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Normalized Solutions for Schrödinger-Bopp-Podolsky Systems in Bounded Domains with General Nonlinearities

Kai Sheng

TL;DR

The paper addresses the challenge of finding normalized standing wave solutions to the Schrödinger–Bopp–Podolsky system in bounded three-dimensional domains under a mass constraint. It reduces the coupled system to a single functional J by solving the BP equation for φ and employs a perturbation framework with a penalized functional J_{r,μ}, together with Cerami compactness and a Pohožaev-type identity, to prove existence and multiplicity of normalized solutions for small μ under Navier or Neumann boundary conditions. Notably, it handles general nonlinearities f with 2 < p < 6, including the L^2-supercritical range (10/3 < p < 6) without the Ambrosetti–Rabinowitz condition, and shows the existence of a normalized ground state on star-shaped domains. The results extend the theory of normalized solutions in bounded domains for nonlocal BP-coupled systems and provide a variational framework that accommodates nonconstant charge distributions q(x) and general nonlinearities.

Abstract

In this paper, by adapting the perturbation method, we study normalized standing wave solutions for the following nonlinear Schrödinger-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. 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.

Normalized Solutions for Schrödinger-Bopp-Podolsky Systems in Bounded Domains with General Nonlinearities

TL;DR

The paper addresses the challenge of finding normalized standing wave solutions to the Schrödinger–Bopp–Podolsky system in bounded three-dimensional domains under a mass constraint. It reduces the coupled system to a single functional J by solving the BP equation for φ and employs a perturbation framework with a penalized functional J_{r,μ}, together with Cerami compactness and a Pohožaev-type identity, to prove existence and multiplicity of normalized solutions for small μ under Navier or Neumann boundary conditions. Notably, it handles general nonlinearities f with 2 < p < 6, including the L^2-supercritical range (10/3 < p < 6) without the Ambrosetti–Rabinowitz condition, and shows the existence of a normalized ground state on star-shaped domains. The results extend the theory of normalized solutions in bounded domains for nonlocal BP-coupled systems and provide a variational framework that accommodates nonconstant charge distributions q(x) and general nonlinearities.

Abstract

In this paper, by adapting the perturbation method, we study normalized standing wave solutions for the following nonlinear Schrödinger-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. 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.

Paper Structure

This paper contains 13 sections, 21 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Background on the Bopp--Podolsky theory
  3. Motivation and main results
  4. Preliminaries
  5. Notation and functional spaces
  6. The variational setting
  7. Functional reduction
  8. The existence and multiplicity results
  9. The perturbation functional
  10. Compactness of Cerami sequences
  11. A non-existence result.
  12. Proofs of Theorems \ref{['thm:1.1']}--\ref{['thm:1.3']}
  13. Existence of ground states

Key Result

Theorem 1.1

Suppose that ($f_1$)--($f_3$) hold or ($f_1$) and ($f_4$) hold. Then there exists $\mu_{0}>0$, depending only on $\Omega$, $p$, and $q(x)$, such that, for every $0<\mu<\mu_{0}$ problem 1.1 under 1.2--1.3 admits a solution

Theorems & Definitions (40)

  • Theorem 1.1: Existence result with boundary conditions (I)
  • Theorem 1.2: Existence result with boundary conditions (II)
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • ...and 30 more