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 $a,\rho >0$ and our unknowns are $u,\varphi :{\mathbb{R}}^3\to {\mathbb{R}}^3$ 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 Schrödinger-Poisson-Slater system under the same $L^2$-norm constraint.

16 pages

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