论文标题
基态对具有谐波电势的分数非线性椭圆方程的独特性
Uniqueness of ground states to fractional nonlinear elliptic equations with harmonic potential
论文作者
论文摘要
在本文中，我们证明了基础状态的独特性，具有谐波潜力的以下分数非线性椭圆方程，$ qur（-Δ）^s U+ \ left（ω+ | x | x |^2 \右） $ω>-λ_{1，s} $，$ 2 <p <\ frac {2n} {（n-2s）^ +} $，$λ_{1，s}> 0 $是$（ - δ）^s + | x | x | x |^2 $的最低特征。分数laplacian $（ - δ）^s $的特征为$ \ MATHCAL {f}（（ - δ）^{s} u）（ξ）= |ξ|^{2S} \ Mathcal {f}（f}（f}（ξ）（U）（U）（U）（U）$ n $ n $ deN $ n $ n $，for n $ n $ n $ n $，这在\ cite {ss}中解决了一个关于接地状态的独特性的悬而未决的问题。
In this paper, we prove the uniqueness of ground states to the following fractional nonlinear elliptic equation with harmonic potential, $$ (-Δ)^s u+ \left(ω+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\,\, \R^n, $$ where $n \geq 1$, $0<s<1$, $ω>-λ_{1,s}$, $2<p<\frac{2n}{(n-2s)^+}$, $λ_{1,s}>0$ is the lowest eigenvalue of $(-Δ)^s + |x|^2$. The fractional Laplacian $(-Δ)^s$ is characterized as $\mathcal{F}((-Δ)^{s}u)(ξ)=|ξ|^{2s} \mathcal{F}(u)(ξ)$ for $ξ\in \R^n$, where $\mathcal{F}$ denotes the Fourier transform. This solves an open question in \cite{SS} concerning the uniqueness of ground states.
