Multiplicity of normalized solutions for a class of nonlinear Schrodinger-Poisson-Slater equations

In this paper, we prove a multiplicity result of solutions for the following stationary Schr\"odinger-Poisson-Slater equations \begin{equation}\label{eq-abstract} -\Delta u - \lambda u + (\left | x \right |^{-1}\ast \left | u \right |^2) u - |u|^{p-2}u = 0 \ \mbox{ in } \ \mathbb{R}^{3}, \end{equation} where $\lambda\in \R$ is a parameter, and $p\in (2,6)$. The solutions we obtained have a prescribed $L^2$-norm. Our proofs are mainly inspired by a recent work of Bartsch and De Valeriola [7].