论文标题
Schrödinger方程的归一化阳性解决方案在无界域中具有电势
Normalized positive solutions for Schrödinger equations with potentials in unbounded domains
论文作者
论文摘要
本文介绍了Schrödinger方程的规定$ L^2 $规范的正面解决方案的存在$$其中$ω= \ Mathbb {r}^n $或$ \ MATHBB {r}^n \setMinusΩ$是一个紧凑的集合，$ρ> 0 $，$ v \ ge 0 $（也允许$ v \ equiv 0 $），允许），$ p \ in \ in \ in \ weft（2,2，+frac \ frac 4 n n n n n n n n \ right）。当$ v $验证合适的衰减假设$（d_ρ）$，或者如果$ \ | v \ | _ {l^q} $很小，对于某些$ q \ ge \ ge \ frac n2 $（$ q> 1 $ n = 2 $）时如果满足衰减假设$（d_ρ）$，则需要对$ v $进行少量假设。没有关于$ \ mathbb {r}^n \setMinusΩ$的假设。解决方案$ \ bar u $是一个界面状态，不存在基础状态解决方案，直到自主情况$ v \ equiv 0 $和$ω= \ mathbb {r}^n $。
The paper deals with the existence of positive solutions with prescribed $L^2$ norm for the Schrödinger equation $$ -Δu+λu+V(x)u=|u|^{p-2}u,\qquad u\in H^1_0(Ω),\quad\int_Ωu^2dx=ρ^2,\quadλ\in\mathbb{R}, $$ where $Ω=\mathbb{R}^N$ or $\mathbb{R}^N\setminusΩ$ is a compact set, $ρ>0$, $V\ge 0$ (also $V\equiv 0$ is allowed), $p\in \left(2,2+\frac 4 N\right)$. The existence of a positive solution $\bar u$ is proved when $V$ verifies a suitable decay assumption $(D_ρ)$, or if $\|V\|_{L^q}$ is small, for some $q\ge \frac N2$ ($q>1$ if $N=2$). No smallness assumption on $V$ is required if the decay assumption $(D_ρ)$ is fulfilled. There are no assumptions on the size of $\mathbb{R}^N\setminusΩ$. The solution $\bar u$ is a bound state and no ground state solution exists, up to the autonomous case $V\equiv 0$ and $Ω=\mathbb{R}^N$.
