Normalized solutions for Sobolev critical Schr\"odinger-Bopp-Podolsky systems

TL;DR

This paper investigates the existence of normalized ground state solutions for a Sobolev critical Schrödinger-Bopp-Podolsky system with a mass constraint, employing a constraint minimizing approach to establish local minimizers and solutions.

Contribution

It introduces a new constraint minimizing method to prove the existence of normalized ground states for the Sobolev critical Schrödinger-Bopp-Podolsky system.

Findings

Existence of a local minimizer for the system.

Existence of normalized ground state solutions.

Application of a constraint minimizing approach.

Abstract

We study the Sobolev critical Schr\"odinger-Bopp-Podolsky system \begin{gather*} -\Delta u+\phi u=\lambda u+\mu|u|^{p-2}u+|u|^4u\quad \text{in }\mathbb{R}^3, -\Delta\phi+\Delta^2\phi=4\pi u^2\quad \text{in } \mathbb{R}^3, \end{gather*} under the mass constraint \[ \int_{\mathbb{R}^3}u^2\,dx=c \] for some prescribed $c>0$, where $2<p<8/3$, $\mu>0$ is a parameter, and $\lambda\in\mathbb{R}$ is a Lagrange multiplier. By developing a constraint minimizing approach, we show that the above system admits a local minimizer. Furthermore, we establish the existence of normalized ground state solutions.