In this paper, we intend to study the following Schrödinger-Bopp-Podolsky with doubly critical growth in \(\mathbb {R}^3\) : \(\begin{aligned} {\left\{ \begin{array}{ll}-\Delta u-\phi |u|^{3}u=\lambda u+\mu |u|^{q-2}u+|u|^{4}u & \text{ in } \mathbb {R}^3, \\ -\Delta \phi +a^{2}\Delta ^{2}\phi =4\pi ^{2}|u|^{5}& \text{ in } \mathbb {R}^3, \end{array}\right. } \end{aligned}\) and prescribed mass \(\int _{\mathbb {R}^3} |u|^{2}dx=\varrho ^2,\) where \(\mu , a, \varrho >0,\) \(\lambda \in \mathbb {R}\) , \(q\in (2, 6)\) . For the \(L^2\) -subcritical case, we show the existence of multiple normalized solutions by using the truncation technique and the genus theory. For the \(L^2\) -supercritical case, we obtain a couple of normalized solutions by developing the auxiliary functional. For both cases, in order to overcome the loss of compactness of the energy functional due to the doubly critical growth, the concentration-compactness principle is needed to overcome this difficulty. As far as we know, this paper is the first attempt that the multiplicity of normalized solutions to the Schrödinger-Bopp-Podolsky system with doubly critical growth. The most obvious and important feature is that we establish some new techniques to prove our results.
