Multiple normalized solutions for a Sobolev critical Schr\"{o}dinger-Poisson-Slater equation

Abstract

We look for solutions to the Schr\"{o}dinger-Poisson-Slater equation $$- \Delta u + \lambda u - \gamma (|x|^{-1} * |u|^2) u - a |u|^{p-2}u = 0 \quad \text{in} \quad \mathbb{R}^3, $$ which satisfy \begin{equation*} \int_{\mathbb{R}^3}|u|^2 \, dx = c \end{equation*} for some prescribed $c>0$. Here $ u \in H^1(\mathbb{R}^3)$, $\gamma \in \mathbb{R},$ $ a \in \mathbb{R}$ and $p \in (\frac{10}{3}, 6]$. When $\gamma >0$ and $a > 0$, both in the Sobolev subcritical case $p \in (\frac{10}{3}, 6)$ and in the Sobolev critical case $p=6$, we show that there exists a $c_1>0$ such that, for any $c \in (0,c_1)$, the equation admits two solutions $u_c^+$ and $u_c^-$ which can be characterized respectively as a local minima and as a mountain pass critical point of the associated {\it Energy} functional restricted to the norm constraint. In the case $\gamma >0$ and $a < 0$, we show that, for any $p \in (\frac{10}{3},6]$ and any $c>0$, the equation admits a solution which is a global minimizer. Finally, in the case $\gamma <0$, $a >0$ and $p=6$ we show that it does not admit positive solutions.