Existence and multiplicity results for the zero mass Schr\"{o}dinger-Bopp-Podolsky system with critical growth

TL;DR

This paper establishes the existence and multiplicity of solutions for a zero mass Schrödinger-Bopp-Podolsky system with critical growth, using a new functional framework and critical point theory.

Contribution

It introduces a novel functional framework and proves existence and multiplicity results for the system with critical growth and parameters.

Findings

Existence of positive ground state solutions for p in (3,6).

Multiplicity of solutions for p in (4,6).

Application of critical point theorems to the system.

Abstract

In this paper we study the following zero mass Schr\"{o}dinger-Bopp-Podolsky system with critical growth \[ \begin{cases} -\Delta u +q^2\phi u=\mu|u|^{p-2}u+|u|^4u\\ -\Delta \phi+a^2\Delta^2\phi=4\pi u^2, \end{cases} \] where $a>0$, $q\neq0$, $\mu>0$ 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}.