Normalized solutions for the Schrödinger-Poisson system with doubly critical growth

Abstract

In this paper we are concerned with normalized solutions to the Schrödinger-Poisson system with doubly critical growth \[\begin{cases}-\Delta u-\phi |u|^3u=\lambda u+\mu|u|^{q-2}u+|u|^4u, &x \in \mathbb R^{3},\\ -\Delta \phi=|u|^5, &x \in \mathbb R^{3},\end{cases} \]and prescribed mass\[ \int_{\mathbb R^3}|u|^2dx=a^2,\]where $a> 0$ is a constant, $\mu> 0$ is a parameter and $2< q< 6$. In the $L^2$-subcritical case, we study the multiplicity of normalized solutions by applying the truncation technique, and the genus theory; and in the $L^2$-supercritical case, we obtain a couple of normalized solutions by developing a fiber map. Under both cases, to recover the loss of compactness of the energy functional caused by the critical growth, we need to adopt the concentration-compactness principle. Our results complement and improve some related studies for the Schrödinger-Poisson system with nonlocal critical term in the literature.