Multiple normalized solutions for Schrödinger-Maxwell equation with Sobolev critical exponent and mixed nonlinearities

In this paper, we study the Schrödinger-Maxwell equation with critical growth$\{\begin{array}{c}-\Delta u+\lambda u+\kappa \left(\right|x{|}^{-1}⁎\left|u{|}^2\right)u=|u{|}^{4}u+\mu |u{|}^{q-2}u\text{in}{R}^3,\\ \underset{R^3}{\int }|u{|}^{2}dx=a^2,\end{array}$ where $a>0$ is prescribed, $\kappa >0$, $\mu \in R$, $2<q<6$, ⁎ denotes the convolution and $\lambda \in R$ appears as a Lagrange multiplier. Motivated by the works of Wei and Wu (2022) [41] for $\kappa =0$ and Bellazzini and Siciliano (2011) [5] for homogeneous nonlinearity, we get two normalized solutions when $q\in (2,\frac{8}{3})$ and $\mu >0$, one of which is ground state normalized solution. It is worth emphasizing that the method of Schwarz spherical rearrangement is invalid for the case of $\kappa >0$, different from the case of $\kappa =0$, and it is hard to establish the strictly subadditive inequality of least energy to exclude the dichotomy of minimizing sequence standardly. To our knowledge, the existence of second solution for the above problem has not been addressed in the current literatures. Moreover, we will show a nonexistence result of positive normalized solution when $\mu \le 0$ and $q\in (2,6)$, which can be regarded as a generalization and improvement of Jeanjean and Le (2021) [21] from the case of $\mu =0$ to the case of $\mu \ne 0$.