论文标题
$ \ mathbb {r}^3 $中，最少能源解决方案的生存和限制行为对约束的schrödinger-bopp-Podolsky系统
Existence and limit behavior of least energy solutions to constrained Schrödinger-Bopp-Podolsky systems in $\mathbb{R}^3$
论文作者
论文摘要
考虑以下以下schrödinger-bopp-podolsky系统在$ \ mathbb {r}^3 $下的$ l^2 $ -norm约束，\ [ \ begin {case} -ΔU +ωu + ϕu = u | u |^{p-2}，\ newline -Δϕ+ a^2δ^2 ϕ =4πu^2，\ newline \ | u \ | _ {l^2} =ρ， \ end {cases} \]其中$ a，ρ> 0 $，我们的未知数为$ u，ϕ \ colon \ mathbb {r}^3 \ to \ mathbb {r}^3 $ and $ω\ in \ mathbb {r} $。我们证明，如果$ 2 <p <3 $（分别，$ 3 <p <10/3 $），$ρ>0 $的$足够小（分别，足够大），那么该系统承认最少的能量解决方案。此外，我们证明，如果$ 2 <p <14/5 $和$ρ>0 $足够小，那么最少的能量解决方案是径向对称的，并且作为$ a \ to $ a \ to $ $ a \ to $，它们会收敛于Schrödinger-Poisson-Slater System在相同$ l^2 $ -NOMM构造下的最小能量解决方案。
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 $a,ρ>0$ and our unknowns are $u,ϕ\colon\mathbb{R}^3\to\mathbb{R}^3$ and $ω\in\mathbb{R}$. We prove that if $2<p<3$ (resp., $3<p<10/3$) and $ρ>0$ is sufficiently small (resp., sufficiently large), then this system admits a least energy solution. Moreover, we prove that if $2<p<14/5$ and $ρ>0$ 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.
