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

TL;DR

This paper investigates the existence, nonexistence, and multiplicity of prescribed mass standing wave solutions for a Schrödinger-Maxwell equation with combined nonlinearities, analyzing their behavior and stability under various parameter limits.

Contribution

It introduces new results on normalized solutions for the Schrödinger-Maxwell equations with combined nonlinearities, including asymptotic analysis and stability considerations.

Findings

Existence of normalized solutions under certain conditions.

Nonexistence results for specific parameter ranges.

Asymptotic behavior of solutions as parameters vary.

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.