Existence and qualitative properties of ground state solutions for the Schr\"{o}dinger-Bopp-Podolsky system

TL;DR

This paper proves the existence, uniqueness, and qualitative properties of ground state solutions for a nonlinear, nonlocal Schr"{o}dinger-Bopp-Podolsky system, including symmetry, positivity, and decay characteristics.

Contribution

It establishes the existence of ground state solutions using variational methods and analyzes their qualitative properties under electrostatic conditions.

Findings

Existence of nontrivial solutions via mountain-pass lemma

Ground state solutions are positive, radially symmetric, and exponentially decaying

Asymptotic behavior of solutions with respect to parameter a

Abstract

This paper concerns the existence and related properties of solutions to the Schr\"{o}dinger-Bopp-Podolsky system, which reduces to a nonlinear and nonlocal partial differential equation describing a Schr\"{o}dinger field coupled with its electromagnetic field in Bopp-Podolsky theory under purely electrostatic conditions. Firstly, by applying the mountain-pass lemma, we obtain the existence of nontrivial solutions. Then, through some estimates of the ground state energy, we prove the existence of ground state solutions. By exploring the relationship between solutions and paths associated with critical points, we further demonstrate that the obtained solutions are ground states of mountain-pass type. Additionally, the positivity, radial symmetry, rotational invariance, and exponential decay of the ground state solutions are considered. Finally, in the radial case, we explore the asymptotic behavior of the obtained solutions with respect to $a$.