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

TL;DR

This paper investigates the existence, symmetry, and limit behavior of least energy solutions to a constrained Schr"odinger-Bopp-Podolsky system in three-dimensional space, revealing conditions on parameters for solutions and their convergence as a parameter tends to zero.

Contribution

It establishes existence and symmetry results for least energy solutions under various parameter regimes and analyzes their limit behavior as the parameter approaches zero.

Findings

Existence of least energy solutions for small and large $\rho$ depending on $p$.

Radial symmetry of solutions for certain $p$ and small $\rho$.

Solutions converge to Schr"odinger-Poisson-Slater solutions as $a \to 0$.

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.