Unified results for existence and compactness in the prescribed fractional Q-curvature problem

Abstract

In this paper we study the problem of prescribing fractional Q-curvature of order $2\sigma$ for a conformal metric on the standard sphere $\mathbb {S}^n$ with $\sigma \in (0,n/2)$ and $n\ge 3$. Compactness and existence results are obtained in terms of the flatness order $\beta$ of the prescribed curvature function K. Making use of integral representations and perturbation result, we develop a unified approach to obtain these results when $\beta \in [n-2\sigma ,n)$ for all $\sigma \in (0,n/2)$. This work generalizes the corresponding results of Jin-Li-Xiong (Math Ann 369:109–151, 2017) for $\beta \in (n-2\sigma ,n)$.

Appendix A

In this section, we review some results about the blow up profiles for integral equations obtained in Jin-Li-Xiong [27]. For any $x\in \mathbb {R}^{n}$ and $r>0,$ ${B}_{r}(x)$ denotes the ball in $\mathbb {R}^{n}$ with radius r and center x, and $B_{r}:=B_{r}(0).$

Let $\Omega$ be a domain in $\mathbb {R}^{n}$ and $K_{i}$ are nonnegative bounded functions in $\mathbb {R}^{n}.$ Let $\{\tau _{i}\}_{i=1}^{\infty }$ be a sequence of nonnegative constants satisfying $\lim _{i \rightarrow \infty } \tau _{i}=0$, and set

$$\begin{aligned} p_{i}=\frac{n+2 \sigma }{n-2 \sigma }-\tau _{i}. \end{aligned}$$

Suppose that $0 \le u_{i} \in L_{{loc}}^{\infty }(\mathbb {R}^{n})$ satisfies the nonlinear integral equation

$$\begin{aligned} u_{i}(x)=\int _{\mathbb {R}^{n}} \frac{K_{i}(y) u_{i}(y)^{p_{i}}}{|x-y|^{n-2 \sigma }}\, \textrm{d}y \quad \text{ in } \,\Omega . \end{aligned}$$

(A.1)

We assume that $K_{i} \in C^{1}(\Omega )$ $(K_{i}\in C^{1,1}(\Omega )$ if $\sigma \le 1/2$) and, for some positive constants $A_{1}$ and $A_{2}$,

$$\begin{aligned} 1 / A_{1} \le K_{i}, \, \Vert K_{i}\Vert _{C^{1}(\Omega )} \le A_{2},\,\left( \Vert K_{i}\Vert _{C^{1,1}(\Omega )} \le A_{2}\, \text{ if } \,\sigma \le \frac{1}{2}\right) . \end{aligned}$$

(A.2)

Proposition A.1

(Pohozaev type identity) Let $u \ge 0$ in $\mathbb {R}^{n}$, and $u \in C(\overline{B}_{R})$ be a solution of

$$\begin{aligned} u(x)=\int _{B_{R}} \frac{K(y) u(y)^{p}}{|x-y|^{n-2 \sigma }} \,\textrm{d}y+h_{R}(x), \end{aligned}$$

where $1<p \le \frac{n+2 \sigma }{n-2 \sigma },$ and $h_{R}(x) \in C^{1}(B_{R}),$ $\nabla h_{R} \in L^{1}(B_{R}).$ Then

$$\begin{aligned}&\Big (\frac{n-2 \sigma }{2}-\frac{n}{p+1}\Big ) \int _{B_{R}} K(x) u(x)^{p+1} \,\textrm{d}x -\frac{1}{p+1} \int _{B_{R}} x \nabla K(x) u(x)^{p+1} \,\textrm{d}x \\&\quad = \frac{n-2 \sigma }{2} \int _{B_{R}} K(x) u(x)^{p} h_{R}(x) \,\textrm{d}x +\int _{B_{R}} x \nabla h_{R}(x) K(x) u(x)^{p} \,\textrm{d}x \\&\qquad -\frac{R}{p+1} \int _{\partial B_{R}} K(x) u(x)^{p+1} \, \textrm{d}s. \end{aligned}$$

Proposition A.2

Suppose that $0 \le u_{i} \in L_{{loc}}^{\infty }(\mathbb {R}^{n})$ satisfies (A.1) with $K_{i}$ satisfying (A.2). Suppose that $x_{i} \rightarrow 0$ is an isolated blow up point of $\{u_{i}\}$, i.e., for some positive constants $A_{3}$ and $\bar{r}$ independent of i,

$$\begin{aligned} |x-x_{i}|^{2 \sigma /(p_{i}-1)}u_{i}(x) \le A_{3}\quad \text{ for } \text{ all } \, x \in B_{\bar{r}} \subset \Omega . \end{aligned}$$

Then for any $0<r<\bar{r}/3$, we have the following Harnack inequality

$$\begin{aligned} \sup _{B_{2r}(x_{i}) \backslash \overline{B_{r / 2}(x_{i})}} u_{i} \le C \inf _{B_{2 r}(x_{i})\backslash \overline{B_{r/2}(x_{i})}} u_{i}, \end{aligned}$$

where C is a positive constant depending only on $\sup _{i}\Vert K_{i}\Vert _{L^{\infty }(B_{\bar{r}}(x_{i}))}, n, \sigma , \bar{r}$ and $A_{3}.$

Proposition A.3

Under the hypotheses in Proposition A.2. Then for every $R_{i} \rightarrow \infty$, $\varepsilon _{i} \rightarrow 0^{+},$ we have, after passing to a subsequence (still denoted as $\{u_{i}\},$ $\{x_{i}\},$ etc.), that

$$\begin{aligned}{} & {} \Vert m_{i}^{-1} u_{i}(m_{i}^{-(p_{i}-1) /2\sigma }\cdot +x_{i}) -(1+k_{i}|\cdot |^{2})^{(2 \sigma -n) / 2}\Vert _{C^{2}(B_{2 R_{i}}(0))} \le \varepsilon _{i},\\{} & {} \quad r_{i}:=R_{i} m_{i}^{-(p_{i}-1) / 2 \sigma } \rightarrow 0 \quad \text{ as } \, i \rightarrow \infty , \end{aligned}$$

where $m_{i}:=u_{i}(x_{i})$ and $k_{i}:=({K_{i}(x_{i}) \pi ^{n/2}\Gamma (\sigma )}/{\Gamma (\frac{n}{2}+\sigma )})^{1/\sigma }.$

Proposition A.4

Under the hypotheses of Proposition A.2, there exists a positive constant $C=C(n, \sigma , A_{1}, A_{2}, A_{3})$ such that,

$$\begin{aligned} u_{i}(x) \ge C^{-1} m_{i}(1+k_{i} m_{i}^{(p_{i}-1) / \sigma }|x-x_{i}|^{2})^{(2 \sigma -n) / 2}\quad \text { for all } \, |x-x_{i}| \le 1. \end{aligned}$$

In particular, for any $e \in \mathbb {R}^{n},|e|=1$, we have

$$\begin{aligned} u_{i}(x_{i}+e) \ge C^{-1} m_{i}^{-1+((n-2 \sigma ) / 2 \sigma ) \tau _{i}}, \end{aligned}$$

where $\tau _{i}=(n+2 \sigma ) /(n-2 \sigma )-p_{i}$.

Proposition A.5

Under the hypotheses of Proposition A.2 with $\bar{r}=2,$ and in addition that $x_{i} \rightarrow 0$ is also an isolated simple blow up point with constant $\rho ,$ we have

$$\begin{aligned} \tau _{i}=O(u_{i}(x_{i})^{-c_{1}+o(1)})\quad \text {and}\quad u_{i}(x_{i})^{\tau _{i}}=1+o(1), \end{aligned}$$

where $c_{1}=\min \{2,2 /(n-2 \sigma )\}$. Moreover,

$$\begin{aligned} u_{i}(x) \le C u_{i}^{-1}(x_{i})|x-x_{i}|^{2 \sigma -n} \quad \text{ for } \text{ all } \,|x-x_{i}| \le 1. \end{aligned}$$

Proposition A.6

Under the hypotheses of Proposition A.5, let

$$\begin{aligned} T_{i}(x):=&u_{i}(x_{i}) \int _{B_{1}(x_{i})} \frac{K_{i}(y) u_{i}(y)^{p_{i}}}{|x-y|^{n-2 \sigma }} \, \textrm{d} y + u_{i}(x_{i}) \int _{\mathbb {R}^{n} \backslash B_{1}(x_{i})} \frac{K_{i}(y) u_{i}(y)^{p_{i}}}{|x-y|^{n-2 \sigma }}\, \textrm{d} y\\ =&:T_{i}^{\prime }(x)+T_{i}^{\prime \prime }(x). \end{aligned}$$

Then, after passing to a subsequence,

$$\begin{aligned} T_{i}^{\prime }(x) \rightarrow a|x|^{2 \sigma -n} \quad \text{ in } \, C_{loc}^{2}(B_{1} \backslash \{0\}) \end{aligned}$$

and

$$\begin{aligned} T_{i}^{\prime \prime }(x) \rightarrow h(x) \quad \text{ in } \, C_{l o c}^{2}(B_{1}) \end{aligned}$$

for some $h(x) \in C^{2}(B_{2})$, where

$$\begin{aligned} a=\Big (\frac{\pi ^{n / 2} \Gamma (\sigma )}{\Gamma (\frac{n}{2}+\sigma )}\Big )^{-\frac{n}{2 \sigma }} \int _{\mathbb {R}^{n}}\Big (\frac{1}{1+|y|^{2}}\Big )^{\frac{n+2 \sigma }{2}} \mathrm {~d} y \lim _{i \rightarrow \infty } K_{i}(0)^{\frac{2 \sigma -n}{2 \sigma }}. \end{aligned}$$

Consequently, we have

$$\begin{aligned} u_{i}(x_{i}) u_{i}(x) \rightarrow a|x|^{2 \sigma -n} +b(x) \quad \text{ in } \, C_{l o c}^{2}(B_{1} \backslash \{0\}). \end{aligned}$$

Proposition A.7

Under the hypotheses of Proposition A.2, we have

$$\begin{aligned} \int _{|y-y_{i}| \le r_{i}}|y-y_{i}|^{s} u_{i}(y)^{p_{i}+1}\,\textrm{d}y= {\left\{ \begin{array}{ll} O(u_{i}(y_{i})^{-2 s /(n-2 \sigma )}), &{} -n<s<n, \\ O(u_{i}(y_{i})^{-2 n /(n-2 \sigma )} \log u_{i}(y_{i})), &{} s=n, \\ o(u_{i}(y_{i})^{-2 n /(n-2 \sigma )}), &{} s>n, \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} \int _{r_{i}<|y-y_{i}| \le 1}|y-y_{i}|^{s} u_{i}(y)^{p_{i}+1}\,\textrm{d}y= {\left\{ \begin{array}{ll} o(u_{i}(y_{i})^{-2 s /(n-2 \sigma )}), &{} -n<s<n, \\ O (u_{i}(y_{i})^{-2 n /(n-2 \sigma )} \log u_{i} (y_{i})), &{} s=n, \\ O(u_{i}(y_{i})^{-2 n /(n-2\sigma )}), &{} s>n.\end{array}\right. } \end{aligned}$$