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On a nonlinear Schrödinger-Bopp-Podolsky system in the zero mass case: functional framework and existence

Erasmo Caponio, Pietro d'Avenia, Alessio Pomponio, Gaetano Siciliano, Lianfeng Yang

Abstract

In this paper, we consider in $\mathbb{R}^3$ the following zero mass Schrödinger-Bopp-Podolsky system \[ \begin{cases} -Δu +q^2φu=|u|^{p-2}u\\ -Δφ+a^2Δ^2φ=4πu^2 \end{cases} \] where $a>0$, $q\ne 0$ and $p\in (3,6)$. Inspired by [Ruiz, Arch. Ration. Mech. Anal. 198 (2010)], we introduce a Sobolev space $\mathcal{E}$ endowed with a norm containing a nonlocal term. Firstly, we provide some fundamental properties for the space $\mathcal{E}$ including embeddings into Lebesgue spaces. Moreover a general lower bound for the Bopp-Podolsky energy is obtained. Based on these facts, by applying a perturbation argument, we finally prove the existence of a weak solution to the above system.

On a nonlinear Schrödinger-Bopp-Podolsky system in the zero mass case: functional framework and existence

Abstract

In this paper, we consider in the following zero mass Schrödinger-Bopp-Podolsky system where , and . Inspired by [Ruiz, Arch. Ration. Mech. Anal. 198 (2010)], we introduce a Sobolev space endowed with a norm containing a nonlocal term. Firstly, we provide some fundamental properties for the space including embeddings into Lebesgue spaces. Moreover a general lower bound for the Bopp-Podolsky energy is obtained. Based on these facts, by applying a perturbation argument, we finally prove the existence of a weak solution to the above system.

Paper Structure

This paper contains 5 sections, 28 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. The electrostatic potential equation
  3. Properties of the Bopp-Podolsky energy
  4. The functional setting
  5. The existence result

Key Result

Theorem 1.1

We have where Moreover, for each $u\in {\mathcal{E}}$, $\phi_u$ is the unique weak solution in $\mathcal{A}$ of $-\Delta \phi_u+a^2\Delta^2\phi_u=4\pi u^2$ in ${\mathbb{R}^3}$, and

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • ...and 43 more