Existence and multiplicity of normalized solutions for a class of fractional Schrödinger–Poisson equations

Abstract

We consider the fractional Schrödinger-Poisson equation
$\{\begin{array}{ll}(-\mathrm{\Delta }{)}^{s}u-\lambda u+\varphi u=|u{|}^{p-2}u,& x\in {\mathbb{R}}^3,\\ (-\mathrm{\Delta }{)}^t\varphi =u^2,& x\in {\mathbb{R}}^3,\end{array}$
where $s,t\in (0,1)$ satisfy $2s+2t>3$, $p\in (\frac{4s+6}{3},2_s^{\ast })$ and $\lambda \in \mathbb{R}$ is an undetermined parameter. We deal with the case where the associated functional is not bounded below on the $L^2$-unit sphere and show the existence of infinitely many solutions $(u,\lambda )$ with $u$ having prescribed $L^2$-norm.