Normalized solutions for Schrödinger-Bopp-Podolsky systems in bounded domains

TL;DR

This work analyzes normalized (i.e., $|u|_2=1$) solutions to a Schrödinger–Bopp–Podolsky system in bounded domains, where a nonconstant coupling $q(x)$ links a nonlinear Schrödinger field to a second-order electrostatic potential. It develops two variational frameworks corresponding to Dirichlet and Neumann boundary conditions for the electrostatic potential, employing a reduction via the auxiliary map $\Phi(u)$ that solves the $\phi$-equation and Lusternik–Schnirelmann theory on constrained manifolds. For the Dirichlet case, the reduced energy $J$ on the $L^2$-sphere $B$ admits infinitely many critical points with $\omega_n\to\infty$ and $\|u_n\|\to\infty$, plus a ground state; for the Neumann case, after homogenization and a constrained reduction on a manifold $M$, one likewise obtains infinitely many critical points and corresponding solutions. The results extend multiplicity theory for gauge-coupled nonlinear PDEs in bounded domains and illustrate how variational methods with topological constraints yield rich solution structures in Schrödinger–BP systems.

Abstract

We consider an elliptic system of Schrödinger-Bopp-Podolsky type in a bounded and smooth domain of R3 with a non constant coupling factor. This kind of system has been introduced in the mathematical literature in [14] and in the last years many contributions appeared. In particular here we present the results in [2] and [34] which show existence of solutions by means of the Ljusternik-Schnirelmann theory under different boundary conditions on the electrostatic potential.

Theorems & Definitions (41)

• theorem 1: Existence result with boundary conditions (I)
• theorem 2: Existence result with boundary conditions (II)
• lemma 1
• proof
• lemma 2
• proof
• theorem 3
• proof
• proposition 1
• proof