Normalized solutions for Schr\"{o}dinger-Bopp-Podolsky system

Abstract

In this paper, we study the following energy functional originates from the Schr\"{o}dinger-Bopp-Podolsky system $$I(u)=\frac{1}{2}\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx+\frac{1}{4}\int_{\mathbb{R}^{3}} \phi_{u}u^{2}dx-\frac{1}{p}\int_{\mathbb{R}^{3}}|u|^{p}dx$$ constrained on $B_{\rho}=\left\{u\in H^{1}(\mathbb{R}^{3},C):\ \left\|u\right\|_{2}=\rho\right\},$ where $\rho>0.$ As such constrained problem $I(u)$ is bounded from below on $B_{\rho}$ when $p\in(2,\frac{10}{3}).$ We use minimizing method to get a normalized solution.