Prescribed mass standing waves for Schr\"{o}dinger-Maxwell equations with combined nonlinearities

Abstract

In the present paper, we study the following Schr\"{o}dinger-Maxwell equation with combined nonlinearities \begin{align*} \displaystyle - \Delta u+\lambda u+ \left(|x|^{-1}\ast |u|^2\right)u =|u|^{p-2}u +\mu|u|^{q-2}u\quad \text{in} \ \mathbb{ R}^3 \quad \quad \text{and}\quad \quad \int_{\mathbb{R}^3}|u|^2dx=a^2, \end{align*} where $a>0$, $\mu\in \mathbb{R}$, $2<q\leq \frac{10}{3}\leq p<6$ with $q\neq p$, $\ast$ denotes the convolution and $\lambda\in \mathbb{R}$ appears as a Lagrange multiplier. Under some mild assumptions on $a$ and $\mu$, we prove some existence, nonexistence and multiplicity of normalized solution to the above equation. Moreover, the asymptotic behavior of normalized solutions is verified as $\mu\rightarrow 0$ and $q\rightarrow \frac{10}{3}$, and the stability/instability of the corresponding standing waves to the related time-dependent problem is also discussed.