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

Abstract

Consider the following Schr\"odinger-Bopp-Podolsky system in $\mathbb{R}^3$ under an $L^2$-norm constraint, \[ \begin{cases} -\Delta u + \omega u + \phi u = u|u|^{p-2},\newline -\Delta \phi + a^2\Delta^2\phi=4\pi u^2,\newline \|u\|_{L^2}=\rho, \end{cases} \] where $a,\rho>0$ and our unknowns are $u,\phi\colon\mathbb{R}^3\to\mathbb{R}^3$ and $\omega\in\mathbb{R}$. We prove that if $2<p<3$ (resp., $3<p<10/3$) and $\rho>0$ is sufficiently small (resp., sufficiently large), then this system admits a least energy solution. Moreover, we prove that if $2<p<14/5$ and $\rho>0$ 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 Schr\"odinger-Poisson-Slater system under the same $L^2$-norm constraint.