Existence and multiplicity results for the zero mass Schrödinger-Bopp-Podolsky system with critical growth

TL;DR

The paper addresses the zero-mass Schrödinger-Bopp-Podolsky system with critical growth in $\mathbb{R}^3$, where the electrostatic potential satisfies a higher-order equation and the mass term vanishes. A new variational framework in the space $E$ is developed to handle the nonlocal BP energy and lack of compactness, incorporating a Nehari-Pohozaev manifold, a mountain-pass structure, and Lions-type concentration-compactness arguments; multiplicity is obtained via Perera's abstract critical point theorem. The main results are the existence of a positive ground state for $p\in(3,6)$ and, for the more restrictive $p\in(4,6)$, the existence of $m$ pairs of nontrivial solutions with positive energy for all sufficiently large $\mu$ (i.e., $\mu\ge\mu_m$). These contributions advance the theory of zero-mass nonlinear Schrödinger equations coupled with Bopp-Podolsky electromagnetism and provide a variational/topological framework for obtaining multiplicity in critical-growth regimes, particularly in radial settings.

Abstract

In this paper we study the following zero mass Schrödinger-Bopp-Podolsky system with critical growth \[ \begin{cases} -Δu +q^2φu=μ|u|^{p-2}u+|u|^4u\\ -Δφ+a^2Δ^2φ=4πu^2, \end{cases} \] where $a>0$, $q\neq0$, $μ>0$ is a parameter and $p\in(3,6)$. By introducing a new functional framework developed by Caponio et al. \cite{Cd}, we first establish the existence of positive ground state solutions for the case of $p\in(3,6)$. Moreover, for the case of $p\in(4,6)$, multiplicity results are obtained by applying an abstract critical point theorem due to Perera \cite{Pe}.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• Lemma 2.6
• proof