Normalized Solutions for Schrödinger-Bopp-Podolsky Systems with Critical Choquard-Type Nonlinearity on Bounded Domains
Li Chen, Li Wang
TL;DR
The paper studies normalized solutions to a Schrödinger-Bopp-Podolsky system on a bounded domain with a critical Choquard-type nonlinearity. It reduces the problem by solving for the electrostatic potential $\phi$ as $\Phi(u)$, yielding a single-variable energy $J$ on the mass-constrained sphere $S_{r,b}$, and employs a truncation technique to obtain a coercive, bounded-below functional $J^T$ with a $(PS)_d$ condition. A genus-based minimax scheme applied to $J^T$ generates negative critical levels $d_k$, which correspond to critical points of the truncated functional; these are shown to be critical for the original functional, establishing the existence of at least $k$ normalized solutions for sufficiently small mass $b$. The results highlight multiplicity phenomena for nonlocal, gauge-field coupled systems on bounded domains under Navier-type constraints and nonuniform charge distributions $q(x)$, driven by a critical Choquard nonlinearity.
Abstract
In this paper, we study normalized solutions for the following critical Schrödinger-Bopp-Podolsky system: $$-Δu + q(x)φu = λu + |u|^{p-2}u + \bigl(I_α* |u|^{3+α}\bigr)|u|^{1+α}u,\quad \text{in } Ω_r,$$ $$-Δφ+ Δ^2φ= q(x)u^2, \ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\ \text{ in } Ω_r,$$ 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 $\int_{Ω_r} |u|^2\, \mathrm d x = b^2$ for some $b > 0$. Here $α> 0$, $I_α$ 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.