Asymptotic profile of least energy solutions to the nonlinear Schrödinger-Bopp-Podolsky system

Abstract

Consider the following nonlinear Schrödinger--Bopp--Podolsky system in ${\mathbb{R}}^3$: \[ \begin{cases} - Δv + v + ϕv = v |v|^{p - 2}; \\ β^2 Δ^2 ϕ- Δϕ= 4 πv^2, \end{cases} \] where $\beta >0$ and $3<p<6$, the unknowns being $v$, $\varphi :{\mathbb{R}}^3\to \mathbb{R}$. 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 Schrödinger--Poisson system in ${\mathbb{R}}^3$: \[ \begin{cases} - Δv + v + ϕv = v |v|^{p - 2}; \\ - Δϕ= 4 πv^2. \end{cases} \]

12 pages, comments are welcome