice of the ground state energy. Furthermore, the asymptotic behavior of $u_{\lambda }$ as $\lambda \to 0^{+}$ is also analyzed.

KEYWORDS:

• Laplacian
• Schrödinger-Poisson system
• sign-changing solution
• asymptotic behavior

AMS SUBJECT CLASSIFICATIONS:

• 35J20
• 35J60

Acknowledgements

The author would like to express his thanks to anonymous referees for their valuable comments and suggestions which improved the work.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant number 11671162] and the excellent doctorial dissertation cultivation [grant number 2015YBYB022] from Central China Normal University.