论文标题
关于分数nirenberg问题的密度和多样性
On the density and multiplicity of solutions to the fractional Nirenberg problem
论文作者
论文摘要
本文致力于建立一些关于分数nirenberg问题的密度和多样性的结果，这等同于研究共同不变的方程$p_σ（v）= k v^{\ frac {n+2σ}} {n+2σ} {n-2σ}} $ nitional sphere $（$ nitive $ nitive） （0,1）$和$ n \ geq 2 $，其中$p_σ$是订单$2σ$和$ k $的交织操作员是规定的曲率函数。更具体地说，通过使用变分胶合方法，对冒泡行为，扩展公式的精致分析以及吹大的分析参数，我们获得了无限的多个多重倾斜溶液的存在。特别是，我们显示了$ c^{0} $拓扑中$ g_0 $的度量的平滑曲率函数。此外，相关的分数拉普拉斯方程$（-Δ）^σu = k（x）
This paper is devoted to establishing some results on the density and multiplicity of solutions to the fractional Nirenberg problem which is equivalent to studying the conformally invariant equation $P_σ(v)=K v^{\frac{n+2σ}{n-2σ}}$ on the standard unit sphere $(\mathbb{S}^n,g_0)$ with $σ\in (0,1)$ and $n\geq 2$, where $P_σ$ is the intertwining operator of order $2σ$ and $K$ is the prescribed curvature function. More specifically, by using the variational gluing method, refined analysis of bubbling behavior, extension formula, as well as the blow up analysis arguments, we obtain the existence of infinitely many multi-bump solutions. In particular, we show the smooth curvature functions of metrics conformal to $g_0$ are dense in the $C^{0}$ topology. Moreover, the related fractional Laplacian equations $(-Δ)^σ u=K(x) u^{\frac{n+2σ}{n-2σ}}$ in $\mathbb{R}^n$, with $K(x)$ being asymptotically periodic in one of the variables, are also studied and infinitely many solutions are obtained under natural flatness assumptions.
