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.