Ground state sign-changing solutions for a class of nonlinear fractional Schr\"odinger-Poisson system in $\mathbb{R}^{3}$

Abstract

In this paper, we are concerned with the existence of the least energy sign-changing solutions for the following fractional Schr\"{o}dinger-Poisson system: \begin{align*} \left\{ \begin{aligned} &(-\Delta)^{s} u+V(x)u+\lambda\phi(x)u=f(x, u),\quad &\text{in}\, \ \mathbb{R}^{3},\\ &(-\Delta)^{t}\phi=u^{2},& \text{in}\,\ \mathbb{R}^{3}, \end{aligned} \right. \end{align*} where $\lambda\in \mathbb{R}^{+}$ is a parameter, $s, t\in (0, 1)$ and $4s+2t>3$, $(-\Delta)^{s}$ stands for the fractional Laplacian. By constraint variational method and quantitative deformation lemma, we prove that the above problem has one least energy sign-changing solution. Moreover, for any $\lambda>0$, we show that the energy of the least energy sign-changing solutions is strictly larger than two times the ground state energy. Finally, we consider $\lambda$ as a parameter and study the convergence property of the least energy sign-changing solutions as $\lambda\searrow 0$.