Existence and qualitative properties of ground state solutions for the Schrödinger-Bopp-Podolsky system
Sheng Wang, Juan Huang
TL;DR
We address existence and qualitative properties of ground state solutions for the Schrödinger-Bopp-Podolsky system on $\\mathbb{R}^3$ with $\\omega>0$, $\\mu>0$, $a>0$, and nonlinearity exponent $p\\in(2,6]$. A variational energy $J$ is formulated with the nonlocal potential $\\phi_u=\\mathcal{K}*|u|^2$, and solutions are obtained via a mountain-pass argument using a Cerami sequence; ground states are characterized by a Nehari minimax on $\\mathcal{N}$, and the mountain-pass level is shown to coincide with the ground-state energy. The paper proves positivity, radial symmetry, rotational invariance (up to a phase), and exponential decay of ground states, and establishes strong radial convergence as the Bopp-Podolsky parameter $a$ tends to zero, yielding a Schrödinger-Poisson-Slater limit. In the radial setting, strong convergence results are obtained: $(u^a,\\phi^a)\\to (u^0,\\phi^0)$ in $H^1_r\\times D^{1,2}_r$, with $(u^0,\\phi^0)$ solving the limit equation $-\\Delta u^0+\\omega u^0-\\mu(1/|x|*|u^0|^2)u^0=|u^0|^{p-2}u^0$ and $\\phi^0=|x|^{-1}*|u^0|^2$. The results extend variational methods to nonlocal gauge-coupled systems and provide detailed asymptotics and symmetry properties relevant to the physical Bopp-Podolsky model.
Abstract
This paper concerns the existence and related properties of solutions to the Schrödinger-Bopp-Podolsky system, which reduces to a nonlinear and nonlocal partial differential equation describing a Schrödinger 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 $a$.