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 Schrodinger-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, $p\in (2,6)$. In contrary to the existing results on the multiplicity of solutions to (0.1) in the literature, such as Ambrosetti and Ruiz [4], Siciliano [22], and also the recent [11], the solutions we ob- tained have a prescribed L2-norm. Our proofs are inspired by a recent work of Bartsch and De Valeriola [6].