Sign-changing solutions for a fractional Schrödinger–Poisson system

ABSTRACT

In this paper, we deal with the existence and multiplicity of $\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{l}$ sign-changing solutions for fractional Schrödinger–Poisson system: (℘) $\left\{\begin{array}{l}(-\mathrm{\Delta }{)}^{s}u+u+\varphi u=f(u),\\ (-\mathrm{\Delta }{)}^{\alpha }\varphi =u^2,\end{array}\right.\text{in}{\mathbb{R}}^3$(℘) where $s\in (\frac{3}{4},1)$, $\alpha \in (0,1)$ and f is a continuous function. Based on perturbation approach and the method of invariant sets of descending flow, we obtain the existence and multiplicity of $\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{l}$ sign-changing solutions of system ($\mathcal{P}$). In addition, by applying the constrained variational method incorporated with Brouwer degree theory, we prove that system ($\mathcal{P}$) possesses at least one $\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{l}$ ground state sign-changing solution. Furthermore, we show that the least energy of sign-changing solutions exceed twice than the least energy, and when f is odd, system ($\mathcal{P}$) admits infinitely many nontrivial solutions.

KEYWORDS:

• Fractional Schrödinger–Poisson system
• sign-changing solutions
• perturbation approach
• invariant sets of descending flow
• Brouwer degree theory

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35J20
• 35J60

Disclosure statement

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

Additional information

Funding

This research was partially supported by National Natural Science Foundation of China 12071486 and the Fundamental Research Funds for the Central Universities of Central South University 2019zzts210.