Normalized Solutions for Schrödinger-Bopp-Podolsky Systems with Critical Choquard-Type Nonlinearity on Bounded Domains

Abstract

In this paper, we study normalized solutions for the following critical Schrödinger-Bopp-Podolsky system: $$-\Delta u+q(x)\varphi u=\lambda u+|u{|}^{p-2}u+(I_{\alpha }\ast |u{|}^{3+\alpha })|u{|}^{1+\alpha }u,\text{in }{\Omega }_r,$$ $$-\Delta \varphi +{\Delta }^2\varphi =q(x)u^2,\text{ in }{\Omega }_r,$$ where ${\Omega }_r\subset {\mathbb{R}}^3$ is a smooth bounded domain, $p\in \left(2,\frac{8}{3}\right)$, $q(x)\in C({\bar{\Omega }}_r)\backslash \{0\}$ and $\lambda \in \mathbb{R}$ is the Lagrange multiplier associated with the constraint ${\int }_{{\Omega }_r}|u{|}^2\mathrm{d}x=b^2$ for some $b>0$. Here $\alpha >0$, $I_{\alpha }$ denotes the Riesz potential, and the domain parameter $r$ reflects the size of ${\Omega }_r$ whose precise definition will be given in Section 3. By applying a special minimax principle together with a truncation technique, we prove that there exists $b^{\ast }>0$ such that the system admits multiple normalized solutions whenever $b\in (0,b^{\ast })$ under Navier boundary conditions.

20 pages