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: $$-Δu + q(x)φu = λu + |u|^{p-2}u + \bigl(I_α* |u|^{3+α}\bigr)|u|^{1+α}u,\quad \text{in } Ωr,$$ $$-Δφ+ Δ^2φ= q(x)u^2, \ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\ \text{ in } Ω_r,$$ where $Ω_r \subset \mathbb R^3$ is a smooth bounded domain, $p \in \left(2, \frac{8}{3}\right)$, $q(x) \in C(\barΩ_r) \backslash {0}$ and $λ\in \mathbb R$ is the Lagrange multiplier associated with the constraint $\int{Ωr} |u|^2\, \mathrm d x = b^2$ for some $b > 0$. Here $α> 0$, $Iα$ denotes the Riesz potential, and the domain parameter $r$ reflects the size of $Ω_r$ 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 $b^* > 0$ such that the system admits multiple normalized solutions whenever $b \in (0, b^*)$ under Navier boundary conditions.