Existence and limit behavior of least energy solutions to constrained Schrödinger-Bopp-Podolsky systems in $\mathbb{R}^3$

Abstract

Consider the following Schrödinger-Bopp-Podolsky system in $\mathbb{R}^3$ under an $L^2$-norm constraint, \[ \begin{cases} -Δu + ωu + ϕu = u|u|^{p-2},\newline -Δϕ+ a^2Δ^2ϕ=4πu^2,\newline \|u\|_{L^2}=ρ, \end{cases} \] where $a,ρ>0$ and our unknowns are $u,ϕ\colon\mathbb{R}^3\to\mathbb{R}^3$ and $ω\in\mathbb{R}$. We prove that if $2 0$ is sufficiently small (resp., sufficiently large), then this system admits a least energy solution. Moreover, we prove that if $2 0$ is sufficiently small, then least energy solutions are radially symmetric up to translation and as $a\to 0$, they converge to a least energy solution of the Schrödinger-Poisson-Slater system under the same $L^2$-norm constraint.