在~1969--1970 年, L. Nirenberg 提出下面的问题: 给定二维标准球面~$(\mathbb{S}^2,g_{\mathbb{S}^2})$ 上的光滑函数~$K$(如果需要, 可以假设 ~$K$ 在某个意义下非常接近某个正常数), 是否存在共形于~$g_{\mathbb{S}^2}$ 且标量曲率或~Gauss曲率等于~$K$ 的度量~$g$? 如果设~$g=e^{2w}g_{\mathbb{S}^2}$,那么这个问题等价于求解下面的二阶椭圆型偏微分方程\[-\Delta_{g_{\mathbb{S}^2}}w +1=K e^{2w},\quad \mbox{于~}\mathbb{S}^2,\]这里~ $\Delta_{g_{\mathbb{S}^2}}$ 是~ $(\mathbb{S}^2, g_{\mathbb{S}^2})$ 上的~ Laplace-Beltrami 算子. Nirenberg 问题的高维情形, $n\geq 3$, 等价于求解\[-\Delta_{g_{\Sn}}v+c(n)R_{g_{\Sn}}v=c(n)Kv^{\frac{n+2}{n-2}}, \quad\mbox{于 }\ \Sn,\]其中~$c(n)=(n-2)/(4(n-1)), R_{g_{\Sn}}=n(n-1)$ 是~ $(\Sn, g_{\Sn})$ 的标量曲率, $v=e^{\frac{n-2}{4}w}$. 近半个世纪来, Nirenberg 问题引起广泛的研究, 如见~ Moser, Chang-Yang, Chang-Liu, Han, Escobar-Schoen, Schoen-Zhang, Bahri-Coron, Li 等的相应参考文献.共形不变算子, 如共形~Laplace 算子~$L_g:=-\Delta_{g}+c(n)R_{g}$, 是几何学与物理学中非常重要的研究课题. 早在~20 世纪初, 人们发现无质量粒子~(massless particles)在弯曲的时空中满足的方程具有共形不变性.在粒子物理中有这样一个问题, 除了共形~Laplace 算子以外, 在共形~Riemannian 流形上是否还存在其它非平凡的共形不变算子;如果存在, 怎么完全地构造这些共形不变算子? 注意到粒子物理中~AdS/CFT 对偶原理, ~C. Fefferman 和~ R. Graham 于 ~1985 年提出通过构造外围度量的方法去构造所有的共形算子, 但是这些都是形式的想法,没有严格的数学推导. 基于这个想法, R. Graham 和~M. Zworski 于~ 2003 年发现渐近~Einstein流形上的~Scattering Matrices 与共形边界上的共形不变算子的关系, 构造了亚纯族的共形算子~ $P_\sigma^g$, $\sigma\in \mathbb{C}$.特别地, $P_1^g$ 等于共形~Laplace 算子~$-\Delta_{g}+c(n)R_g$,所以~ $P_1^g(1)=c(n)R_g$ 与标量曲率仅相差常数~$c(n)$. 更值得注意地, 类似于~Gauss-Bonnet 定理, 在 ~$2n+1$ 维渐近~Einstein~ 流形~$X$的共形边界~$\pa X$ 上的积分~$\int_{\pa X}P_{n}^g(1)\,\ud vol_g$是拓扑不变量. 受~Yamabe 问题启发, 我们可以问是否存在共形度量使得~$P_\sigma^g(1)$ 是常数?这方面的研究可以参见~ Qing-Raske, Gonz\'alez-Qing, Gonz\'alez-Mazzoe-Sire 的最近工作.我们问: 对于球面~$\Sn$ 上的光滑函数~$K$, 是否存在共形于~$g_{\Sn}$ 的度量~$g$ 使得~$P_\sigma^g(1)=K$? 当~$\sigma\in (0,n/2)$ 时, 这个问题等价于求解方程\be\label{zy1}P^{g_{\Sn}}_{\sigma}(v)=Kv^{\frac{n+2\sigma}{n-2\sigma}},\quad \mbox{于 }\ \mathbb{S}^n,\ee其中\[P^{g_{\Sn}}_\sigma=\frac{\Gamma(B+\frac{1}{2}+\sigma)}{\Gamma(B+\frac{1}{2}-\sigma)},\quad B=\sqrt{-\Delta_{g_{\Sn}}+\left(\frac{n-1}{2}\right)^2},\]$\Gamma(\cdot)$ 是~ Gamma 函数, 而~ $\Delta_{g_{\Sn}}$ 是~ $(\Sn, g_{\Sn})$ 上的~ Laplace-Beltrami 算子. $\sigma=1$ 的情形就是上面提到的~ Nirenberg 问题.\medskip本论文关心~ $0<\sigma<1$ 的情形, 得到如下主要结果:\begin{itemize} \item 建立~ Kazdan-Warner~ 型的恒等式, 从而给出~\eqref{zy1} 存在性的必要条件; \item 考虑球极对称的~ $K$, 即~ $K(-\xi)=K(\xi)$ 对任意~ $\xi\in\Sn$, 建立~ Moser, Escobar-Schoen 型的存在性结果; \item 利用非线性泛函分析方法, 建立扰动结果, 即~ $\|K-1\|_{L^\infty(\Sn)}$ 充分小意味着解的存在性; \item 在假设~ $K$ 满足自然的平坦的条件下, 证明解的存在性和解族的紧致性.\end{itemize}不同于~ $\sigma=1$ 的情形, $P^{g_{\Sn}}_\sigma$ 现在是~ $2\sigma$ 阶的非局部椭圆算子. 在处理非局部算子过程中, 本论文得到一系列具有自身意义的结果,如线性非局部方程正解的~ Harnack 不等式, 局部 ~Schauder 估计, B\^ocher 型孤立奇异解的分类; Liouville 型定理等等.在本论文的最后一章, 我们还研究了球面上的分数阶 ~Yamabe 流, 证明了其长时间的存在性及收敛性. 作为应用, 我们建立了一种分数阶多孔质方程的解的渐近消失行为.

In 1969--1970, L. Nirenberg posed the following problem: Given a (positive) smooth function $K$ on standard $2$-sphere$(\mathbb{S}^2,g_{\mathbb{S}^2})$ ("close" to the constant function, if we want), is it the scalar curvature (or Gauss curvature) of a metric$g$ conformal to $g_{\mathbb{S}^2}$. If we let $g=e^{2w}g_{\mathbb{S}^2}$,then the Nirenberg problem is equivalent to solving the equation\[-\Delta_{g_{\mathbb{S}^2}}w +1=K e^{2w},\quad \mbox{on }\mathbb{S}^2,\]where $\Delta_{g_{\mathbb{S}^2}}$ is the Laplace-Beltrami operator on $(\Sn, g_{\mathbb{S}^2})$.The Nirenberg problem and its higher dimension cases, i.e.,\[-\Delta_{g_{\Sn}}v+c(n)R_0v=c(n)Kv^{\frac{n+2}{n-2}}, \quad\mbox{on }\ \Sn\mbox{ for }n\geq 3,\]where $c(n)=(n-2)/(4(n-1)), R_0=n(n-1)$ is the scalar curvature of $(\Sn, g_{\Sn})$ and $v=e^{\frac{n-2}{4}w}$, have drawn a wide range of attention, seethe work of Moser, Chang-Yang, Chang-Liu, Han, Escobar-Schoen, Schoen-Zhang, Bahri-Coron, Li and etc.Studying conformal invariant operators, such as the conformal Laplace operator $L_g:=-\Delta_{g}+c(n)R_{g}$,is an important topic to geometers and physicists. In the very early part of 20 century, it was shown that the equations of massless particles on curvedspace-time exhibit conformal invariance. A problem from particle physics says that,besides from the conformal Laplace operator, are there other nontrivial conformal invariant operators? If there are, how to construct them?With noticing the principle of the AdS/CFT correspondence, C. Fefferman and R. Graham in 1985 proposed a way to construct all conformal invariant operators by constructing some proper ambient metrics.Based on this idea, R. Graham and M. Zworski in 2003 discovered the relationship between scattering matrices on asymptotical Einsteinmanifolds and conformally invariant operators on their conformal infinity,and thus constructed a meromorphic family of conformally invariant operators $P_\sigma^g$, $\sigma\in \mathbb{C}$.In particular, $P_1^g$ is equal to the conformal Laplace operator $-\Delta_{g}v+c(n)R_g$,and, upon a constant depending only on the dimension, $P_1^g(1)$ is the scalar curvature. Furthermore,similar to the Gauss-Bonnet theorem, on the conformal infinity $\pa X$ of $2n+1$-dimensional asymptotical Einsteinmanifold $X$, the integral $\int_{\pa X}P_{n}^g(1)\,\ud vol_g$is topologically invariant. Motivated by the Yamabe problem, we can ask is there conformal metric $g $ such that$P_\sigma^g(1)$ is a constant?The study of this question can be found in the recent work of Qing-Raske, Gonz\'alez-Qing, Gonz\'alez-Mazzoe-Sire and so on.We ask: Given a smooth function $K $ on $\Sn$, is there a metric $g$ conformal to$g_{\Sn}$ such that $P_\sigma^g(1)= K$. When $\sigma\in (0,n/2)$, this problem is equivalent to solving the equation\be\label{ab1}P^{g_{\Sn}}_{\sigma}(v)=Kv^{\frac{n+2\sigma}{n-2\sigma}},\quad \mbox{on }\mathbb{S}^n\eewhere $\sigma\in (0,n/2)$\[P^{g_{\Sn}}_\sigma=\frac{\Gamma(B+\frac{1}{2}+\sigma)}{\Gamma(B+\frac{1}{2}-\sigma)},\quad B=\sqrt{-\Delta_{g_{\Sn}}+\left(\frac{n-1}{2}\right)^2},\]$\Gamma(\cdot)$ is the Gamma function and $\Delta_{g_{\Sn}}$ is the Laplace-Beltrami operator on $(\Sn, g_{\Sn})$. When $\sigma=1$,this problem coincides with the Nirenberg problem mentioned above.The thesis is devoted to the case $0<\sigma<1$ and includes the following main results:\begin{itemize} \item Derive a Kazdan-Warner type identity which is a necessary condition for existence; \item For the antipodally symmetric $K$, i.e., $K(\xi)=K(-\xi)$ $\forall~\xi\in \Sn$, Moser, Escobar-Schoen type existence result holds; \item By the nonlinear functional analysis approach, a perturbation result is established, namely, smallnessof $\|K-1\|_{L^\infty(\Sn)}$ implies the existence; \item With some natural flatness assumptions on $K$, existence and compactness of solutions are established.\end{itemize}Different from the case $\sigma=1$, now $P^{g_{\Sn}}_\sigma$ is a nonlocal elliptic operator of order $2\sigma$.In dealing with the nonlocal operators, this thesis alsoobtains some results of independent interest, for instance, Harnack inequality, local Schauder estimates, B\^ocher type theorem, Liouville theorem and etc.In the last chapter of thesis, we also studied a frational Yamabe flow on $\Sn$, and obtained the long time existence and convergence. As an application, weestablished the extinction behavior of solutions a fractional porous media equation.