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,

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u + \omega u + \phi u = u|u|^{p-2},\\ -\Delta \phi + a^2\Delta ^2\phi =4\pi u^2,\\ \Vert u\Vert _{L^2}=\rho , \end{array}\right. } \end{aligned}$$

where $a,\rho >0$ are fixed, with our unknowns being $u,\phi :{\mathbb {R}}^3\rightarrow {\mathbb {R}}$ 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 $a\rightarrow 0$, they converge to a least energy solution of the Schrödinger–Poisson–Slater system under the same $L^2$-norm constraint.