Ground state sign-changing solutions for a class of nonlinear fractional Schrödinger–Poisson system in $${\mathbb {R}}^{3}$$R3

Abstract

In this paper, we are concerned with the existence of the least energy sign-changing solutions for the following fractional Schrodinger–Poisson system: $$\begin{aligned} \left\{ \begin{aligned}&(-\Delta )^{s} u+V(x)u+\lambda \phi (x)u=f(x, u),\quad&\text {\mathbb {R}}^{3},\\&(-\Delta )^{t}\phi =u^{2},&\text {in}\, {\mathbb {R}}^{3}, \end{aligned} \right. \end{aligned}$$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 $$\lambda \searrow 0$$.