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 \(\beta \in (n-2\sigma ,n)\).
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Funding
Y. Li was supported by Science Foundation of China University of Petroleum, Beijing (No.2462023SZBH012), China Postdoctoral Science Foundation (2023M743879) and Postdoctoral Fellowship Program of CPSF (GZC20233106). Z. Tang was supported by National Science Foundation of China (12071036, 12126306). N. Zhou was supported by China Postdoctoral Science Foundation (2023TQ0167, 2023M741993, GZC20231344)
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Appendix A
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
Suppose that \(0 \le u_{i} \in L_{{loc}}^{\infty }(\mathbb {R}^{n})\) satisfies the nonlinear integral equation
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 \(A_{2}\),
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
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
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,
Then for any \(0<r<\bar{r}/3\), we have the following Harnack inequality
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
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,
In particular, for any \(e \in \mathbb {R}^{n},|e|=1\), we have
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
where \(c_{1}=\min \{2,2 /(n-2 \sigma )\}\). Moreover,
Proposition A.6
Under the hypotheses of Proposition A.5, let
Then, after passing to a subsequence,
and
for some \(h(x) \in C^{2}(B_{2})\), where
Consequently, we have
Proposition A.7
Under the hypotheses of Proposition A.2, we have
and
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Li, Y., Tang, Z., Wang, H. et al. Unified results for existence and compactness in the prescribed fractional Q-curvature problem. Nonlinear Differ. Equ. Appl. 31, 38 (2024). https://doi.org/10.1007/s00030-024-00927-6
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DOI: https://doi.org/10.1007/s00030-024-00927-6