Multiple normalized solutions for a Sobolev critical Schrödinger-Poisson-Slater equation

Abstract

We look for solutions to the Schrödinger-Poisson-Slater equation $$-\Delta u+\lambda u-\gamma (|x{|}^{-1}\ast |u{|}^2)u-a|u{|}^{p-2}u=0\text{in}{\mathbb{R}}^3,$$ 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.

This version is the final one, corresponding to the paper now published in Journal of Differential Equations