Normalized Solutions for Schr\"odinger-Bopp-Podolsky Systems with Critical Choquard-Type Nonlinearity on Bounded Domains

TL;DR

This paper establishes the existence of multiple normalized solutions for a critical Schr"odinger-Bopp-Podolsky system with nonlinearities involving Riesz potentials on bounded domains, using minimax and truncation techniques.

Contribution

It introduces a novel approach combining minimax principles and truncation to find multiple solutions for a complex nonlinear PDE system with critical nonlinearity.

Findings

Existence of multiple normalized solutions for small mass parameter b

Solutions are obtained under Navier boundary conditions

The method applies to systems with critical Choquard-type nonlinearities

Abstract

In this paper, we study normalized solutions for the following critical Schr\"odinger-Bopp-Podolsky system: $$-\Delta u + q(x)\phi u = \lambda u + |u|^{p-2}u + \bigl(I_\alpha * |u|^{3+\alpha}\bigr)|u|^{1+\alpha}u,\quad \text{in } \Omega_r,$$ $$-\Delta\phi + \Delta^2\phi = q(x)u^2, \ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\ \text{ in } \Omega_r,$$ 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 $b^* > 0$ such that the system admits multiple normalized solutions whenever $b \in (0, b^*)$ under Navier boundary conditions.