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

Abstract

We consider the fractional Schrödinger-Poisson equation
\(\begin{cases}(-\Delta)^su-\lambda u+\phi u=|u|^{p-2}u,& x\in\mathbb{R}^3,\\ (-\Delta)^t\phi=u^2,& x\in\mathbb{R}^3,\end{cases}\)
where \(s,t\in(0,1)\) satisfy \(2s+2t>3\), \(p\in(\frac{4s+6}{3},2^*_s)\) 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.