Asymptotic profile of least energy solutions to the nonlinear Schr\"odinger-Bopp-Podolsky system

TL;DR

This paper studies the asymptotic behavior of least energy solutions to a nonlinear Schr"odinger-Bopp-Podolsky system in three dimensions, showing convergence to solutions of a related Schr"odinger-Poisson system as a parameter tends to zero.

Contribution

It establishes the limiting profile of least energy solutions for the Bopp-Podolsky system as the parameter beta approaches zero.

Findings

Solutions converge to Schr"odinger-Poisson solutions as beta -> 0

Characterizes the asymptotic profile of least energy solutions

Provides rigorous proof of convergence in the specified limit

Abstract

Consider the following nonlinear Schr\"odinger--Bopp--Podolsky system in $\mathbb{R}^3$: \[ \begin{cases} - \Delta v + v + \phi v = v |v|^{p - 2}; \\ \beta^2 \Delta^2 \phi - \Delta \phi = 4 \pi v^2, \end{cases} \] where $\beta > 0$ and $3 < p < 6$, the unknowns being $v$, $\phi \colon \mathbb{R}^3 \to \mathbb{R}$. We prove that, as $\beta \to 0$ and up to translations and subsequences, least energy solutions to this system converge to a least energy solution to the following nonlinear Schr\"odinger--Poisson system in $\mathbb{R}^3$: \[ \begin{cases} - \Delta v + v + \phi v = v |v|^{p - 2}; \\ - \Delta \phi = 4 \pi v^2. \end{cases} \]