Normalized Solutions for Schr\"odinger-Bopp-Podolsky Systems in Bounded Domains with General Nonlinearities

TL;DR

This paper establishes the existence and multiplicity of normalized standing wave solutions for a Schr"odinger-Bopp-Podolsky system in bounded domains, using a perturbation approach and considering various boundary conditions and nonlinearities.

Contribution

It introduces a perturbation framework to find normalized solutions for the Schr"odinger-Bopp-Podolsky system with general nonlinearities in bounded domains.

Findings

Existence of normalized solutions for all masses in a certain interval.

Multiple solutions when the nonlinearity is odd.

Existence of a normalized ground state in star-shaped domains.

Abstract

In this paper, by adapting the perturbation method, we study normalized standing wave solutions for the following nonlinear Schr\"odinger-Bopp-Podolsky system: - Delta u + q(x) phi u = omega u + f(u) in Omega, - Delta phi + a^2 Delta^2 phi = q(x) u^2 in Omega, where Omega is a smooth bounded domain in R^3, a > 0, and omega is the Lagrange multiplier associated with the L^2 mass constraint integral over Omega of u^2 equals mu, and f: R -> R is a continuous function satisfying some technical conditions. We introduce a perturbation framework for the problem and investigate normalized solutions. In particular, we prove the existence of normalized solutions for all masses mu in an interval (0, mu_0), under either Navier or Neumann boundary conditions for phi. Moreover, when f is odd, we obtain multiplicity of normalized solutions; and if Omega is star-shaped, we further obtain a normalized ground state solution.