Normalized solutions for Schrödinger-Bopp-Podolsky system

Abstract

In this paper, we study the following energy functional originates from the Schrödinger-Bopp-Podolsky system $$I(u)=\frac{1}{2}{\int }_{{\mathbb{R}}^3}|\nabla u{|}^{2}dx+\frac{1}{4}{\int }_{{\mathbb{R}}^3}{\varphi }_u{u}^{2}dx-\frac{1}{p}{\int }_{{\mathbb{R}}^3}|u{|}^{p}dx$$ constrained on $B_{\rho }=\left\{u\in H^1({\mathbb{R}}^3,C):{\Vert u\Vert }_2=\rho \right\},$ 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.

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• Normalized solutions for Sobolev critical Schrödinger-Bopp-Podolsky systems